Investigation of progressive damage

mechanisms in aerospace grade composites

Yash Guha

Space Engineering, master's level (120 credits)

2017

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

1

Investigation of progressive damage

mechanisms in aerospace grade

composites

MASTER’S THESIS

Submitted by

Yash Guha

Supervised by

Associate. Prof. Andrejs Pupurs

And

Prof. Janis Varna

Division of Material Science

October 2017

Division of Materials Science

Department of Engineering Science and Mathematics

Luleå University of Technology

2

ABSTRACT

In order to advance the understanding of micro-damage occurring in polymer composite

materials during quasi-static and fatigue loading, an experimental study along with finite

element analysis has been conducted. The two primary progressive damage mechanisms;

transverse matrix cracks and interlaminar delaminations, their initiation, progression and their

effects in reduction of global laminate properties are studied using carbon/epoxy symmetric

cross-ply laminates in two configuration; (0, 904)𝑆 and (0,90)𝑆. Tensile quasi-static and

tension-tension fatigue tests are performed on multiple specimens from each laminate

configurations to quantitatively measure the transverse crack density, interlaminar

delamination growth, and longitudinal modulus reduction due to these two damage

mechanisms. Predictions of these observed parameters based on Weibull distribution and

Loukil-Varna equation for crack density and effective modulus respectively are compared

with the experimental observations.

Further, a two-dimensional finite element model is developed in a Generalized Plain Strain

(GPS) state for calculating the Energy Release Rates (ERR) for interlaminar delamination

progression in a damaged laminate. Residual thermal stresses are included in the model to

analyze their effects in delamination growth. Analysis was performed on two different

laminate configurations similar to the tested specimens, and three different crack densities are

considered to understand delamination onset and progression. The ERR distributions obtained

from these analyses along with the experimental results of this study and some previous

studies are used to propose a hypothesis for initiation and progression of interlaminar

delaminations in a cross-ply laminate during quasi-static and fatigue loads.

Keywords: cross-ply laminate, transverse cracks, crack density, interlaminar delamination,

effective modulus, energy release rate, fracture toughness

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PREFACE

This project work titled ‘Investigation of progressive damage mechanisms in aerospace

grade composites’ is the Master’s thesis course work for 30 ECTS in compliance with the

Erasmus+ Master’s Degree Program in Space Science and Technology. The thesis work has

been carried out at division of Material Science, Department of Engineering Sciences and

Mathematics, Luleå University of Technology (LTU).

The project has been supervised by Associate Professor Andrejs Pupurs, and Professor Janis

Varna at LTU.

I would like to thank both my supervisors for their valuable time, guidance and support during

the course of this project. They not only finessed all my irrational ideas, but also motivated

me to pursue this research topic in future. My interesting discussions with them about

fundamental principles of mechanics will guide me through all my future endeavors.

I would also like to thanks all the technical and administrative staff members at Department of

Engineering Sciences and Mathematics, who helped me feel comfortable in this department.

A special thanks to the two colleagues at division of Mechanics of Solid Materials, Dmitrij

Ramanenka and Stefan Golling, who helped me on multiple occasions, including weekends,

in testing specimens at their lab.

Of course, I owe my gratitude to all my friends in Luleå, especially my roommates Samuel

Konatham and Samundra Rijal, who always stood beside me and gave their constructive

criticism whenever needed.

Finally, I would like to dedicate this work to my parents, my two sisters and my fiancé

without whom I wouldn’t be here to explore my scientific curiosity. I am always thankful for

their patience, emotional support and love.

Yash Guha

Lulea Technical University, October 2017

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TABLE OF CONTENTS

ABSTRACT ............................................................................................................................................ 2

PREFACE ............................................................................................................................................... 3

TABLE OF CONTENTS ........................................................................................................................ 4

1. INTRODUCTION .......................................................................................................................... 5

2. THEORETICAL BACKGROUND ................................................................................................ 7

3. EXPERIMENTAL METHODS ................................................................................................... 10

3.1 Material and specimen configuration .................................................................................... 10

3.2 Quasi-static test ..................................................................................................................... 12

3.3 Fatigue test ............................................................................................................................ 13

3.4 Fiber and matrix volume fraction .......................................................................................... 14

4. RESULTS AND DISCUSSION ................................................................................................... 15

4.1 Quasi-static tests .................................................................................................................... 15

4.1.1 Crack density ................................................................................................................. 15

4.1.2 Effective modulus .......................................................................................................... 17

4.1.3 Weibull parameters ........................................................................................................ 22

4.2 Fatigue Tests .......................................................................................................................... 26

4.2.1 Crack density ................................................................................................................. 26

4.2.2 Interlaminar Delamination ............................................................................................. 29

4.2.3 Oblique Cracks .............................................................................................................. 34

4.2.4 Effective modulus .......................................................................................................... 36

4.2.5 Fatigue Power Law parameters ..................................................................................... 43

5. PARAMETRIC FINITE ELEMENT ANALYSIS ....................................................................... 47

5.1 Energy Release rate (Mechanical loads) ............................................................................... 49

5.2 Energy Release rate (Thermo-mechanical loads) .................................................................. 52

5.3 Modulus reduction due to delamination ................................................................................ 53

6. CONCLUSIONS AND RECOMMENDATIONS ....................................................................... 56

7. REFERENCES ............................................................................................................................. 58

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1. INTRODUCTION

The increasing use of composite materials mainly in aerospace, automotive and wind power

industries has created a need to understand the behavior of these materials under cyclic

mechanical and thermal loadings. As the name suggests, composite materials are

heterogeneous and anisotropic in nature. The probabilistic nature of the strength of composite

materials presents a difficulty in understanding the initiation and propagation of damages. The

difficulty is even more severe in case of cyclic loadings, due to introduction of an

irreversibility in composite behavior, which makes the interaction of various damage

mechanisms much more complex. In order to use composite materials safely and to their

maximum capabilities, their damage mechanisms at micro level need to be clearly understood,

both for static and cyclic loads. There have been many experimental research campaigns

conducted in the past to understand the damage mechanisms that have incrementally

improved our understanding about this subject. This work is an attempt to quantitatively

advance the knowledge about the effects of these damages on thermo-mechanical properties,

and further, to qualitatively understand one of the two primary damage mechanisms at a

micro-level during cyclic loading using fracture mechanics approach.

The two primary damage mechanisms that severely affect the global mechanical and thermal

properties of composite materials are matrix cracking and interlaminar delamination. These

two mechanisms are the primary focus of this study. A brief description of these mechanisms,

their structure and their contribution in reduction of mechanical properties of composite

materials is discussed in Chapter 2. Further, an overview of the previous experimental work

done on this topic is presented in this chapter. It includes some of the analytical approaches

and empirical equations along with some finite element studies that are developed to predict

the damaged state, i.e. onset and progression of these damage mechanisms, and their effects

on the thermo-mechanical properties of composite materials under quasi-static and fatigue

loads. Chapter 3 describes in detail the experimental methods applied in this research. This

chapter includes the manufacturing methods for the test specimens, their preparation,

experimental setup and the test parameters used for both quasi-static and fatigue tests. Chapter

4 is focused on the experimental test results and their comparisons with the models and

equations discussed in Chapter 2. In order to study these mechanisms, both experimental and

finite element analysis methods are used in this work. Chapter 5 is focused on a 2D finite

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element analysis of a damaged laminate. A fracture mechanics approach is applied in this

analysis to study the delamination growth. The focus is on calculating the Energy Release

Rate (ERR) associated with these interlaminar delaminations initiated at the tips of the matrix

cracks. ERR is calculated for different delamination lengths using virtual crack closure

technique (VCCT) in a parametric manner for two different cross-ply configurations and three

different matrix crack densities. Lastly, Chapter 6 concludes the experimental and FE analysis

results with a recommendation for future works and improvements in experimental testing

and FE analysis.

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2. THEORETICAL BACKGROUND

Composite materials are a combination of two or more constituent materials with different

physical and chemical properties. This heterogeneous nature of composite materials

complicates the understanding of their damage mechanisms, because the damage is not only

governed by the individual constituents (matrix and fibers), but also by the properties of the

interface between them. All macro damages, which eventually lead to the failure in a

composite material, involve the micro-level interface debonding between fiber and matrix.

These interface debonds can coalesce into macro level damages, which can cause significant

changes in thermo-mechanical properties or catastrophic failure of composite materials.

Most practical applications of composite materials use a laminated configuration with fibers

oriented at different angles to the applied load. This is because the strength and stiffness of a

fibrous ply in off-loading axes are very low, and it is imperative to use differently angled plies

in order to design a durable composite structure. Thus, a study of damage initiation and

progression in laminated composites is of great value. The most fundamental laminate

configuration to analyze for composites is a symmetric cross-ply laminate. It consists only of

symmetric 0° and 90° layers, and is geometrically and mathematically easier to study. The

understanding of damage mechanisms studied through these simple configurations can further

extended to more intricate layup sequences.

Typical damage mechanisms that appear in cross-ply laminates are matrix dominated cracks.

The various damage mechanisms, their cause, and their characteristics based on [1] and

references therein, that are generally observed (not necessarily in this order) during tensile

quasi-static or fatigue loadings are:

1. Transverse cracks in 90° plies, which are initiated by the micro-level fiber matrix

debonding due to a load transverse to the fibers-matrix interface. These micro-cracks

grow up to form through-thickness cracks in transverse layers. They are generally

initiated at the free surfaces in the transverse ply, and in most quasi-static load cases,

immediately propagate across the width of the laminate. In case of fatigue loads, their

growth can be slower across the specimen’s width. Once a transverse crack is initiated,

they grow in number as the loads are increased, mostly maintaining a near uniform

crack spacing between them. For a given applied stress, the number of cracks per unit

length is inversely proportional to the total thickness of the transverse layer, with a

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maximum crack spacing nearly of the size of transverse layer thickness. The primary

effect of these transverse cracks is reduction of longitudinal modulus of the laminate.

