MASTER'S THESIS

Models for bending stiffness in laminateswith intralaminar and interlaminar damage

Hiba Ben Kahla2014

Master of Science (120 credits)Materials Engineering

Luleå University of TechnologyDepartment of Engineering Sciences and Mathematics

Models for bending stiffness in laminates

with intralaminar and interlaminar damage

MASTER THESIS

Hiba Ben Kahla

Supervised by:

Prof. Janis Varna, Division of Materials Science

June 2014

Luleå University of Technology

Department of Engineering Science and Mathematics

Division of Materials Science

“Life is like riding a bicycle. To keep your balance, you must keep moving.”

-Albert Einstein

i

Preface

Titled “Models for bending stiffness in laminates with intralaminar and interlaminar damage”,

the master thesis work presented in this report was carried out at the Division of Materials

Sciences of Department of Engineering Sciences and Mathematics of Luleå University of

Technology during the first semester of 2014.

This thesis is the report of a long process. It cannot express the joy for the research, the hope

for good results and the sadness and tiredness with each failed attempt.

The successful completion of this thesis would not have been possible without the support and

the encouragement from many individuals who contributed in diverse ways to the conduct of

the thesis. I owe you a great deal of gratitude.

I would like to thank everyone who has helped and supported me during the writing of my

Master’s thesis and I will name a few here.

First of all I would like to express my gratitude to my supervisor Janis Varna who has shared

his valuable knowledge. I was so glad to discuss with you and I was proud to take from your

valuable time despite your tight schedule, proud because I learnt directly from you. Your

advice, guidance, consistency, support, patience, valuable feedback and professional touch

have been so appreciated during this period.

Moreover, I would like to give some special thanks to Professor Roberts Joffe, Doctor

Andrejs Purpurs, Doctor Mohamed Loukil and Hana Zrida for their support. I never forget

that you were welcoming my questions and helping me when I need support to make the work

advance.

I would also like to thank AMASE Program for giving me the opportunity to do my Master

studies in this amazing atmosphere.

I want to thank my dear friends in luleå. You were always there for me to support me and to

make me see the light again if I got stuck with my research and homesickness.

Last, but not least, I highly appreciate continuous support of my dear friends from Tunisia and

my loving family for which words are not enough to express my “THANKS”. My

appreciation especially goes to the most important people in my life, my parents and my

brother, for their unconditional love and support: The hardest part of studying abroad was

missing you.

Luleå University of Technology, June 2014

Hiba BEN KAHLA

Abstract

Failure process of composite laminate under mechanical loading involves

accumulation of intra- and interlaminar damage. Matrix cracking parallel to the fibers is

generally the first mode observed and these cracks are either arrested at the interface or

cause delamination due to high interlaminar stress at the ply interface. In plane-stiffness

degradation due to presence of interlaminar damage introduced from transverse cracks

in cross ply laminates is analysed and its dependency on delamination length and on the

crack density is examined. Simple approach based on Classical Laminate Theory,

effective stiffness of the damage layer and GLOB-LOC approach applied for cross ply

laminates with intralaminar and interlaminar damage undergo tensile load is suggested

to estimate bending stiffness of those laminates. The results derived from tensile loading

are compared to the results obtained from 3-D finite element simulation for 4-point

bending test.

Keywords: damaged laminate, bending stiffness, effective stiffness, intralaminar cracks,

delamination.

iv

v

Nomenclature

The symbols and abbreviations used in this thesis are listed here.

List of symbols:

𝐸𝑇 Transverse modulus of the damaged layer in the local coordinate system

𝐸𝐿 Longitudinal modulus of the damaged layer in the local coordinate system

𝜈𝐿𝑇 Poisson’s’ ratio of the damaged layer

𝐸𝑥𝐿𝐴𝑀 x- Modulus of the damaged laminate

𝐸𝑦𝐿𝐴𝑀 y- Modulus of the damaged laminate

𝜈𝑥𝑦𝐿𝐴𝑀 Poisson’s ratio of the damaged laminate

U2an Normalized average crack opening displacement

𝜌 Crack density (cracks/mm)

𝑘𝑥𝑦 Twisting curvature

𝑘𝑥 Curvature in the x direction

ky Curvature in the y direction

c11 Bending stiffness (N.m)

List of abbreviations:

COD Crack Opening Displacement

FE Finite Element

FEM Finite Element Methods

CLT Classical Laminate Theory

RVE Representative Element Volume

vi

vii

Tables of contents

Preface ..................................................................................................................................................................................... i

Abstract ................................................................................................................................................................................ iii

Nomenclature ...................................................................................................................................................................... v

Chapter 1: Introduction ............................................................................................................................. 1

1. Background ....................................................................................................................................................... 1

2. Objectives ........................................................................................................................................................... 1

3. Methodology ..................................................................................................................................................... 2

4. Thesis outline ................................................................................................................................................... 2

Chapter 2: Literature Review .................................................................................................................. 3

1. Preamble ............................................................................................................................................................ 3

2. Damage mechanisms: ................................................................................................................................... 7

2.1. Matrix crack (Intralaminar cracking) ................................................................................................. 8

2.2. Delamination (Interlaminar cracking) ............................................................................................... 9

2.3. Fiber breaks ................................................................................................................................................ 10

2.4. Development of damage in laminated composites .................................................................... 10

3. Stiffness reduction in damaged laminates ........................................................................................ 11

Chapter 3: Effect of local delamination on in-plane stiffness ................................................... 13

1. Methodolody .................................................................................................................................................. 13

2. Results and Discussion .............................................................................................................................. 15

Chapter 4: Models for bending stiffness of laminates with intralaminar and interlaminar damage .............................................................................................................................. 17

1. Methodolody .................................................................................................................................................. 17

1.1. Effective stiffness of the damaged layer ................................................................................... 17

1.2. Bending stiffness ................................................................................................................................ 19

1.3. Models ..................................................................................................................................................... 21

2. Results and Discussion .............................................................................................................................. 22