2. Longitudinal cracks/ Splitting in 0° plies, that can occur at very high strain levels

due to the difference in the two in-plane Poisson’s ratios; 𝜈12 is general higher than

𝜈21. This difference, during application of higher strains, results in high transverse

tensile stresses in 0° plies and cause them to split.

3. Interlaminar delaminations, which are initiated by the high stresses at the free edges

or due to a transverse crack reaching the interface, which also results in very high

shear and out of plane (normal to the thickness) stresses at the interface. Further

propagation of these delamination is either due to further increase in these stresses in

case of a quasi-static load, or in case of fatigue loads, it is due to an irreversible

change in the interface that affects the fracture toughness or interlaminar shear

strength of the interface. Propagation of these delamination cracks is more easily

understood using an energy approach, rather than using stress distribution. Thus, linear

elastic fracture mechanics principles to evaluate/calculate Energy Release Rates

(ERR) have been used in past to analyze these cracks.

Apart from these matrix-dominated damage mechanisms, a cross-ply laminate can also have

fiber fractures in 0° plies due to tensile loading. It is also a commonly observed damage

mechanism during high strain quasi-static loads or high cycle fatigue loads. These fiber

fractures are generally a result of high stress concentrations ahead of a transverse crack tip,

which exceed the fiber’s ultimate strength, and can result in localized fiber fracture in the

adjacent ply.

Among all the above mentioned damage mechanisms, most of earlier work has been focused

on transverse cracks because of their significant contribution in reducing the effective

stiffness and strength of laminates. They also aid initiation and progression of several other

damage mechanisms like fiber fracture and interlaminar delaminations, which makes their

understanding dually important. Systematic experimental observations of transverse cracking

in cross-ply laminates started as early as 1977 by Garrett and Bailey [2]; who analyzed the

development of transverse cracking in glass fiber laminates under quasi-static loadings.

Following that study, there have been many experimental campaigns to analyze the

progression of transverse cracks and delamination in composite laminates using different

materials, configuration and load types. Alongside these experimental observations, many

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different analytical models to predict initiation and progression of these transverse cracks and

delaminations have also been developed. A detailed overview of major experimental work

and analytical models is presented in an excellent review by (Berthelot, 2003) [1]. In addition

to the prediction of progression of these damage mechanisms, several models also predict

their effects on laminate properties. One such parameter is effective longitudinal modulus that

represents the reduction in longitudinal modulus of the damaged laminate. A simple and

effective method of predicting effective modulus was recently developed by (Varna and

Loukil, 2016) [3], which is also used in this study for comparison with experimental results.

Further, there are several experimental and analytical studies explicitly to understand

delamination growth in recent times. Most of the analytical studies have been focused on

calculating the energy release rate associated with delamination growth. (O’Brien, 1982) [4]

used a quasi-3D stress analysis to obtain stress distributions in laminate and strain energy

release rates (ERR) for delamination growth. Further, (Nairn and Hu, 1992) [5] extended their

variational analysis to include delaminations and calculated ERR for initiation and growth of

such delamination initiated from transverse crack tips. (Takeda et al) [6,7] used extended

shear-lag analysis for the same purpose. All these analytical models explain the initiation of

these delaminations at the transverse crack tips due to high normal stresses at the interface,

and further grow due to increasing ERR due to increase in applied strain in a quasi-static

loading. However, they do not capture the variation of ERR at lower delamination lengths,

and also unable to separate the ERR for Mode-I and II. Thus, a need for finite element

analysis for such delamination was generated. In recent times, 2 dimensional analysis of

delamination progression has been done by (Paris et al., 2010) [8,9]. A generalized plane

strain model was used for calculating ERR in Mode-I and II for different delamination

lengths, including lower delamination sizes, in case of a (0,90)𝑆 cross-ply laminate. A similar

FE model is used in this study for two different laminate configurations to obtain ERR and

modulus reduction due to delamination growth.

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3. EXPERIMENTAL METHODS

There are different ways in which the damage progression in composites can be

experimentally studied. Methods can differ depending on the configuration of the specimen

and the tested material. For instance, in case of glass/epoxy composites, the refractive index

of the fiber is similar to the epoxy matrix, which makes laminates opaque and facilitates the

observation of damage in-situ for both flat and tube specimens. While for carbon/epoxy

composites, the laminate is completely non-transparent. So non-destructive evaluation

methods like acoustic emission, X-radiography, or a replica method can be used in-situ, or

edge view microscopy can be done after removing the specimen from the testing machine. In

replica method, a replicating material is applied on the edges of the specimen, and the

damaged state is replicated on that material and can be seen without taking out the specimen.

But, it is not always reliable. In this study, only edge view optical microscopy is used to

observe damage, by removing the specimen out of the machine after each test step.

This work is focused on testing of carbon/epoxy flat specimens cut from two different

laminate configurations. The two types of tests conducted in this study are 1) quasi-static

tensile test and 2) tension-tension fatigue test. The details of material, specimen configuration

and different tests are described in further sub-sections.

3.1 Material and specimen configuration

The test specimens are obtained from the laminates of two different configurations, i.e.

(0, 904)𝑆 and (0,90)𝑆. The raw material for these laminates is a high strength carbon/epoxy

prepreg, with commercial name: HexPly® M10/38%/UD300/CHS. The material has been

previously characterized at Luleå Technical University. The mechanical and thermal

properties at lamina level for the material are given in Table 1.

The laminates were prepared by manual layup of the prepreg, and were cured at 120 [°C] for

1 hour using a hot-press with vacuum. Specimens were cut from these laminates using a

diamond cutting disc. The general dimensions of specimens are 13.5 [mm] in width and 200

[mm] in length, with thickness depending on the number of layers used in each configuration.

Nominal ply thickness is 0.29 [mm]. Glass/epoxy tabs are bonded at the end of these

specimens using epoxy adhesive Araldite 2011 A and B, to avoid damage due to gripping

pressure during the test. A gauge length of 110 [mm] is available for damage observation in

all the specimens, although only 50 [mm] of this gauge length in the middle of the specimen

11

is used to observe damages. As mentioned earlier that carbon/epoxy laminates are not

transparent, and in order to facilitate the edge-view microscopy, the specimens are polished at

both edges, initially using different grade of Silicon-Carbide (SiC) sand papers, and finally

using liquids with suspended diamond particles up to a fine polishing level of 1 [µm].

Table 1 Thermo-mechanical properties of the orthotropic lamina of the tested carbon/epoxy material

Property Notation Value Units

Longitudinal modulus 𝐸𝐿 115 [GPa]

Transverse modulus 𝐸𝑇 7.9 [GPa]

In-plane Shear modulus 𝐺12 3.8 [GPa]

Out-of-plane Shear modulus 𝐺23 2.7 [GPa]

Major Poisson’s ratio 𝜈12 0.35 -

Out of plane Poisson’s ratio 𝜈23 0.45[1]

-

Difference between Longitudinal and Thermal expansion

coefficient

𝛼𝐿 − 𝛼𝑇 3.63×10-5

[1/℃]

[1] Out-of-plane Poisson’s ratio is not measured. It is assumed to be 0.45

For (0, 904)𝑆 and (0,90)𝑆 configurations, 11 and 9 samples respectively are used for testing.

Before performing the tests, all the specimens were subjected to a quasi-static loading up to

0.3 [%] strain level to measure their longitudinal modulus. A common ramp rate of 1

[mm/min] is used for both loading and unloading ramps. The strain is measured using an

extensometer of 50 [mm] gauge length. All the tests were performed at room temperature and

no visible damage is introduced during these modulus measurements.

Out of the 11 specimens in (0, 904)𝑆 configuration, 3 are tested for quasi-static tension tests,

while the remaining 8 are tested for tension-tension fatigue tests with 4 different strain levels.

Similarly, out of 9 specimens in (0,90)𝑆 configuration, 3 are tested for quasi-static tension

tests, while the remaining 6 are tested for tension-tension fatigue tests with 3 different strain

levels.

The cross-sectional area, modulus and the type of test conducted on each specimen are

summarized in Table 2.

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3.2 Quasi-static test

The quasi-static tests are performed on 3 specimens each from both layup configurations.

During these tests, specimens were loaded in tension to pre-defined strain levels in a

displacement control mode. All specimens were tested starting from 0.3 [%] maximum strain

to various levels in increasing steps and then unloaded. A ramp rate of 1 [mm/min]

corresponding to a strain rate of approximately 1 [% per min] was used for both loading and

unloading. The strain is measured using an extensometer of 50 [mm] gauge length. All the

tests were performed at room temperature. The different strain levels for each specimen are

mentioned in Table 2.

Due to the limitation on maximum load capacity and availability of the equipment, these

quasi-static tests were performed on three different machines; Instron® ElectroPuls™ E10000,

Specimen No Modulus [GPa] Area [mm2] Type of test performed

1 28.67 37.91

6 30.63 38.91

11 31.45 38.56

2 29.31 39.63

7 29.42 38.30

3 33.02 38.53

8 30.23 40.21

4 30.81 38.21

9 31.91 39.67

5 31.32 38.38

10 31.09 38.38

1 56.31 16.86

6 59.63 16.42

11 63.40 16.47

2 58.07 17.21

10 59.82 16.67

3 59.28 16.79

9 60.45 17.15

4 64.30 15.53

8 58.10 17.95

(0,904)s

Quasi-static tests from 0.3 to 1.4 [%] strain

Fatigue tests at e max = 0.3 [%] up to 1 million cycles

Fatigue tests at e max = 0.5 [%] up to 1 million cycles

Fatigue tests at e max = 0.6 [%] up to 1 million cycles

Quasi-static tests from 0.3 to 1.2 [%] strain

Fatigue tests at e max = 0.3 [%] up to 1 million cycles

Fatigue tests at e max = 0.4 [%] up to 1 million cycles

Fatigue tests at e max = 0.5 [%] up to 1 million cycles

Fatigue tests at e max = 0.6 [%] up to 1 million cycles

(0,90)s

Table 2 Details of specimen properties, dimension, and type of tests performed

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Instron® 3366 and Instron

® 1272. The ramp rate and the extensometer gauge length is same

for all the three machines.

After every strain level, the specimens were removed from the machine and were observed in

an optical microscope to count the number of cracks in 90° layers. These cracks are counted

at both free edges for an identified gauge length of 50 [mm] at the center, where the

extensometer is fixed. An average of the number of cracks on both the edges is then used to

calculate the crack density in [cracks/mm] corresponding to each strain level. The longitudinal

modulus is measured from the slope of stress-strain curve (in the range of 0.05-0.25 [%]

strain) during unloading to calculate the effective longitudinal modulus after each step. Due to

the removal of specimens after every step, the alignment of the specimens before every quasi-

static test step can vary, which can result in a variation of up to 5-10 [%] in the measured

modulus of the specimens. The transverse crack position within the gauge length was also

noted to see the evolution of these cracks.

3.3 Fatigue test

Fatigue tests are performed up to 1 million cycles on 8 and 6 specimens for (0, 904)𝑆 and

(0,90)𝑆 configuration respectively. Two specimens were tested for each strain level to avoid

the probable outlier behavior of any specimen. The different strain levels for each specimen

are mentioned in Table 2.

An Instron ElectroPuls™ E10000 machine was used to perform all the fatigue tests. All

specimens are tested for tension-tension cyclic loads with a load ratio (minimum load in the

cycle to maximum load in the cycle), 𝑅 = 0.1, and a frequency of 5 [Hz]. Strain is measured

using an extensometer with a gauge length of 50 [mm]. Modulus of the specimens was

calculated from an applied quasi-static ramp up to 0.3 [%] strain level before and after the

cyclic loads.

Similar to quasi-static tests, the specimens were taken out of the clamps after a specified

number of cycles, and were observed in the optical microscope to study the crack evolution.

Interlaminar delamination is also a significant damage mechanism during fatigue loading. So,

the delamination lengths at the transverse crack tips after each step are also measured for two

specimens in (0, 904)𝑆 configuration. All the specimens are tested for same total number of

cycles, but the number of steps to reach 1 million cycles was different for each strain level to

capture the damage evolution more closely, especially for higher strain levels.

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3.4 Fiber and matrix volume fraction

In order to check the consistency of matrix and fiber volume fraction in all the specimens,

volume fraction of the matrix and fiber, along with void content was measured after

completion of all the tests. This was done by a matrix digestion procedure, based on ASTM-

D3171-15 ‘Standard Test Methods for Constituent Content of Composite Materials’.

After completion of all the tests, specimen length of 110 [mm] was further cut into three parts.

The central gauge length (50 [mm]) of each specimen was kept aside to perform further

microscopic investigations. The two outer pieces were used to measure the fiber volume

fraction by burning a piece from each specimen in a furnace at 450 [℃] for 5 hours. At these

conditions, the epoxy resin becomes volatile and is removed completely, and only fibers are

left from each piece. The difference in mass of the specimen piece before and after the

burning gives matrix mass, and thus the fiber and matrix volume fractions are calculated. The

summary of volume fractions and void content is given in Table 3.

Table 3 Volume fraction for fiber and matrix measured from the specimens cut from (0,904)s laminate

(𝟎, 𝟗𝟎𝟒)𝑺 Volume fraction Void content

Sample no Fiber Resin (%)

1 0.57 0.42 1.29

2 0.52 0.48 0.77

3 0.54 0.46 0.52

4 0.53 0.47 0.79

5 0.54 0.46 0.88

6 0.53 0.47 0.59

7 0.52 0.48 0.66

8 0.53 0.46 1.01

9 0.53 0.46 1.44

10 0.55 0.42 1.83

15

4. RESULTS AND DISCUSSION

4.1 Quasi-static tests

The two primary parameters that are measured from these quasi-static tests are ‘Crack

Density’ and ‘Effective Stiffness’. These two parameters help understand the advent of

damage and its effects on mechanical properties in the composite. Thus, they are discussed in

detail in separate sections. Delaminations are not a significant damage mechanism for quasi-

static tests, and are not discussed in this section.

4.1.1 Crack density

Crack density, denoted by 𝜌𝑐, is defined as the number of cracks per unit length of the

specimen. In this case, it can be written as

𝜌𝑐 =𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑟𝑎𝑐𝑘𝑠

50 𝑚𝑚 [𝑐𝑟𝑎𝑐𝑘𝑠/𝑚𝑚]

Fig. 1 and 2 show the crack density evolution in the three specimens as a function of applied

axial strain for (0, 904)𝑆 and (0,90)𝑆 configurations respectively.

Fig. 3 shows the comparison of crack density evolution in the two configurations. Following

observations can be made from this comparison:

The damage initiation strain is slightly higher for (0,90)𝑆 configuration; 0.7 [%] for

(0,90)𝑆, as compared to 0.6 [%] for (0, 904)𝑆 configuration.

Crack density shows no tendency towards saturation level in (0,90)𝑆, even up to a

strain level of 1.4 [%], while in case of (0, 904)𝑆, the crack density is slightly tending

towards a saturation level.

Crack density in (0,90)𝑆 is higher than the (0, 904)𝑆 configuration, which is

consistent with the observations in previous studies [1,2]. It shows that lower the

thickness of transverse layer, lower is the crack spacing, i.e. higher the crack density.

In addition to these observations, small interlaminar delaminations up to a maximum length of

0.5 [mm] are also observed in case of (0, 904)𝑆 after 1.2 [%] strain level. Whereas, no such

delaminations are observed for (0,90)𝑆 laminates up to a strain level of 1.4 [%].

Fiber breaks were observed in (0,90)𝑆 specimens at 1.4 [%], from which it can be concluded

that the specimens were very close to failure but no saturation in crack density was observed.

16

Thus, the slight tendency towards saturation of crack density in case of (0, 904)𝑆 at a strain

level of 1.2 [%] can be attributed to the presence of delaminations in this case.

The observed crack density is an input for further calculations for estimating Weibull

parameters for the material and estimating the reduced effective stiffness of the transverse

layer as per recently developed Loukil-Varna equation. These are discussed in further sub-

sections.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.3 0.5 0.7 0.9 1.1 1.3 1.5

Cra

ck d

ensi

ty (

1/m

m)

Strain (%)

(0,90)s Specimen 1

Specimen 6

Specimen 11

Statistical average

Figure 2 Crack density vs. applied strain during quasi-static test for (0,90)s laminate

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.3 0.5 0.7 0.9 1.1 1.3

Cra

ck d

ensi

ty (

1/m

m)

Strain (%)

(0,904)s Specimen 1

Specimen 6

Specimen 11

Statistical Average

Figure 1 Crack density vs. applied strain during quasi-static test for (0,904)s

laminate

17

4.1.2 Effective modulus

The development of transverse cracks during quasi-static or fatigue loadings results in

reduction of the longitudinal modulus of the laminate. This is associated with the reduced

ability of transverse ply to transfer stresses, and is partially due to release of thermal strains in

0° layers. This reduction in longitudinal modulus is measured in terms of effective modulus.

Fig. 4 and 5 show the effective Young’s modulus in longitudinal direction as a function of

increasing strain for (0, 904)𝑆 and (0,90)𝑆 configurations respectively. These experimental

values are compared with the lower bound value obtained from Ply Discount Model (PDM).

This model assumes that the ability of transverse layers to carry loads is completely

diminished after initiation of the first crack. This value is calculated using the LAP software

that uses Classical Laminate Theory (CLT) to calculate the laminate properties, using the

input thermo-mechanical lamina properties given in Table 1.

It can be seen from these two figures that the reduction in modulus is higher for (0, 904)𝑆

laminates, which can be understood by the fact that for this configuration, transverse layers

contribute up to 22 [%] to the total longitudinal modulus of the laminate, as compared to 7

[%] for (0,90)𝑆 laminates. Once the transverse layer is damaged, the maximum reduction

should be represented by PDM. The effective modulus values for (0,90)𝑆 laminate are within

the scatter range of modulus measurements. As mentioned earlier that the specimens are

removed from the test setup after each strain level, there is a probable error associated with

the measurement of modulus due to slight change in alignment.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Cra

ck d

ensi

ty (

1/m

m)

Strain (%)

(0,90)s

(0,904)s

Figure 3 Comparison of crack density evolution between (0,90)s and (0,904)s

laminates under quasi-static loads

18

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

Eff

ecti

ve

mo

du

lus

Strain (%)

(0,904)s

Specimen 1

Specimen 6

Specimen 11

Ply discount model

Figure 5 Effective modulus change for three samples in (0,904)s configuration during quasi-static

tests

0.8

0.85

0.9

0.95

1

1.05

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Eff

ecti

ve

Mod

ulu

s

Strain (%)

(0,90)s

Specimen 1

Specimen 6

Specimen 11

Ply Discount Model

Figure 4 Effective modulus change for three samples in (0,90)s configuration during quasi-static

tests

19

Further, a comparison of these experimental values for both configurations with prediction

values based on Loukil-Varna equation [3] is shown in Fig. 6 and 7 for (0, 904)𝑆 and (0,90)𝑆

configurations respectively. It was shown by Loukil and Varna that the macroscopic

properties of a damaged laminate with intralaminar cracks (transverse cracks in a cross-ply

laminate) are primarily governed by the effective transverse modulus change. This effective

transverse modulus as a function of normalized crack density is same for glass and

carbon/epoxy laminates, as is given by:

𝐸𝑇𝑒𝑓𝑓

= 𝐸𝑇 (3

5𝑒−2.5𝜌𝑐𝑛 +

2

5𝑒−0.9𝜌𝑐𝑛) (1)

Thus by using eq. (1), lamina transverse modulus as a function of crack density (or

corresponding strain level in this case) is used as an input in the Classical Laminate Theory to

predict the effective longitudinal modulus for each specimen. LAP software is used to

calculate reduced stiffness values using the input of effective transverse modulus of the

lamina. Here the transverse crack density does not include the effect of delaminations present

between 0° and 90° layers in case of higher stress levels or for higher cyclic loads.