2.1. Effect of the delamination length on transverse effective modulus.............................. 22

2.2. Effect of the delamination length on bending stiffness ...................................................... 25

2.3. Comparison to 3-D FE calculation of bending stiffness for 4-point bending test ... 30

Conclusions ........................................................................................................................................................................ 31

Future work ....................................................................................................................................................................... 31

References .......................................................................................................................................................................... 32

viii

1

Chapter 1: Introduction

1. BACKGROUND

Composites are mainly used to reduce the weight of structures, as they show

better strength to weight ratio. Generally, they find use with helicopters, light air craft,

commuter planes and sailplanes due to their advantage in low weight and fatigue

strength. However, their heterogeneous and anisotropic structure make more difficult

the understanding and characterization of their damage under mechanical loading,

which is internal and occurs on multiple scales. With the increase of their use, the

identification of the mechanical behavior of these materials has now become a major

issue. The most common modes of damage are matrix cracking, delaminations and

fibers break. Several researchers are studying the effect of these different modes of

damage on the stiffness of the composite. Most of them have been focusing on the matrix

cracking which is usually the first mode of damage to occur. In this framework, the

thesis work is focused on the effect of the local delaminations initiated from transverse

cracks on in-plane and bending stiffness for specific kinds of composites: cross-ply

laminates.

2. OBJECTIVES

To make the best use of the laminated composites, it is very important to well

understand the properties degradation induced by different modes of damage. In this

context, the first objective of this work is to study the effect of local delamination on

in-plane stiffness for cross-ply laminates. Then, the second part of the work is to predict

the bending stiffness of cross-ply laminates with intralaminar and interlaminar damage

using the results of the first part dedicated to in-plane stiffness. The accuracy of

prediction has to be evaluated later by comparing the results already found to direct

3-D FE simulation of 4-point bending test. In order to achieve the thesis objectives, the

methodology summarized in the nest paragraph is suggested.

2

3. METHODOLOGY

The Thesis work consists of two parts. The first part is dedicated to study the effect

of the local delamination initiated from transverse cracks in 90°-layer on in-plane

stiffness within a cross-ply laminate. Since Stiffness reduction could be described in

terms of crack opening displacement (COD), the COD is determined using finite element

methods (FEM) and its dependency on the delamination length is studied. Then, in the

second part, the idea is, using the results obtained in the FE simulation of a tensile test

done in the first part, to predict the bending stiffness of the laminate based on classical

laminate theory (CLT) and GLOB-LOC approach. The effective stiffness of the damaged

layer has to be determined to be used later in bending simulations. The accuracy of

those results is to be evaluated by comparing them with bending stiffness results from

FE simulation of 4-point bending test.

4. THESIS OUTLINE

This thesis can be divided into 4 Chapters.

Chapter One is about the introduction, main objectives and scope of performed research

work.

Chapter Two is a literature review regarding the background of the work.

Chapter Three is dedicated to the effect of the local delamination initiated from

transverse cracks in 90°-layer on in-plane stiffness within a cross-ply laminate: the

procedure of the work is described and the results are discussed.

Chapter Four is a detailed description of the methodology of the second part of the work

related to models for bending stiffness in damaged laminates and it discusses the main

results.

At the end, the most important conclusions are mentioned and some recommendations

for future research work are suggested.

3

Chapter 2: Literature Review

1. PREAMBLE

A composite material is a material consisting of at least two distinct phases or

constituents. This description doesn’t define rigorously a composite. The constituents

have to have significantly different physical properties and combining them gives the

composite unique and better properties than the properties that has each component

separately. One constituent is termed the reinforcing phase (discontinuous phase) and

the one in which it is planted is called the matrix (continuous phase).

The matrix plays several important roles in the composites structure, such as

holding the fibers together and oriented, transferring stress between fibers, and

protecting them from adverse environment.

Matrix materials used in composites are typically ceramics, metals, or polymers.

Polymeric matrices are those used in this work.

The reinforcing phase material may be in form of particles, flakes or fibers (short

or continuous long fibers).

The properties of a composite depend on the properties of the constituent

materials, the geometric distribution of fibers, the volume ratio of the fibers, the nature

of the fibers / matrix interfaces, the manufacturing process...

There are different composites which perform each one on a useful purpose so

they can be used in various applications. Composites have been used widely in the

industry due to their performance in term of their strength while maintaining light

weight.

The aeronautics industry especially takes benefits from the performance of

composites to develop the plane structure replacing metallic materials with the lighter

ones. Fig. 1 shows the evolution of composite applications at airbus.

4

Figure 1.Evolution of composite application at Airbus [1]

Fibers are the common reinforcement due to their effectiveness. They dominate

most of the characteristics and properties of the composites. It is important to select the

proper fibers in order to obtain the properties estimated for the final product.

Fiber reinforcements are divided into continuous or discontinuous. Continuous

long fiber composites are those used for advanced applications such as aircraft, space

shuttles and load bearing structural parts. They are also those used in this work. They

have generally excellent specific properties. The Table 1 lists the properties of often

used fibers comparing them with steel and aluminum. It shows that fibers have much

higher specific properties compared to metals but a rigorous comparison needs taking

the properties of matrix into account.

There are a large variety of fibers with different properties that can be used as

reinforcements: glass, carbon, aramid, high performance polyethylene (HPPE), ceramic

fibers, and metal fibers.