Fig. 6 shows that the trends for effective moduli match in all cases, but the numerical values

don’t match in most cases. As mentioned earlier, it can be due to a slight error in alignment of

test specimen, which can reflect in erroneous measurement of the modulus of the undamaged

laminate, which can later translate into these offsets from the predicted values in Fig. 6 a) and

b). In a broader sense, Loukil Varna equation is an easy to use and very effective tool to

predict the effects of transverse cracks in any laminate configuration.

20

0.7

0.8

0.9

1

1.1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Eff

ecti

ve

mod

ulu

s

Strain (%)

a) Loukil-Varna

Specimen 1

Ply discount

0.7

0.8

0.9

1

1.1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Eff

ecti

ve

mo

du

lus

Strain (%)

b) Loukil-Varna

Specimen 6

Ply discount

0.7

0.8

0.9

1

1.1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Eff

ecti

ve

mo

du

lus

Strain (%)

c) Loukil-Varna

Specimen 11

Ply discount

Figure 6 Comparison of predicted and experimental effective longitudinal modulus for (0,904)s

configuration.

21

0.8

0.85

0.9

0.95

1

1.05

1.1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Eff

ecti

ve

Sti

ffn

ess

Strain (%)

a)

Loukil-Varna

Specimen 1

Ply discount

0.8

0.85

0.9

0.95

1

1.05

1.1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Eff

ecti

ve

Sti

ffn

ess

Strain (%)

b)

Loukil-Varna

Specimen 6

Ply discount

0.8

0.85

0.9

0.95

1

1.05

1.1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Eff

ecti

ve

Sti

ffn

ess

Strain (%)

c)

Loukil-Varna

Specimen 11

Ply discount

Figure 7 Comparison of predicted and experimental effective longitudinal modulus for (0,90)s

configuration

22

4.1.3 Weibull parameters

The strength of a composite material is known to have a statistical nature that can be

represented by a distribution function given by a two parameter Weibull distribution. Using a

similar approach, the probability of failure of the transverse layer for an applied mechanical

stress can be written as:

𝑃𝑓 = 1 − 𝑒𝑥𝑝 (−𝜎90

𝜎0)

𝑚

(2)

Where,

𝑃𝑓 − Probability of failure

𝑚 − Weibull shape parameter

𝜎0 − Weibull scale parameter

𝜎90𝑇ℎ − Thermal stress in 90° layer

(σ90Th = 21.89 MPa and 25.68 MPa for (0, 904)S and (0,90)S configurations respectively)

𝜎90𝑀𝑒𝑐ℎ − Applied Mechanical stress

𝜎90 = 𝜎90𝑀𝑒𝑐ℎ + 𝜎90

𝑇ℎ , Resultant stress in 90° layer

Eq. (2) can also be written in a logarithmic form as following:

ln(− ln(1 − 𝑃𝑓)) = 𝑚 ln 𝜎90 − 𝑚 ln 𝜎0 … … … … … … … … … (3)

The probability of failure in above equation can also be written in terms of the experimental

crack density, 𝜌𝑐 and the maximum possible crack density, 𝜌𝑐,𝑚𝑎𝑥 in the transverse layer.

𝑃𝑓 = 𝜌𝑐

𝜌𝑐,𝑚𝑎𝑥

From experiments in the past, it has been observed that the transverse crack density tends to

saturate towards 𝜌𝑐,𝑚𝑎𝑥, which is a function of the total thickness of transverse layer, 𝑡90,

given as

𝜌𝑐,𝑚𝑎𝑥 = 1

𝑡90

Using the calculated probability of failure from experimental crack density, a linear fit

between ln(− ln(1 − 𝑃𝑓)) and ln 𝜎90, gives the values for 𝑚 and 𝜎0.

23

Using the above method, Weibull parameters are calculated for the three specimens tested in

quasi-static load for both layup configurations. The scale and shape parameters for both

configurations are compiled in Table 4

Table 4 Weibull parameters obtained from quasi-static tests

Specimen No. (𝟎, 𝟗𝟎𝟒)𝑺 (𝟎, 𝟗𝟎)𝑺

𝑚 𝜎0 𝑚 𝜎0

[-] [MPa] [-] [MPa]

1 6.30 99.20 8.98 141.71

6 7.38 105.25 6.10 132.41

11 6.70 108.36 4.04 148.49

Average 6.79 104.27 6.37 140.87

Standard deviation 0.543 4.657 2.48 8.06

Once the scale and shape parameter are known for these specimens, probability of failure for

a similar configuration and material system can be predicted for different average applied

stresses. Based on this probability of failure, the crack density as a function of applied

transverse stress can also be predicted. This predicted crack density (𝜌𝑐∗), and the experimental

crack density (𝜌𝑐), are plotted for all three specimens of each configuration in Fig. 8 and 9,

along with the linear fit for Eq. (3) for each specimen.

24

y = 6.3015x - 28.969 R² = 0.9875

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

4.2 4.3 4.4 4.5 4.6 4.7 4.8

ln(-

ln(1

-Pf)

) [-

]

ln(s90) [MPa]

0.0

0.1

0.2

0.3

0.4

0.5

0.0 30.0 60.0 90.0 120.0

rc

[1/m

m]

s90 [MPa]

(0,904)s

Specimen 1

Self-prediction

y = 7.376x - 34.345 R² = 0.9892

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

4.2 4.3 4.4 4.5 4.6 4.7 4.8

ln(-

ln(1

-Pf)

) [-

]

ln(s90) [MPa]

0.0

0.1

0.2

0.3

0.4

0.0 30.0 60.0 90.0 120.0

rc

[1/m

m]

s90 [MPa]

(0,904)s

Specimen 6

Self Prediction

y = 6.7005x - 31.395 R² = 0.9332

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

4.4 4.5 4.6 4.7 4.8

ln(-

ln(1

-Pf)

) [-

]

ln(s90) [MPa]

0.0

0.1

0.2

0.3

0.4

0.0 30.0 60.0 90.0 120.0

rc

[1/m

m]

s90 [MPa]

(0,904)s

Specimen 11

Self prediction

Figure 8 Weibull parameters and associated comparison between experimental and predicted crack density for (0,90)s

configuration

25

y = 9.0193x - 44.703 R² = 0.978

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

4.5 4.6 4.7 4.8 4.9 5.0

ln(-

ln(1

-Pf)

) [-

]

ln(s90) [MPa]

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.0 30.0 60.0 90.0 120.0 150.0

rc

[1/m

m]

s90 [MPa]

(0,90)s

Specimen 1

Self prediction

y = 6.0973x - 29.791 R² = 0.983

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

4.2 4.4 4.6 4.8 5.0

ln(-

ln(1

-Pf)

) [-

]

ln(s90) [MPa]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 30.0 60.0 90.0 120.0 150.0

rc [

1/m

m]

s90 [MPa]

(0,90)s

Specimen 6

Self prediction

y = 4.0388x - 20.196 R² = 0.993

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

4.4 4.5 4.6 4.7 4.8 4.9 5.0

ln(-

ln(1

-Pf)

) [-

]

ln(s90) [MPa]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.0 30.0 60.0 90.0 120.0 150.0

rc [

1/m

m]

s90 [MPa]

(0,90)s

Specimen 11

Self prediction

Figure 9 Weibull parameters and associated comparison between experimental and predicted crack density for (0,904)s

configuration

26

4.2 Fatigue Tests

The same two parameters i.e. crack density (𝜌𝑐) and effective longitudinal modulus are

measured for fatigue tests as well. During fatigue, interlaminar delaminations also play a vital

role in crack progression and change in mechanical properties of test specimens. Thus, for the

case of fatigue tests, these delaminations are also observed for some of the (0, 904)𝑆

specimens, where change in delamination lengths are more noticeable during these test steps.

4.2.1 Crack density

Crack density for fatigue test specimens is measured in the same manner as quasi-static test

specimens. In fatigue tests, transverse crack observations can be divided in two parts; a) lower

strain levels, i.e. max = 0.3 [%] and 0.4 [%], and b) higher strain levels, i.e. max = 0.5 [%] and

0.6 [%] strain levels.

a) Lower strain levels: As mentioned in Table 2, two specimens are tested at 0.3 [%] strain

levels for (0,90)𝑆 configuration, and both specimens did not show any transverse crack

after 1 million cycles. Further, for case of (0, 904)𝑆 configuration, two specimen each are

tested for both lower strain levels, and the crack density plots are shown in Fig. 10 and 11.

Cracks initiate around 104 cycles. In general, crack density does not tend to saturate,

except in the case of Specimen 7 of (0, 904)𝑆 configuration, which seems like an outlier

behavior.

0

0.05

0.1

0.15

0.2

1 10 100 1000 10000 100000 1000000

rc

[1/m

m]

Number of cycles

(0,904)s, max = 0.3% Specimen 2

Specimen 7

Figure 10 Crack density progression during lower strain level fatigue tests

27

b) Higher strain levels: For these cases, more number of steps are used to reach 1 million

cycles, and crack density is measured after every step. The crack density evolution as a

function of number of cycles is shown in Fig. 12. Some important observations that can be

made from these plots are:

The number of cycles to initiate cracks is around 104 cycles for both

configurations, which is similar to tests at lower strain levels.

The crack density reaches a saturation state for case of (0, 904)𝑆 configuration

around 4×105 cycles for max = 0.5 [%], and around 2×10

5 cycles for max = 0.6

[%]. However, for (0,90)𝑆 case, cracks tend not to saturate even around 1 million

cycles.