5

Material Tensile modulus

(E) (GPa)

Tensile strength

(𝝈𝒖) (MPa)

Density (ρ)

(𝒈 𝒄𝒎𝟑⁄ )

Specific modulus

(𝑬 𝝆⁄ ) 𝟏𝟎𝟔 𝑵𝒎 𝒌𝒈⁄

Specific strength (𝝈𝒖 𝝆⁄ )

𝟏𝟎𝟔 𝑵𝒎 𝒌𝒈⁄ E-glass 72.4 3.5 2.54 28.5 1.38 S-glass 85.8 4.6 2.48 34.5 1.85

Graphite (high Modulus)

390 2.1 1.9 205.0 1.1

Graphite (high tensile strength)

240 2.5 1.9 126.0 1.3

Boron 385 2.8 2.63 146.0 1.1 Kevlar 49 130 2.8 1.5 87.0 1.87

Steel 210.0 0.34-2.1 7.8 26.9 0.043-0.27 Aluminum alloys 70.0 0.14-0.62 2.7 25.9 0.052-0.23

Table 1.Properties of fibers and metals [2]

Fiber-reinforced composites generally have many interesting advantages over

traditional materials; the most important one is their light weight. Low density leads to

high specific strength and specific stiffness. In addition, polymer laminates offer design

flexibility [3p130]. That justifies the fact that in recent decades, there is a rapid growth

in the use of fiber reinforced polymer (FRP) composites in advanced applications such

as aerospace, aeronautics, automobile and marine industries.

Fiber reinforced composites can be classified as single layer and multilayer

composite. Composites used in structural applications are mostly multi-layered. Each

layer is a single composite: same orientation of fibers and same properties within the

whole thickness of the layer.

The multi-layered composites could be consisting of layers made up of different

constituent materials; the composite is called in this case hybrid laminates. If the

constituents in each layer are the same the composite is termed laminate.

A laminated composite is fabricated by stacking plies in different orientations, as

it is shown in Fig. 2.a each unidirectional ply has a fiber direction given by the angle

between the fibers and an arbitrary axis and the denomination of a laminate is based on

the sequence of the different angles of layers. This assemblage of layers provides

required engineering properties, including in-plane stiffness, bending stiffness, strength,

and thermal expansion and by proper combination of constituting layers a balance of

such properties can be achieved such as light weight, high strength, high stiffness, wear

6

resistance, corrosion resistance, unusual thermal expansion characteristics, appearance,

etc.

Figure 2. a. Laminated composite [4] ; b. Cross-ply laminate [5]

A cross-ply laminate, as shown in Fig. 2.b, consists of an arbitrary number of layers of

the same material but with alternating orientations of 0° and 90°. They present

interesting properties. Table 2 shows the properties of a cross ply laminate for different

fibers comparing with metals. It is clear that except Glass epoxy cross-ply, laminates

have higher specific properties which explain the use of composites in structural

components when low weight and high stiffness are needed.

Although they are rarely used in practical applications, cross ply laminates are excellent

for academic studies, because they are relative simple to be studied.

Material Fiber Volume Fraction 𝑽𝒇(%)

Tensile modulus

(E) (GPa)

Tensile strength

(𝝈𝒖) (MPa)

(𝒈 𝒄𝒎𝟑⁄ )

Density (ρ)

𝟏𝟎𝟔 𝑵𝒎 𝒌𝒈⁄

Specific modulus

(𝑬 𝝆⁄ ) 𝟏𝟎𝟔 𝑵𝒎 𝒌𝒈⁄

Specific strength (𝝈𝒖 𝝆⁄ )

Mild steel Aluminum

210 450-830 7.8 26.9 58-106

2024-T4 73 410 2.7 27.0 152 6061-T6 69 260 2.7 25.5 96 E-glass-epoxy

57 21.5 570 1.97 10.9 260

Kevlar 49-epoxy

60 40 650 1.4 29.0 460

Carbon fiber-epoxy

58 83 380 1.54 53.5 240

Boron epoxy

60 106 380 2.0 53.0 190

Table 2.Properties of conventional structural materials and cross-ply fiber composites[2]

7

2. DAMAGE MECHANISMS:

During its life, composites structure is subjected to impacts by foreign objects. These

impacts (for example during take offs and landings…) occur during manufacturing,

service, and maintenance operations. The impact on a fiber-reinforced composite differs

from the damage on a metallic structure. It is more dangerous because it creates internal

damage that can be observable as shown in Fig. 3 but most of the time it is not visible

and cannot be detected by visual inspection which is not the case for metallic materials.

Figure 3. Damage of aileron due to hail impact [6]

The study of damage on laminate composite structure shows that generally the

damage involves three types of physics phenomena [7] which are: matrix cracking,

delamination and fiber fracture as shown in Fig.4.

Figure 4. Damage mechanism of laminate composite structure subjected to impact loading [3p166]

8

2.1. Matrix crack (Intralaminar cracking)

The matrix cracking or intralaminar cracking or micro-cracking refer to the same

phenomenon. It was due to big difference mechanical properties between matrix and

fibers. It happens generally parallel with fiber direction of plies as it is shown in Fig. 5.

The properties in the longitudinal direction for fiber-reinforced composites are

superior over transverse direction’s properties which are generally low, which develops

easily cracks along fibers. Intralaminar cracks are usually the first mode of damage and

they induce other modes of damage (such as delamination and fiber breaks) which are

more dangerous and may lead to composite’s failure.

Figure 5. Matrix cracking [8]

Many researchers are studying matrix cracking to be able to understand the micro-

cracking process and to design laminates which are more resistant to this mode of

damage. In fact intralaminar cracks cause degradation on the thermo-mechanical

properties of the material and in particular the stiffness, so it is important to know how

such damage occurs and how does it grow or propagate.

Cracking occurs by irreversible separation of a continuous solid in two parts, called

the crack faces, which introduces a discontinuity in the direction of the displacement.

The relative movement of the faces of the crack can decompose into two contributions

as shows Fig. 6: the crack opening displacement (COD), normal to the surface of the

crack and crack sliding displacement (CSD) along this surface.

9

These two parameters, during loading, reduce the average stress in the damaged layer,

thus causes degradation on the thermo-mechanical properties of the laminate.

So the key to understand more the micro-cracking process and its effects on the

laminate’s structure is to investigate the COD and the CSD.

Figure 6. Crack opening and crack sliding displacements

2.2. Delamination (Interlaminar cracking)

Delamination, which is a crack in the interfacial plane between two plies on

contact in a laminate, leads to separation of the plies as is shown in Fig.7. The most

worrying thing about this mode of damage is that its growth under loading may lead to

the deterioration of the mechanical properties and to catastrophic failure of the

composite.

Delamination could reduce the role of fiber strength and make the weaker matrix

properties handle the structural strength.

Many research were focused on intralaminar cracking, however, transverse

cracks often induce interlaminar delamination which relieve the local strain

concentration at the transverse crack tips, but which cause other problems in the load-

carrying plies.

10

Figure 7. Delamination [8]

2.3. Fiber breaks

When laminate is loaded in fiber direction the individual fibers fail at their weaker

flaws, the fiber/matrix interface debonds affecting other nearby fibers and may break

some. Within a laminate, the stress on fibers is intensified in the proximity of ply crack

in the adjacent layers and the failure of a laminated composite finally comes from

breakage of fibers. The Fig.8 shows fiber breaks.

Figure 8. Fiber breaks [8]

2.4. Development of damage in laminated composites

The first mode of damage to appear is generally matrix cracking. Matrix cracks

initiate from the location of defects: voids, areas with high fiber volume fraction or resin

rich area and then as the applied load is increased more and more, cracks appear.

Multiple matrix cracking dominate in the layers where fibers are disposed transversally

to the applied load direction. At the beginning, the cracks are isolated from each other

and there is no interaction between them. When the number of cracks increases, they

become closer and closer and they start interacting. Applying more load causes initiation

11

of cracks transverse to the primary matrix cracks then small and isolated interlaminar

cracks could appear. Those interlaminar cracks can merge into strip like zones leading

to large scale delamination. This appears as a consequence of the loss of the integrity of

the laminate in these regions and the damage develops unstably and involves extensive

fiber breaks.

3. STIFFNESS REDUCTION IN DAMAGED LAMINATES

When a laminates composite is subjected to thermal or mechanical loading, the

different kinds of microdamage modes may derive causing degradation of thermo-

elastic properties without leading to final failure. Transverse ply cracking is the

principle cause of stiffness reduction in composites. Relative displacements of crack

surfaces during loading reduce the average strain and stress in the damaged layer which

causes the laminate stiffness degradation. Over the years many researches, based on

analytical approaches and/or numerical methods, have been developed which is attempt

to model, with different degrees of accuracy the laminate reduced stiffness properties.

The analytical analyses include shear lag method [9, 14], continuum damage model [15,

16], variational approaches [17, 18,19,20] …

The most common approach and the simplest way to determine the stiffness

reduction of layers after being damaged is called ‘ply-discount model’. That approach

assumes that cracked layer is unable to carry any load therefore the stiffness matrix of

the damaged layer is changing to zero. This approach overestimates the reduction of the

properties and it doesn’t depend on the crack density and therefore it is more suitable at

high crack densities.

Most of the models developed were confined to cross ply laminates under uniaxial

tensile loading. But that doesn’t preclude that some studies are focused on the

investigation of the stiffness reduction due to intralaminar cracking in unbalanced

laminates under general in-plane loading [21-24], matrix cracking in the off-axis plies [25],

multilayer matrix cracking of angle-ply and quasi-isotropic laminates [26-28], and

transverse cracking interacting with edge and local delamination [29-33].

The stiffness degradation due to cracks in layers is uniquely related to the opening

and the sliding of crack surfaces. The relative displacement of both crack faces reduce

the average stress between cracks therefore the contribution of the damage layer in

12

carrying the applied load is reduced. Hence, the crack density, the crack opening

displacement (COD) and the crack sliding displacement (CSD) are the micromechanical

parameters governing the stiffness reduction. Since the COD is proportional to the

applied load and ply thickness in linear elastic problem, it has to be normalized to be

used in stiffness modeling.

A theoretical framework, called GLOB-LOC approach [34], presents a unique

relationship between the damaged laminate thermo-elastic properties and the

micromechanical properties. The largest advantage of this model is the transparency of

the derivations and the simplicity of application. Only the average values of COD and

CSD are used in the stiffness expression and the details of the relative displacement

profile are not important.

In this work, we are interested only in cross-ply laminates. The GLOB LOC model in

case of cross-ply laminates with only one cracked 90-layer leads to these expressions:

𝐸𝑥𝐿𝐴𝑀

𝐸𝑥0𝐿𝐴𝑀=

11 + 2𝜌𝑡90(𝑡90 ℎ0⁄ )𝑢2𝑎𝑛𝑐2

(1)

𝐸𝑦𝐿𝐴𝑀

𝐸𝑦0𝐿𝐴𝑀=

11 + 2𝜌𝑡90 (𝑡90 ℎ0⁄ ) 𝑢2𝑎𝑛𝑐4

(2)

𝜈𝑥𝑦𝐿𝐴𝑀

𝜈𝑥𝑦0𝐿𝐴𝑀 =1 + 2𝜌𝑡90(𝑡90 ℎ0⁄ ) 𝑢2𝑎𝑛𝑐1 �1 − 𝜈𝐿𝑇

𝜈𝑦𝑥0𝐿𝐴𝑀�

1 + 2𝜌𝑡90(𝑡90 ℎ0⁄ ) 𝑢2𝑎𝑛𝑐2

(3)

𝑐1 = 𝐸𝑇𝐸𝑥0𝐿𝐴𝑀

1−𝜈𝐿𝑇𝜈𝑥𝑦0𝐿𝐴𝑀

(1−𝜈𝐿𝑇𝜈𝑇𝐿)2 𝑐2 = 𝑐1�1 − 𝜈𝐿𝑇𝜈𝑥𝑦0𝐿𝐴𝑀�

𝑐4 =𝐸𝑇𝐸𝑦0𝐿𝐴𝑀

�𝜈𝐿𝑇−𝜈𝑦𝑥0𝐿𝐴𝑀�

2

(1 − 𝜈𝐿𝑇𝜈𝑇𝐿)2

(4)

h0 denotes the half thickness of the laminate, h0 = t90 + t0. Quantities with subscript

x, y and ‘LAM’ are laminate constants; and quantities with additional index ‘0’ are used

for quantities of the laminate before being damaged. 𝑢2𝑎𝑛 is the normalized average

crack opening displacement.