Fatigue crack density at saturation in case of (0, 904)𝑆 configuration at max = 0.6

[%] is very close to the average crack density of 0.347 [1/mm] at 1.2 [%] strain for

quasi-static test.

The damaged state of the laminates in terms of crack density for these fatigue test

specimens at max = 0.6 [%] strain level is very close to the damaged state at the

end of quasi-static tests, which shows that near failure states can be reached in

fatigue tests after 1 million cycles at such high strain levels.

0

0.05

0.1

0.15

0.2

0.25

0.3

1 10 100 1000 10000 100000 1000000

rc

[1/m

m]

Number of cycles

(0,904)s, max = 0.4% Specimen 3

Specimen 8

Figure 11 Crack density progression during lower strain level fatigue tests

28

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 10 100 1000 10000 100000 1000000

rc

[1/m

m]

Number of cycles

max = 0.5% (0,90)s_Specimen 3

(0,90)s_Specimen 9

(0,90_4)s_Specimen 4

(0,90_4)s_Specimen 9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100 1000 10000 100000 1000000

rc

[1/m

m]

Number of cycles

max = 0.6% (0,90)s_Specimen 4

(0,90)s_Specimen 8

(0,90_4)s_Specimen 5

(0,90_4)s_Specimen 10

Figure 12 Crack density evolution during fatigue tests at higher strain levels

29

4.2.2 Interlaminar Delamination

Significant delaminations initiated at the tips of transverse cracks are observed for (0, 904)𝑆

configuration during fatigue tests for higher strain levels. Appearance of these delaminations

begin after 100×103 cycles for max = 0.5 [%] and around 60×10

3 cycles for max = 0.6 [%]

fatigue tests. This shows that delaminations start to grow before the saturation of transverse

crack density. This is a very significant observation, and is discussed in more detail in chapter

5, along with energy concepts. These delaminations are observed to initiate at the tips of

transverse cracks and can grow on either side of the crack. A typical delamination observed in

these tests is shown in Fig.13.

Figure 13 Interlaminar delamination observed in specimen 4 of (0,904)s configuration during fatigue tests after 1

million cycles

30

These delaminations are then observed carefully after each step of fatigue tests. Based on

these measurements, following observations can be made:

1) Very small or no delaminations are observed for (0,90)𝑆 laminate specimens for all

strain levels, and for (0, 904)𝑆 laminate specimens for lower strain levels.

2) After 1 million cycles, a complete (through-the-length of specimen) delamination at

both the edges of the two specimens of (0, 904)𝑆 configurations tested for max = 0.6

[%] was observed. This shows the severity of the damages due to high cycle fatigue

loads at higher strain levels.

3) The initiation of delamination is simultaneous with the increasing crack density, i.e.

the delaminations do not start to grow after the saturation of transverse crack density.

Table 5 shows the average delamination length (𝑙𝑑), crack density, (𝜌𝑐) and average

crack spacing (𝑙𝑐𝑟𝑎𝑐𝑘𝑠) with number of cycles for the two specimens of (0, 904)𝑆

configuration, tested at max = 0.5 [%]. These average lengths are calculated by first

taking average of all the individual delamination lengths at tips of all the transverse

cracks at top and bottom 0°-90° interfaces, and then averaged further for both free

edges of specimen.

Table 5 Delamination growth during fatigue tests for (0,904)s specimens tested for max = 0.5 [%]

No of cycles Specimen 9 Specimen 4

N 𝝆𝒄 [1/mm] 𝒍𝒄𝒓𝒂𝒄𝒌𝒔 [mm] 𝒍𝒅 [mm] 𝝆𝒄 [1/mm] 𝒍𝒄𝒓𝒂𝒄𝒌 [mm] 𝒍𝒅 [mm]

311111 0.26 3.84 0.59 0.24 4.16 0.66

411111 0.30 3.33 0.68 0.30 3.33 1.11

611111 0.30 3.33 0.80 0.30 3.33 1.41

811111 0.30 3.33 0.89 0.30 3.33 1.48

1011111 0.30 3.33 1.05 0.30 3.33 1.60

Even higher delamination lengths were observed for the two specimens tested at max =

0.6 [%], but all the delaminations were not clearly visible due to relatively improper

edge polishing in those two specimens, and hence they are not reported here

quantitatively. However, as mentioned above, complete edge delaminations were

observed for these two specimens of (0, 904)𝑆.

4) Higher delamination lengths are observed at the tips of angled cracks (termed as

oblique cracks) next to a transverse crack. The structure and growth of these oblique

cracks is discussed in Section 4.2.3.

31

5) Growth of delaminations associated with individual transverse cracks suggests that

these delaminations grow logarithmically as a function of number of cycles. Fig. 14

shows average (of top and bottom 0°-90° interface) delamination lengths at three

different transverse crack tips in specimen 9 of (0, 904)𝑆 configuration. A similar

logarithmic relation between delamination length and number of cycles has been

presented in [10], for a case of single fiber-matrix debond study under tension-tension

cyclic loading. Although, these two cases cannot be compared directly, but a similarity

is worth mentioning.

6) A majority of delaminations do not grow symmetrically on either side of the

transverse crack tip. However, an S-shape pattern was observed for a majority of

delaminations, i.e. if a delamination grows on the right side of the transverse crack at

the top interface of 0° and 90° layers, there will be no or very little growth of

delamination on the left side of that transverse crack at top interface. Following an S-

type shape, the delamination on the bottom interface will grow essentially on the left

side of the transverse crack, with little or no growth on the right side of the transverse

crack at bottom interface. This is shown in Fig.15. The reason for such an S-shape is

unclear at the moment. But, this observation generates a need to consider the stress

states on the either side of the transverse crack separately, and not symmetrically, as it

is usually done is most studies.

32

y = 0.000196ln(x) - 0.002327 R² = 0.962290

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

3.0E-04

3.5E-04

4.0E-04

0 200000 400000 600000 800000 1000000 1200000

Del

am

ina

tio

n l

eng

th,

l d [

m]

Number of cycles

Crack 6

y = 0.00019255ln(x) - 0.00196440 R² = 0.89458768

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

6.0E-04

7.0E-04

8.0E-04

0 200000 400000 600000 800000 1000000 1200000

Del

am

ina

tio

n l

eng

th,

l d [

m]

Number of cycles

Crack 2

y = 0.000487ln(x) - 0.005819 R² = 0.961150

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

6.0E-04

7.0E-04

8.0E-04

9.0E-04

1.0E-03

0 200000 400000 600000 800000 1000000 1200000

Dela

min

ati

on

len

gth

, l d

[m

]

Number of cycles

Crack 12

Figure 14 Delamination growth at transverse crack tips during fatigue tests for specimen 9

of (0,904)s configuration tested at max = 0.5 [%]

33

Figure 15 Typical S-shape of

delamination-transverse crack

combination

A) and B) show the top and

bottom interface respectively at

higher magnifications.

34

4.2.3 Oblique Cracks

Another observation exclusive to fatigue of (0, 904)𝑆 specimens was development of oblique

cracks next to a transverse crack. These oblique cracks are observed in all the four (0, 904)𝑆

specimens tested for higher strain levels. Such oblique cracks were also observed for all the

three specimens of (0, 904)𝑆 configuration under quasi-static tests. A typical oblique crack is

shown in Fig 16.

These oblique cracks have some interesting characteristics. They start at the interface between

0° and 90° layers, at an angle ≥ 50° from the interface plane next to a transverse crack. As the

number of cycles increase, they propagate towards the neighboring transverse crack. In some

cases, they merge into the neighboring transverse crack, as shown in Fig 17a. While, in some

cases, they propagate away from the neighboring transverse crack and form a curved

transverse crack right next to the already existing neighboring transverse crack, as shown in

Fig 17b, or in quasi-static specimens, they stop mid-way as shown in Fig.17c. These delta

cracks, almost in all cases, have a large interlaminar delamination at their initiation point in

the direction away from the neighboring transverse crack as shown in Fig 17.

These oblique cracks do not have the same shape through the width of the specimen, as

observed in one of the specimens of (0, 904)𝑆 configuration. They either move away from the

parent transverse crack or they move to the other side of that transverse crack.

Figure 16 Typical delta crack observed in (0,904)s specimen during fatigue tests at 0.4 %

maximum strain

35

A)

Figure 17 Different types of oblique cracks observed in (0,904)s specimens during fatigue

tests

36

There are not a lot of studies focused on such oblique cracks. One of the reasons for that is the

fact that these oblique cracks are not observed in all composite materials. One detailed study

on oblique cracks by (Jalalvand et al, 2014) [11] suggests that these oblique cracks are more

likely to occur in materials with higher Mode-II fracture toughness. In this study, they

performed a FE analysis on a (0, 904)𝑆 laminate using cohesive elements at the 0° - 90°

interface, and compared the competitiveness of delaminations and oblique cracks. Their study

shows that at higher crack densities, two different damage mechanisms can occur; namely

delamination and oblique cracks. For materials where the Mode-II fracture toughness was

close to Mode-I fracture toughness values, delamination is more likely to occur, while if

Mode-II fracture toughness is more than twice the Mode-I fracture toughness, then such

oblique cracks are more likely to occur. Their analysis results were found in concurrence with

their experimental test results, in which they performed quasi-static tensile tests on (0, 904)𝑆

specimens prepared from a high strength carbon/epoxy prepreg with Mode-I and Mode-II

fracture toughness values as 200 J/m2 and 1000 J/m

2 respectively. They observed oblique

cracks at higher strain levels, rather than delaminations, which supports their hypothesis about

oblique cracks. The characteristics of oblique cracks found in the experimental study of [11]

are similar to the characteristics presented here in this study, except for the fact that they

didn’t observe any delaminations at the tips of this oblique cracks. This similarity between the

experimental observations in quasi-static tests has led author to assume that the material used

in this study also has a Mode-II fracture toughness value considerably higher than Mode-I.