The stiffness reduction defined by the ratio of the transverse modulus of the

laminate after being damaged divided by the transverse modulus before being damaged

is described in terms of crack density and normalized average crack opening

displacement.

13

Chapter 3

Effect of local delamination on in-plane stiffness

1. METHODOLODY

FE calculations were performed to analyze the dependency of the normalized

average crack opening displacement (COD) on the delamination length for different

crack densities since the stiffness reduction is described in terms of (COD). For all FE

calculations the commercial code ANSYS 14.5 was used. The laminate, subjected to

tensile load, matrix cracks appear and initiate delaminations on the tips of the cracks as

it is shown in Fig.9. A 3-D model was created to model the repeating unit shown in Fig.

10. Transverse cracks are supposed to be uniformly distributed and delaminations are

assumed to be symmetric with respect to the transverse crack plane. t90 and t0 are,

respectively, the 90°-layer and the 0°-layer thicknesses. Lc is the half distance between

two cracks and Ld is the half length of the delamination.

Figure 9. Schematic representation of the tensile loaded laminate in the damage state

The boundary conditions applied to this RVE are: (see Fig.10)

• Constant displacement corresponding to 1% strain was applied to the

representative volume in x-direction and specifically at the surface x= Lc.

• Symmetries are applied to the surface z= t90+t0 and to the surface defined by

(x=0 and t90 < z < t90+t0).

• The bottom surface z= 0 and the surface defined by (x=0 and 0 <z < t90) are

traction free.

14

• Displacement coupling: nodes on the surfaces y=0 and y=w (the width of the

laminate) have the same displacement in y-direction. Therefore the solution does

not depend on y-coordinate and edge effect is eliminated.

Figure 10. Schematic representation of representative volume element FEM model

FE calculations were done for Carbon fiber/Epoxy and Glass fiber/Epoxy, [90 0⁄ ]𝑠 and

[90 02⁄ ]𝑠 laminates. The elastic properties of these materials are described in Table 3.

Table 3. Elastic properties of the UD composites

Series of calculations are done for each material and for each geometrical

configuration to study the effect of the length of the delamination on COD, and for

different crack densities (different values of lc). Then we use the displacement in X-

direction for the nodes at the crack surface to calculate the average value of the COD.

We have fixed an average applied strain. In case of varying element length lc, the same

applied average strain corresponds to different applied loading (laminate stress). That

complicates the analysis and deprives us to see directly the evolution of the parameters

and doing a direct comparison. To overcome this problem and to be used on modeling

the obtained COD is always normalized with respect to the thickness of the cracked layer

and the far field stress in the layer transverse to the crack plane [35].

Then the idea is to know how the length of the delamination affects the normalized

average crack opening displacement. The delamination length ld is parametrically

varied from 0 to 1.1t90.

15

2. RESULTS AND DISCUSSION

Series of calculations are done for [90 0⁄ ]𝑠 and [90 02⁄ ]𝑠Glass fiber/Epoxy and

Carbon Fiber/Epoxy laminates. In Fig. 11 and 12, the evolution of the normalized

average COD versus the length of the delamination is plotted for different crack

densities. The length of the delamination is normalized with respect to the crack size

(t90) .

Figure 11.Evolution of the normalized average crack opening displacement versus normalized delamination length of [𝟗𝟎 𝟎⁄ ]𝐬and [𝟗𝟎 𝟎𝟐⁄ ]𝐬 Glass fiber/Epoxy laminates for different crack

densities

ρ is the crack density ( number of cracks per length unit) expressed on [cracks/mm].

11,21,41,61,8

22,22,42,62,8

3

0 0,2 0,4 0,6 0,8 1 1,2

U2a

n

ld/t90

GF/EP [90/0] 𝑠

ρ = 0.04

ρ = 0.08

ρ = 0.1667

ρ = 0.3333

ρ = 0.64

1

1,4

1,8

2,2

2,6

3

0 0,2 0,4 0,6 0,8 1 1,2

U2a

n

ld/t90

GF/EP [90/02] 𝑠

ρ = 0.04

ρ = 0.08

ρ = 0.1667

ρ = 0.3333

ρ = 0.64

16

Figure 12. Evolution of the normalized average crack opening displacement versus normalized delamination length of [𝟗𝟎 𝟎⁄ ]𝐬and [𝟗𝟎 𝟎𝟐⁄ ]𝐬 Carbon fiber/Epoxy laminates for different crack

densities

The normalized average crack opening displacement is increasing linearly with the

delamination length. It increases slightly faster for GF/EP than for CF/EP and it

increases faster for [90 0⁄ ]𝑠 than [90 02⁄ ]𝑠 .

The normalized average COD evolution with the delamination length is linear with slope

which may depend on the crack densities ρ, the layer properties and on the lay-up.

0,91,11,31,51,71,92,12,32,5

0 0,2 0,4 0,6 0,8 1 1,2

U2a

n

ld/t90

CF/EP [90/0]𝑠

ρ = 0.04

ρ = 0.08

ρ = 0.1667

ρ = 0.3333

ρ = 0.64

0,9

1,1

1,3

1,5

1,7

1,9

2,1

2,3

2,5

0 0,2 0,4 0,6 0,8 1 1,2

U2a

n

ld/t90

CF/EP[90/02] 𝑠

ρ = 0.04

ρ = 0.08

ρ = 0.1667

ρ = 0.3333

ρ = 0.64

17

Chapter 4

Models for bending stiffness of laminates with intralaminar and interlaminar damage

1. METHODOLODY

In this part of the work, first, FE results obtained previously are used to predict the

bending stiffness change.