This assumption is used later in this study to propose the hypothesis for delamination

progression during fatigue loads.

4.2.4 Effective modulus

In addition to transverse cracks, interlaminar delaminations also add to the total modulus

reduction in case of fatigue loads. Further reduction in longitudinal modulus after reaching a

saturation in crack density can be attributed to subsequent increase in number of

delaminations or their lengths. But, it is very difficult to separate the contribution of matrix

cracks and delaminations in macro level modulus reduction of a laminate, experimentally.

However, in section 5.3, modulus reduction due to delaminations is calculated using a FE

analysis on damaged laminate.

Similar to crack density, effective modulus can also be studied separately for lower and higher

strain levels.

37

a) Lower strain levels: Fig. 18 shows a comparison of the experimental effective

modulus with the predicted value using Loukil-Varna equation (marked as

“LV_Specimen no.”) for both configurations. Effective modulus based on Ply

discount model is also included to compare the state of the damage in the laminates.

Although Loukil-Varna equation does not consider the effect of delaminations, but

this comparison still provides valuable insights in the behavior or material at different

strain levels. Using this comparisons, following observations can be made:

From Fig. 18(a), where no cracks are observed, a scatter of modulus and some

values more than 1 are observed due to alignment error can be seen. The

scatter can also be attributed to some micro-damage after a million cycle in

these specimens, which can affect the behavior of these specimens.

For (0, 904)𝑆 configuration, a reduction in modulus of 4 to 8 [%] can be

observed for both strain levels after 1 million cycles. The scatter between the

two specimens tested at same strain levels can be attributed to the presence of

inherent micro-level damages or voids that can vary among specimens, and

also to the slight alignment change during testing.

In case of this study, the prediction based on Loukil-Varna equation in-general

forecast a more severe reduction in modulus than experimentally observed

values, where delaminations are absent/ very small, i.e. for lower strain levels.

Above observation is more significant for the higher strain levels, where the

experimental reduction in modulus is more than the predicted values using

Loukil-Varna equation due to presence of significant delaminations. This

clearly demonstrates that the effects of delaminations are significant, and it is

imperative to include the effects of delaminations in predicting modulus

reduction during fatigue.

38

0.7

0.8

0.9

1.0

1.1

10 100 1000 10000 100000 1000000 10000000

Eff

ecti

ve

mo

du

lus

Number of cycles

(b) (0,904)s, max = 0.3 %

Specimen 2

LV_S2

Specimen 7

LV_S7

Ply Discount

0.7

0.8

0.9

1

1.1

1 10 100 1000 10000 100000 100000010000000

Eff

ecti

ve

mo

du

lus

Number of cycles

(c) (0,904)s, max = 0.4 %

Specimen 3

LV_S3

Specimen 8

LV_S8

Ply Discount

0.90

0.92

0.94

0.96

0.98

1.00

1.02

1.04

1 100 10000 1000000

Eff

ecti

ve

mo

du

lus

Number of cycles

(a) (0,90)s, max = 0.3 %

Specimen 2

Specimen 10

LV_S2 and S10

Ply Discount

Figure 18 Comparison of experimental effective modulus with its predicted values based on

Loukil-Varna equation and ply discount model during fatigue tests at lower strain levels

39

b) Higher strain levels: Fig. 19 and 20 show the same comparison for 0.5 [%] and 0.6

[%] strain levels for both configurations. Following observations can be deduced from

these effective moduli plots:

For (0,90)𝑆 laminates, reduction in modulus after 1 million cycles is very

small; a maximum modulus reduction of 2.5 [%] in case of max= 0.5 [%], and

4.5 [%] in case of max= 0.6 [%] is observed.

For thicker transverse layers, maximum modulus reduction after 1 million

cycles is around 15-20 [%] for both strain levels.

For (0,90)𝑆 laminates, the predictions in modulus reduction based on Loukil-

Varna equation are either very close or higher than the experimentally

observed reductions, for both strain levels. This observation is consistent with

no delamination observations in this case.

In (0, 904)𝑆 laminates, where complete through-the-length delaminations are

present, observed reductions in longitudinal moduli are higher than the values

predicted based on Loukil-Varna equation, see Fig. 20 (b). This observation

along with the fact that observed values for lower strain levels were lower than

predictions, gives us a rough estimate of the modulus reduction due to

delaminations. It is safe to assume that the difference between the observed

and predicted value is the contribution of delaminations.

At those crack densities (0.36 and 0.32 [cracks/mm]), where the delamination

lengths are about the size of crack spacing (2.7 and 2.9 [mm]) and of the order

of transverse layer thickness (2.3 [mm]), the reduction due to such

delaminations is roughly 5 [%]. This experimental observation is in close

correlation with the modulus reduction calculated using FE analysis in section

5.3.

In (0, 904)𝑆 laminates, the effective moduli values observed for max= 0.6 [%]

case are also very close to the values predicted based on Ply Discount Model

(PDM). This shows that the damaged state of transverse layer is very severe,

and it is carrying negligible load in applied direction. This observation also

confirms that there is negligible transfer of shear stresses between the two

layers; a prime assumption of PDM; which is the result of through-the-length

edge delaminations present in these two specimens.

40

From Fig.20 (a), it can be seen that modulus reduction is higher for specimen 4

than specimen 9. This can also be correlated to the average delamination

lengths shown in Table 4 in section 3.2.2, where the average delamination

lengths are higher for specimen 4, resulting in lower effective modulus at the

end of 1 million cycles.

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1 100 10000 1000000

Eff

ecti

ve

mo

du

lus

Number of cycles

(a) (0,90)s, max = 0.5 %

Specimen 3

LV_S3

Specimen 9

LV_S9

Ply Discount

0.92

0.94

0.96

0.98

1

1.02

1.04

1 100 10000 1000000

Eff

ecti

ve

mo

du

lus

Number of cycles

(b) (0,90)s, max = 0.6 %

Specimen 4

LV_S4

Specimen 8

LV_S8

Ply Discount

Figure 19 Comparison of experimental effective modulus with its predicted values based on

Loukil-Varna equation and ply discount model during fatigue tests at higher strain levels for

(0,90)s configuration

41

It is clearly demonstrated in this section that delaminations have significant effect in effective

modulus of damaged laminate, and the effect of these delaminations in prediction of effective

modulus need to be included. This inclusion is recently done by Varna et al (2017). They

included the delamination lengths into their equation for calculating effective transverse

modulus (Eq. (1)).

In this study, a modified representation of normalized crack density, to include the effects of

delaminations on the modulus reduction is presented as follows:

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

100 1000 10000 100000 1000000 10000000

Eff

ecti

ve

Mo

du

lus

Number of cycles

(a) (0,904)s, max = 0.5 %

Specimen 4

LV_S4

Specimen 9

LV_S9

Ply Discount

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

100 1000 10000 100000 1000000 10000000

Eff

ecti

ve

mo

du

lus

Number of cycles

(b) (0,904)s, max = 0.6 %

Specimen 5

LV_S5

Specimen 10

LV_S10

Ply Discount

Figure 20 Comparison of experimental effective modulus with its predicted values based on

Loukil-Varna equation and ply discount model during fatigue tests at higher strain levels for

(0,904)s configuration

42

𝜌𝑐𝑛𝑒𝑓𝑓

= 𝜌𝑐𝑛 (0.73𝑙𝑑

𝑡90+ 1) … … … … … . (4)

Where,

𝑙𝑑 − average delamination length

𝑡90 − thickness of transverse layer

This 𝜌𝑐𝑛𝑒𝑓𝑓

can be used in Eq. (1) in place of 𝜌𝑐𝑛, and modulus reduction including

delaminations can be calculated. A modified comparison including the average delamination

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

100 1000 10000 100000 1000000 10000000

Eff

ecti

ve

Mo

du

lus

No of cycles

(a) (0,904)s, max = 0.5 %

Specimen 9

LV_S9_without

delamination

LV_S9_with

delamination

Ply Discount

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

100 1000 10000 100000 1000000 10000000

Eff

ecti

ve

Mo

du

lus

Number of cycles

(b) (0,904)s, max = 0.5 %

Specimen 4

LV_S4_without

delamination

LV_S4_with

delamination

Ply Discount

Figure 21 Comparison of predicted values based on Loukil-Varna equation ‘with’ and ‘without’ including the

effects of delamination

43

lengths for Specimen 4 and 9 of (0, 904)𝑆 configuration is shown below in Fig. 21.

4.2.5 Fatigue Power Law parameters

Similar to probability distribution function based on Weibull parameters for quasi-static case,

a Power Law representation of probability of failure in case of fatigue is given by following

equation:

𝑃𝑓 = 1 − exp (−𝐿

𝐿0. 𝑁𝑛. (

𝜎90

𝜎0)

𝑚

) … … … … … … … . . (5)

Where,

𝐿 = 𝐿0 − Element length

𝑁 − Number of cycles

𝑛 − Power Law parameter

𝑚 𝑎𝑛𝑑 𝜎0 − Weibull shape and scale parameter

Similar to quasi-static tests, the crack density prediction using above equation are shown in

Fig. 22, 23 and 24 for all the specimens from the two configurations. The scale parameter can

be calculated here for individual specimen but shows a very large variation among specimens.

As it can be seen in Table 6, there is significant variation in Power law parameter (n). It

shows that this equation is just a basic formulation for prediction of crack density. It also does

not include delamination effects and should be a function of many other test parameters like

frequency and load ratio. But, from the Fig. 22, 23 and 24, it can be seen that although the

predicted numerical values are not same as experimentally observed, but the trends of

saturation (or not) in the two cases is replicated in the predicted values. So, it can act as a

basic equation to be used as first calculations for crack density based on these parameters.