Once we have the elastic properties of the damaged laminate, CLT is suggested to

calculate the bending stiffness of the damaged laminate. The damaged layer can be

replaced by non-damaged layer with effective elastic properties. The obtained effective

stiffness will be used to predict the laminate bending stiffness change with delamination

length for different crack densities, hence, the necessity to introduce the effective

stiffness of the damaged layer. The results are compared with 3-D FE simulation of the

4-point bending test.

The work in this part is done with the same laminates used for the first part (the two

different materials and the two different lay-ups).

The effective stiffness of the damaged layer and the bending stiffness are detailed more

below.

1.1. Effective stiffness of the damaged layer

To model laminate undergo 4-point bending test for example the model used is as

it is shown in Fig.13: Under bending, the laminate behaves as unsymmetrical because

cracks in the compressed 90°-layer are closed. But we will calculate the effective

stiffness of the damaged 90°-layer obtained from the in-plane stiffness change of the

[90𝑛 0𝑚⁄ ] assuming that the crack density is the same in both 90°-layers. So the model

used in this paragraph is schematized in Fig 14.

18

The assumed symmetry in the damaged state simplifies the work since for symmetric

damaged laminate the [A]-matrix represents the laminate stiffness matrix [Q]lam. The

undamaged laminate stiffness is denoted[𝑄]0𝐿𝐴𝑀.

Figure 13. Damaged laminate under 4-point brnding test

Figure 14. Damaged laminate under tensile load

When, the two equal surface 90°-layers with index i=1 and i= N are damaged, their

effective stiffness is changing from [𝑄�]1 to [𝑄�]1𝑒𝑓𝑓. According to CLT the damaged

laminate stiffness matrix can be written as:

[𝑄]𝐿𝐴𝑀 =𝑡1ℎ

[𝑄�]1𝑒𝑓𝑓 + �

𝑡𝑘ℎ

𝑁−1

𝑘=2

[𝑄�]𝑘 +𝑡1ℎ

[𝑄�]1𝑒𝑓𝑓 (5)

The undamaged laminate stiffness matrix can be written in a similar way:

[𝑄]0𝐿𝐴𝑀 =𝑡1ℎ

[𝑄�]1 + �𝑡𝑘ℎ

[𝑄�]𝑘

𝑁−1

𝑘=2

+𝑡1ℎ

[𝑄�]1 (6)

Subtracting (4)-(5) we obtain:

[𝑄]0𝐿𝐴𝑀 − [𝑄]𝐿𝐴𝑀 = 2𝑡1ℎ

[𝑄�]1 − 2𝑡1ℎ

[𝑄�]1𝑒𝑓𝑓 (7)

Then the effective stiffness matrix of the damaged k-th layer in global axes is:

[𝑄�]1𝑒𝑓𝑓 = [𝑄�]1 −

ℎ2𝑡1

{[𝑄]0𝐿𝐴𝑀 − [𝑄]𝐿𝐴𝑀} (8)

For the analyzed cross ply laminate in question in this project shown in Fig.10

ℎ = 2𝑡90 + 2𝑡0 (9)

19

[𝑄�]90𝑒𝑓𝑓 = [𝑄�]90 −

ℎ2𝑡90

{[𝑄]0𝐿𝐴𝑀 − [𝑄]𝐿𝐴𝑀}

(10)

The equation (8) is transformed to the local coordinate system using the expression:

[𝑄]1𝑒𝑓𝑓 = [𝑇][𝑄�]1

𝑒𝑓𝑓[𝑇]𝑇 (11)

[T] is the transformation matrix.

The effective compliance matrix of the damaged layer in local axes is calculated by inversion and the effective engineering constants of the damaged layer are:

𝐸𝐿𝑒𝑓𝑓 = 1 𝑆11

𝑒𝑓𝑓; 𝐸𝑇𝑒𝑓𝑓 = 1 𝑆22

𝑒𝑓𝑓; ⁄ 𝜈𝐿𝑇𝑒𝑓𝑓 = −𝐸𝐿

𝑒𝑓𝑓𝑆12𝑒𝑓𝑓; 𝐺𝐿𝑇

𝑒𝑓𝑓 = 1 𝑆66𝑒𝑓𝑓 ⁄⁄ (12)

This analysis is valid for symmetric N-layered laminate.

1.2. Bending stiffness

In a cross-ply laminate subjected to 4-point bending test for example (see Fig.15),

a region with constant applied moment exists leading to constant curvature. Since cracks

emerge only in the 90°-layer on the tensile load, the laminate becomes unsymmetrical in

the damage state. Therefore the B-matrix is not zero and some mid-plane strains

𝜀𝑥0 𝑎𝑎𝑎 𝜀𝑦0 are also present. Due to cracks, A, B and D-matrices change. We will study

the change of the laminate bending resistance with damage development plotting the

bending moment to create unit curvature 𝑀𝑥 𝑘𝑥⁄ versus the delamination length

𝑙𝑑normalized with respect to the damaged ply thickness 𝑡90 for different crack densities.

Boundary condition used with applied 𝑀𝑥 leading to 𝑘𝑥are: 𝑘𝑥𝑦 = 𝑘𝑦 = 𝛾𝑥𝑦0.

20

Figure 15. 4-point bending test

In this case the relevant CLT equations are:

0 = 𝐴11𝜀𝑥0 + 𝐴12𝜀𝑦0 + 𝐵11𝑘𝑥

0 = 𝐴12𝜀𝑥0 + 𝐴22𝜀𝑦0 + 𝐵12𝑘𝑥

𝑀𝑥 = 𝐵11𝜀𝑥0 + 𝐵12𝜀𝑦0 + 𝐷11𝑘𝑥

𝑀𝑦 = 𝐵12𝜀𝑥0 + 𝐵22𝜀𝑦0 + 𝐷12𝑘𝑥

(13)

𝐴𝑖𝑗 = �𝑄𝚤𝚥𝑘����𝑁

𝑘=1

𝑡𝑘

𝐵𝑖𝑗 = �𝑄𝚤𝚥𝑘����𝑁

𝑘=1

𝑧𝑘+12 − 𝑧𝑘2

2

𝐷𝑖𝑗 = �𝑄𝚤𝚥𝑘����𝑁

𝑘=1

𝑧𝑘+13 − 𝑧𝑘3

3

(14)

𝑡𝑘 is the thickness of the ply k, k=1,2..N ; the overbar denotes the stiffness of the layer in

the global coordinate system. Solving (13) with respect to 𝑀𝑥 we can express it in form:

𝑀𝑥 = 𝑐11(𝑙𝑑, 𝜚)𝑘𝑥 (15)

The parameter 𝑐11denotes the bending stiffness and its dependency on delamination

length for different cracks densities will be investigated in this work.

21

1.3. Models

FE Analysis done in the first part provides us with the COD. To obtain the elastic

properties of the damaged laminate 𝐸𝑥𝐿𝐴𝑀; 𝐸𝑦𝐿𝐴𝑀; 𝜈𝑥𝑦𝐿𝐴𝑀, we can inject the value of the

normalized average crack opening displacement obtained from FE calculations for

cross-ply laminate under tensile load in the GLOB-LOC approach expressions (1), (2)

and(3) (the laminate lay-up, layer properties and crack density in the 90°-layer are

known [input data]). Then the effective elastic constants could be calculated following

the steps described by the expressions from (5) to (12). Parametric analysis has shown

that only the effective transverse modulus 𝐸𝑇𝑒𝑓𝑓and the shear modulus 𝐺𝐿𝑇

𝑒𝑓𝑓 of the layer

are reduced. The longitudinal modulus 𝐸𝐿𝑒𝑓𝑓and the Poisson’s ratio 𝜈𝐿𝑇

𝑒𝑓𝑓 are not

changing. Since 𝐺𝐿𝑇𝑒𝑓𝑓 is not affecting the bending resistance of cross ply laminates only

the effect of delamination length on 𝐸𝑇𝑒𝑓𝑓 is studied. The damaged layer is replaced by an

undamaged layer with the effective properties:𝐸𝑇𝑒𝑓𝑓; 𝐸𝐿

𝑒𝑓𝑓𝑎𝑎𝑎 𝜈𝐿𝑇𝑒𝑓𝑓. Then, using CLT, the

bending stiffness is calculated. That will be the first model.

Other methods could be suggested and the difference between the methods is the

way to find out the damaged laminate properties.

We found out in the chapter 3 that variation of the normalized average COD is

linear with the delamination length. What we suggest for the second model is to use the

linearity assumption of the COD to predict its value for different delamination length

without doing the FE calculations. The idea is to determine the COD at the absence of

delamination and for only one other corresponding to a determined delamination

length. Based on the assumption of the linearity the equation of the line could be found

and other COD corresponding to others delamination lengths could be easily found out.

Then the same as we have done on the previous method, using GLOB-LOC approach

expressions, 𝐸𝑥𝐿𝐴𝑀; 𝐸𝑦𝐿𝐴𝑀𝑎𝑎𝑎 𝜈𝑥𝑦𝐿𝐴𝑀 are calculated.

The third method is to calculate the COD using the equation (1) after determining

𝐸𝑥𝐿𝐴𝑀from FEM. Then replacing it by its value in (2) and (3), 𝐸𝑦𝐿𝐴𝑀𝑎𝑎𝑎 𝜈𝑥𝑦𝐿𝐴𝑀 are

determined.

22

The results for bending stiffness obtained using these three methods will be

compared to the results found using a fourth method in which the bending stiffness is

calculated when all damaged laminate properties are determined using FEM.

The properties of the laminate after being damaged in all these cases are obtained

for a cross-ply laminate subjected to tensile load (Fig 15) and based on its effective

properties after being damaged, the bending stiffness is found. To evaluate the accuracy

of these methods and the ability to predict the bending stiffness from results obtained

for a tensile load, it is necessary to compare those results to 3-D FE simulation of

bending test. 4-point bending test is suggested.

It has to be emphasized that the back calculation is unique only if one layer is

damaged.

2. RESULTS AND DISCUSSION

2.1. Effect of the delamination length on transverse effective modulus

The results related to the effective transverse modulus presented below

correspond the first model describes in the previous paragraph.

Plotting the transverse effective modulus versus the normalized delamination length

for different crack densities (see Fig.16 and Fig.17.a), we can come to these conclusions:

• Thickness ratio of plies in cross ply laminates has negligible effect on effective

transverse modulus of the layer regardless of the delamination size and of the

crack density. ( See Fig.17.b)

• The effective transverse modulus of GF decreases with both crack density and

delamination size slightly faster than that of CF fiber reinforced laminates.

• The effective transverse modulus is decreasing linearly with the delamination

length.

23

Figure 16. Reduction of the 90°-layer effective transverse modulus versus normalized delamination length of [𝟗𝟎 𝟎⁄ ]𝐬and [𝟗𝟎 𝟎𝟐⁄ ]𝐬 Glass fiber/Epoxy laminates for different crack

densities corresponding to the first model

0

3

6

9

12

15

0 0,2 0,4 0,6 0,8 1 1,2

E T (G

Pa)