44

y = 0.9958x - 14.866 R² = 0.9716

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

11.50 12.00 12.50 13.00 13.50 14.00

ln(-

ln(1

-Pf)

) [-

]

ln(N) [-]

0.0

0.1

0.2

0.3

0.4

0.5

1 100 10000 1000000

ρc

[1/m

m]

No of cycles

max = 0.5 [%]

Specimen 9

Self Prediction

y = 0.9467x - 13.823 R² = 0.9941

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

8.00 9.00 10.00 11.00 12.00 13.00 14.00

ln(-

ln(1

-Pf)

) [-

]

ln(N) [-]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 100 10000 1000000

ρc

[1/m

m]

No of cycles

max = 0.5 [%]

Specimen 3

Self Prediction

0.0

0.2

0.3

0.5

0.6

0.8

0.9

1.1

1.2

1 100 10000 1000000

rc

[1/m

m]

No of cycles

max = 0.6 [%]

Specimen 8

Self prediction

y = 0.4526x - 6.1323 R² = 0.8954

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

9.00 10.50 12.00 13.50 15.00

ln(-

ln(1

-Pf)

) [-

]

ln(N) [-]

y = 0.6313x - 8.756 R² = 0.8914

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

8.0 10.0 12.0 14.0 16.0

ln(-

ln(1

-Pf)

) [-

]

ln(N) [-]

0.0

0.2

0.3

0.5

0.6

0.8

0.9

1.1

1.2

1 10 100 1000 10000 100000 1000000

rc

[1/m

m]

No of cycles

max = 0.6 [%]

Specimen 4

Prediction

Figure 22 Fatigue Power law parameter and crack density (experimental and predicted) for (0,90)s specimens tested under

fatigue loads

45

y = 0.663x - 10.743 R² = 1

-3.50

-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

8.0 10.0 12.0 14.0 16.0

ln(-

ln(1

-Pf)

) [-

]

ln(N) [-]

0.00

0.02

0.04

0.06

0.08

0.10

1 100 10000 1000000

rc

[1/m

m]

N [-]

max = 0.4 [%]

Specimen 3

Self Prediction

y = 0.6223x - 8.5795 R² = 0.9422

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

8.00 10.00 12.00 14.00 16.00

ln(-

ln(1

-Pf)

) [-

]

ln(N) [-]

0.0

0.1

0.2

0.3

1 100 10000 1000000

rc

[1/m

m]

N [-]

max = 0.4 [%]

Specimen 8

Self Prediction

y = 0.4355x - 5.6549 R² = 0.9576

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

8.0 10.0 12.0 14.0

ln(-

ln(1

-Pf)

) [-

]

ln(N) [-]

0.00

0.10

0.20

0.30

0.40

1 100 10000 1000000

rc

[1/m

m]

N [-]

max = 0.5 [%]

Specimen 4

Self Prediction

y = 0.6686x - 8.2645 R² = 0.9175

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

8.0 10.0 12.0 14.0

ln(-

ln(1

-Pf)

) [-

]

ln(N) [-]

0.0

0.1

0.2

0.3

0.4

0.5

1 100 10000 1000000

rc

[1/m

m]

N [-]

max = 0.5 [%]

Specimen 9

Self Prediction

Figure 23 Fatigue Power law parameter and crack density (experimental and predicted) for (0,904)s specimens tested under

fatigue loads

46

Table 6 Weibull scale parameter and fatigue Power law parameter for crack density during fatigue tests

(𝟎, 𝟗𝟎)𝑺 (𝟎, 𝟗𝟎𝟒)𝑺

𝜺𝒎𝒂𝒙 𝝈𝟎 𝒏 𝝈𝟎 𝒏

[%] [MPa] [-] [MPa] [-]

0.3 123,64 0,48

0.4 177,98 0,62

0.4 241,08 0,66

0.5 1007,56 0,95 195,42 0,67

0.5 1236,76 1,00 135,52 0,44

0.6 413,23 0,63 200,90 0,66

0.6 245,53 0,45 145,89 0,50

Average 725,77 0,76 174,35 0,58

Standard deviation 472,17 0,26 41,84 0,10

y = 0.501x - 5.3179 R² = 0.9923

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

9.0 10.0 11.0 12.0

ln(-

ln(1

-Pf)

) [-

]

ln(N) [-]

0.00

0.10

0.20

0.30

0.40

0.50

1 100 10000 1000000

ρc

[1/m

m]

N [-]

max = 0.6 [%]

Specimen 5

Self Prediction

y = 0.6645x - 7.5992 R² = 0.9855

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

6.0 8.0 10.0 12.0 14.0

ln(-

ln(1

-Pf)

) [-

]

ln(N) [-] 0.0

0.1

0.2

0.3

0.4

0.5

1 100 10000 1000000

ρc [1

/mm

]

N [-]

max = 0.6 [%]

Specimen 10

Self Prediction

Figure 24 Fatigue Power law parameter and crack density (experimental and predicted) for (0,904)s specimens tested under

fatigue loads

47

5. PARAMETRIC FINITE ELEMENT ANALYSIS

The two primary damage mechanisms, namely transverse cracks and delaminations, their

initiation and progression is discussed in detail in chapter 1. The initiation and progression of

transverse cracks and their effects in degradation of laminate properties in cross ply laminates

are well studied. However, the initiation and progression of delaminations is still a current

research topic. From the mechanical test performed on several flat specimens, delamination

lengths at each crack tip were measured and their growth was monitored during the cyclic

loadings. But, this data alone is not sufficient to conclusively predict the initiation and

progression of such delaminations. Thus, a finite element study has been carried out to

calculate the energy release rate (ERR) associated with these delaminations in different

damaged state, and laminate configuration. The laminate configurations studied in these FE

simulations are the two laminate configurations tested during this work, i.e. (0,90)𝑆 and

(0, 904)𝑆.

From the fatigue tests, it has been observed that these delaminations initiate before saturation

of transverse cracks. So, the cases studied in FE analysis correspond to 3 different normalized

crack densities; 𝜌𝑐𝑛 = 0.05, 0.25 𝑎𝑛𝑑 1. These values correspond to three different damaged

states which are observed progressively in fatigue tests.

A fracture mechanics approach has been chosen to study the growth of these delaminations.

The energy release rate (ERR) in Mode-I and Mode-II is calculated using the Virtual Crack

Closure Technique (VCCT) for different delamination lengths. Also the J-integral at the crack

tips is calculated using in-built ANSYS function, which represents the total ERR.

In a practical case, the progression of delamination is a 3-dimensional problem, as the

delamination initiated at the transverse crack tips tend to grow along the length (at the edges)

and the width of the specimen. But, to simplify the model, and to study the longitudinal

growth of delaminations, a 2D model is considered. The geometry and boundary conditions

for the model are shown in Fig 25. The section analyzed here is one quarter of the region

between two transverse cracks. A delamination is embedded in the model between the 0° and

90° layers.

A generalized plane strain (GPS) problem is considered using a Boundary Element Method

(BEM), and contact elements at the embedded delamination crack are employed for the

48

analysis [12, 13]. A detailed review of BEM, and an explanation for the choice of contact

element is clearly identified and explained in [12, 13].

The material properties used in the analysis are same as the material used in the experimental

tests. These properties are mentioned in Table 2. The ply thickness is considered as an

average value of 0.29 [mm], as measured from the test specimens. Energy release rates for

Mode-I and Mode-II are calculated initially ‘with’ and ‘without’ including the residual

thermal stresses using two VCCT approaches:

1) Manually, by using the stress and displacement field near the delamination crack tip to

calculate the work required to close the crack by integrating following

𝐺𝐼 ≈ ∫ 𝜎𝑦 . ∆𝑢𝑦

2. 𝑑𝑟 𝑎𝑛𝑑 𝐺𝐼𝐼 ≈

∫ 𝜏𝑥𝑦 . ∆𝑢𝑥

2. 𝑑𝑟

Where,

𝜎𝑥 𝑎𝑛𝑑 𝜏𝑥𝑦- normal and shear stress arrays respectively, along the integration path

∆𝑢𝑦- displacement in y-axis, between the two sets of nodes of 0° and 90° layer

separated by the delamination crack

∆𝑢𝑥- displacement in x-axis, between the two sets of nodes of 0° and 90° layer

separated by the delamination crack

𝑑𝑟 – small area around the crack tip where the integration is performed

2) In-built VCCT calculation by ANSYS®

Figure 25 FE model used in this study, with boundary conditions

49

5.1 ENERGY RELEASE RATE (MECHANICAL LOADS)

Fig. 26 shows the total ERR distribution (at 1 [%] strain) as a function of delamination length

normalized with respect to thickness of 90° layer for the two laminate configurations. The

thermal residual stresses are neglected in this case. It can be seen from this figure that the

trends are same for the two cases, but total ERR is higher for (0, 904)𝑆 laminate. The general

trends for the total ERR obtained from this analysis are similar to the trends observed by

(París et al., 2010) for (0,90)𝑆 configuration in [8]. But, this numerical comparison between

(0, 904)𝑆 and (0,90)𝑆 configuration suggests that former laminate configuration is more

prone to delamination than latter for same applied strain levels. The comparison shown here is

for a normalized crack density (𝜌𝑐𝑛) of 0.25, but similar trend results are obtained for other

two crack densities as well.

Further, Fig. 27 and 28 show the individual contribution of Mode-I and II ERR for

delamination growth at 𝜌𝑐𝑛 = 1 for (0, 904)𝑆 configuration. It can be said that the growth of

delamination is dominated by Mode-I in the initial progression region (where GI > GII). Also,

it is known that generally Mode-I fracture toughness is lower than Mode-II, which further

supports this conclusion. Beyond that region, the delamination growth is purely in Mode-II, as

GI is close to zero beyond a certain delamination length = 0.2×t90. Further, as the

0

50

100

150

200

250

300

350

400

450

500

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

To

tal

En

erg

y R

ele

ase

Ra

te (

J/m

2)

𝑙𝑑⁄𝑡90

rcn = 0.25

0_90_4_S

0_90_S

Figure 26 Total ERR distribution comparison for delamination growth between (0,90)s and (0,904)s

laminates

50

delamination length tends to the element length (half of crack spacing, 𝑙𝑐), the GII value tends

to zero because a symmetrical delamination growth is considered from neighboring transverse

crack and a very large amount of Energy is required to join the two delamination cracks.

0

50

100

150

200

250

300

350

400

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

GI,

GII

(J/

m2)

𝑙𝑑⁄𝑡90

rcn = 0.25

GI

GII

0

50

100

150

200

250

300

350

400

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

GI,

GII

(J/

m2)

𝑙𝑑⁄𝑡90

rcn = 0.25

GI

GII

Figure 27 ERR in Mode-I and Mode-II for delamination growth in (0,904)s laminates

Figure 28 GI and GII variation at small delamination lengths (Early stages of Fig 27)

51

Fig. 29 and 30 show the effect of crack density on total ERR values for delamination growth.

As it can be seen in this figure, ERR available for delamination growth is higher at low crack

densities (𝜌𝑐𝑛 = 0.05), and decreases as the crack density tends towards maximum value

(𝜌𝑐𝑛 = 1). Based on such an effect of crack density, one could conclude that the delamination

growth should be higher even at lower crack densities. But, such a behavior is not observed in

either quasi-static or fatigue tests. As mentioned in experimental results, delaminations are

only observed in (0, 904)𝑆 specimens under quasi-static tests at higher strain levels, where the

crack density is close to saturation (corresponds to a 𝜌𝑐𝑛 = 0.8). For fatigue specimens, the

delaminations start to grow before saturation of crack density (at 𝜌𝑐𝑛 = 0.6), and continue to

grow further when the crack density is saturated. This represents a contradiction between the

experimental observations and ERR obtained from FE analysis. A possible explanation for

this contradiction is explained later by a hypothesis in conclusions.

0

100

200

300

400

500

600

0 1 2 3 4 5 6 7 8 9 10

To

tal

ER

R (

J/m

2)

ld/t90

(0,904)s RCN 1

RCN 0.25

RCN 0.05

Figure 29 ERR distribution at different crack densities (different damaged stage of laminate)

52

5.2 ENERGY RELEASE RATE (THERMO-MECHANICAL LOADS)

Fig 31 shows the comparison of GI, GII and total G for the case of (0, 904)𝑆 configuration at

𝜌𝑐𝑛 = 1; between ‘with’ and ‘without’ thermal stresses. It can be seen that the trends are not

affected by inclusion of thermal stresses, only the numerical values are higher when thermal

stresses are included.

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

To

tal

ER

R (

J/m

2)

ld/t90

(0,904)s

RCN 1

RCN 0.25

RCN 0.05

0

50

100

150

200

250

0 0.1 0.2 0.3 0.4 0.5

To

tal

ER

R (

J/m

2)

𝑙𝑑⁄𝑡90

G_Without residual thermal

G_With residual thermal stress

Figure 30 Early stages of Fig. 29

Figure 31 Effect of thermal stresses on total ERR for (0,904)s laminate

53

5.3 MODULUS REDUCTION DUE TO DELAMINATION

Effect of delaminations in further reducing the longitudinal modulus of the laminate is

discussed in this section. Two different approaches are used for comparison. In first approach,

the Effective modulus is calculated as follows

𝐸𝐿𝑒𝑓𝑓

= 𝐸𝐿

𝐸𝐿𝑙𝑑=0

Where,

𝐸𝐿 − Longitudinal modulus of damaged laminate with transverse cracks and delamination

𝐸𝐿𝑙𝑑=0 − Longitudinal modulus of the damaged laminate at corresponding rcn at zero

delamination.

Fig. 33 shows the variation of effective modulus for (0, 904)𝑆 calculated using first approach

as a function of delamination length normalized with respect to the half crack spacing (lc). A

further reduction of 5, 15 and 20 [%] is possible due to delamination growth for rcn = 1, 0.25

and 0.05 respectively.

-5

15

35

55

75

95

115

135

155

175

195

0 0.1 0.2 0.3 0.4 0.5

GI a

nd

GII

(J/

m2)

𝑙𝑑⁄𝑡90

(0,904)s, rcn = 1

GI_with thermal stresses

GII_with thermal stresses

GI_without thermal stresses

GII_without thermal stresses

Figure 32 Effect of thermal stresses on GI and GII

54

Another approach to compare these moduli values would be to normalize them with respect to

the Longitudinal modulus of the undamaged laminate. Such a plot is shown in Fig 34. The

advantage of the first approach is that the further reduction (in [%]) in longitudinal modulus

due to delamination can be easily identified, as all the values are normalized with respect to

the modulus at zero delamination, which visually removes the effect of transverse crack

density in the plot. Same conclusion about the percentage reduction in longitudinal modulus

can be made from latter plot, but it is not easily visible. However, the second approach

represents the effect of both crack density and delamination clearly, and shows that all the

curves tend towards the predicted value by Ply discount model, which corresponds to a

0.75

0.8

0.85

0.9

0.95

1

1.05

0 0.2 0.4 0.6 0.8 1

Eff

ecti

ve

mo

du

lus

ld/lc

(0,904)s

RCN=1

RCN=0.25

RCN=0.05

Figure 33 Modulus reduction due to delamination at three different crack densities for (0,904)s

laminates using FE analysis. (Approach 1)

0.75

0.8

0.85

0.9

0.95

1

0 0.2 0.4 0.6 0.8 1

Eff

ecti

ve m

od

ulu

s

ld/lc

(0,904)s RCN=1

RCN=0.25

RCN=0.05

Ply dicount model

Figure 34 Modulus reduction due to delaminations at three different crack densities for (0,904)s

laminates using FE analysis (Approach 2)

55

damaged state where 90° layer is unable to carry any load. Similar two comparisons for

(0,90)𝑆 laminates in Fig. 35 and 36.

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

0 0.2 0.4 0.6 0.8 1

Eff

ecti

ve m

od

ulu

s

ld/lc

(0,90)s

RCN=1

RCN=0.25

RCN=0.05

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 0.2 0.4 0.6 0.8 1

Eff

ecti

ve m

od

ulu

s

ld/lc

(0,90)s RCN=1

RCN=0.25

RCN=0.05

Ply dicount model

Figure 35 Modulus reduction due to delaminations at three different crack densities for (0,90)s

using FE analysis (Approach 1)

Figure 36 Modulus reduction due to delaminations at three different crack densities for (0,90)s

using FE analysis (Approach 2)

56

6. CONCLUSIONS AND RECOMMENDATIONS

There are several general observations about the two damage mechanisms that have already

been published by several authors presented in Chapter 2, and have become generally

accepted facts. Most of the experimental observations of present study are concurrent with

such generally accepted facts, and are mentioned in detail in Chapter 4 of this report. Apart

from these generally accepted observations, following additional conclusions can be made

from this study:

During tension-tension cyclic loadings of these cross-ply laminates, the delaminations

initiate before saturation of transverse crack density. These delaminations seem to

grow as a logarithmic function of number of cycles (𝑁). Delamination length (𝑙𝑑) that

can be represented as

𝑙𝑑 = 𝐴. 𝑙𝑛 𝑁 − 𝐶

Where, A and C are constants that determine the rate of growth of delamination and

initiation of delamination respectively. According to the authors understanding, these

constants should depend on material properties, mainly fracture toughness and also on

the load ratio and maximum applied stress during fatigue tests. But, there is not

enough data available to make any concrete statements based on current study. A

further experimental study specifically focused on delamination growth on different

materials with different fracture toughness can shed more light on this observed

behavior and help identify these constants.

The ERR distribution comparison for the two laminate configurations confirms the

experimental observations that cross-ply laminates with thicker transverse plies are

more prone to delamination growths due to higher available ERR in all the damaged

states.

Residual thermal stresses promote growth of delaminations both in Mode-I in case of

small delamination lengths, and Mode-II for larger delamination lengths.

For typical polymer composites that usually have GIIc > GIc, during tensile quasi-static

loads, delaminations initiated from transverse crack tips can only grow up to a small

delamination lengths dominated by Mode-I. Further growths can occur due to increase

in loads, but at same stress levels further growth of delaminations, as controlled my

Mode-II, is less likely, as GIIc is higher than the available ERR. In this conclusion,

only a vague ‘>, greater than’ sign is used, because a quantitative relation between the

57

two fracture toughness is unknown for the material used in this study. i.e. It is difficult

for the author to conclude that how much the GIIc has to be greater than GIc for such a

statement to be true.

The observation that, ‘during high strain cyclic loads delaminations initiate before

saturation of crack density, but tend to grow further to larger lengths after saturation of

crack density’, is in contradiction with the fact that ERR available for these

delaminations to grow is higher at low crack density, and much lower at higher crack

density. This contradiction leads to following hypothesis to explain the observed

behavior during fatigue tests:

Assuming a stepwise delamination growth in fatigue, once the delaminations are

initiated at the transverse crack tips, they grow up to smaller lengths controlled by

Mode-I, as GIc is smaller than the GI at small delamination lengths. But, in order

for these delaminations to grow further, controlled by Mode-II, without any further

increase in loads (as the case in fatigue tests), the fracture toughness (GIc and GIIc)

of the material must decrease ahead of the delamination crack tips. This will result

in a stepwise growth of delaminations as the no of cycles continue to increase,

because the GII will continue to decrease either with increasing crack density or

with increasing delamination length.

For matrix materials with higher fracture toughness, oblique cracks can be a

competing damage mechanism at higher crack densities. Further, the delaminations

associated with these oblique cracks are very significant. Such oblique cracks or the

associated delaminations are not usually considered in basic studies of cross-ply

laminates, as they are not done in this study as well. But, as the composite industry is

moving towards the resin systems with higher fracture toughness, study of oblique

cracks is much more pertinent. A similar fracture mechanics based approach that is

used in this study should be implemented for progression of such oblique cracks and

the delaminations associated with them.

58

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