Ld/t90

GF/EP [90/0]𝑠

ρ = 0.04

ρ = 0.08

ρ = 0.1667

ρ = 0.3333

ρ = 0.64

0

3

6

9

12

15

0 0,2 0,4 0,6 0,8 1 1,2

E T (G

Pa)

ld/t90

GF/EP [90/02]𝑠

ρ = 0.04

ρ = 0.08

ρ = 0.1667

ρ = 0.3333

ρ = 0.64

24

Figure 17.a. Reduction of the 90°-layer effective transverse modulus versus normalized delamination length of [𝟗𝟎 𝟎⁄ ]𝐬and [𝟗𝟎 𝟎𝟐⁄ ]𝐬Carbon fiber/Epoxy laminates for different crack

densities corresponding to the first model

0

2

4

6

8

0 0,2 0,4 0,6 0,8 1 1,2

E T (G

Pa)

ld/t90

CF/EP [90/0]𝑠

ρ = 0.04

ρ = 0.08

ρ = 0.1667

ρ = 0.3333

ρ = 0.64

0

2

4

6

8

0 0,2 0,4 0,6 0,8 1 1,2

E T (G

Pa)

ld/t90

CF/EP [90/02]𝑠

ρ = 0.04

ρ = 0.08

ρ = 0.1667

ρ = 0.3333

ρ = 0.64

25

Figure 27.b. Non-independence of effective transverse modulus of the layer (GPa) on the thickness ratio regardless the delamination length and the crack density

2.2. Effect of the delamination length on bending stiffness

In Fig. 18,19, 20 and 21; the influence of the delaminations length in bending

stiffness reduction is shown. M1, M2, M3 and M4 correspond to results obtained by

using Method 1, 2, 3 and 4 respectively. These methods are described with details in the

paragraph related to models in the Methodology.

The bending resistance is additionally reduced by the presence of delamination

initiating from cracks. As expected the reduction is much larger due to longer

delamination. The bending stiffness is decreasing linearly with delamination length.

There is a good agreement between the four methods. The assumption of the COD as a

linear function of ld/t90 shows a good agreement with FEM and GLOB-LOC approach

especially at low crack density.

If we focus on how does the stiffness bending evolves with increasing crack

density and/or with longer delaminations (Comparing the three curves of each figure of

the next four figures), the effect of the delamination seems to be relatively smaller

comparing to matrix cracking effect for laminates with thick cracked layer.

26

Figure 18. Reduction of the 90°-layer bending stiffness versus normalized delamination length of

Carbon fiber/Epoxy [𝟗𝟎 𝟎⁄ ]𝐬 laminates

19,8

20

20,2

20,4

20,6

0 0,2 0,4 0,6 0,8 1 1,2

C 11(

Nm

)

ld/t90

CF/EP [90/0]𝑠 ρ=0.1667

C11 M1

C11 M2

C11 M3

C11 M4

18,6

18,8

19

19,2

19,4

19,6

19,8

20

0 0,2 0,4 0,6 0,8 1 1,2

C 11(

Nm

)

ld/t90

CF/EP [90/0]𝑠 ρ=0.3333

C11 M1

C11 M2

C11 M3

C11 M4

18

18,2

18,4

18,6

18,8

19

19,2

0 0,2 0,4 0,6 0,8 1 1,2

C 11(

Nm

)

ld/t90

CF/EP [90/0]𝑠 ρ=0.64

C11 M1

C11 M2

C11 M3

C11 M4

27

Figure 19. Reduction of the 90°-layer bending stiffness versus normalized delamination length of

Carbon fiber/Epoxy [𝟗𝟎 𝟎𝟐⁄ ]𝐬laminates

129

130

131

132

0 0,2 0,4 0,6 0,8 1 1,2

C 11(

Nm

)

ld/t90

CF/EP [90/02]𝑠 ρ=0.64

C11 M1

C11 M2

C11 M3

C11 M4

130

131

132

133

134

0 0,2 0,4 0,6 0,8 1 1,2

C 11(

Nm

)

ld/t90

CF/EP [90/02]𝑠 ρ=0.3333

C11 M1

C11 M2

C11 M3

C11 M4

133

134

135

136

0 0,2 0,4 0,6 0,8 1 1,2

C 11(

Nm

)

ld/t90

CF/EP [90/02]𝑠 ρ=0.1667

C11 M1

C11 M2

C11 M3

C11 M4

28

Figure 20. Reduction of the 90°-layer bending stiffness versus normalized delamination length of Glass fiber/Epoxy [𝟗𝟎 𝟎⁄ ]𝐬 laminates

14

15

16

17

18

0 0,2 0,4 0,6 0,8 1 1,2

C 11(

Nm

)

ld/t90

GF/EP [90/0]𝑠 ρ=0.3333

C11 M1

C11 M2

C11 M3

C11 M4

13

14

15

16

0 0,2 0,4 0,6 0,8 1 1,2

C 11(

Nm

)

ld/t90

GF/EP [90/0]𝑠 ρ=0.64

C11 M1

C11 M2

C11 M3

C11 M4

17,5

18

18,5

19

19,5

20

20,5

0 0,2 0,4 0,6 0,8 1 1,2

C 11(

Nm

)

ld/t90

GF/EP [90/0]𝑠 ρ=0.1667

C11 M1

C11 M2

C11 M3

C11 M4

29

Figure 21. Reduction of the 90°-layer bending stiffness versus normalized delamination length of

Glass fiber/Epoxy [𝟗𝟎 𝟎𝟐⁄ ]𝐬 laminates

74

76

78

80

82

84

0 0,2 0,4 0,6 0,8 1 1,2

C 11(

Nm

)

ld/t90

GF/EP [90/02]𝑠 ρ=0.3333

C11 M1

C11 M2

C11 M3

C11 M4

72

74

76

78

80

0 0,2 0,4 0,6 0,8 1 1,2

C 11(

Nm

)

ld/t90

GF/EP [90/02]𝑠 ρ=0.64

C11 M1

C11 M2

C11 M3

C11 M4

82

84

86

88

90

0 0,2 0,4 0,6 0,8 1 1,2

C 11(

Nm

)

ld/t90

GF/EP [90/02]𝑠 ρ=0.1667

C11 M1

C11 M2

C11 M3

C11 M4

30

2.3. Comparison to 3-D FE calculation of bending stiffness for 4-point bending test

We have found that the 4 methods used come to the almost the same bending

stiffness reduction and that they show high accuracy (The results corresponding to this

models are denoted by II in Fig 22). FE direct calculation (III) were performed for the

same laminates but subjected this time to 4-bending test. Ply-discount model results

(I) are also presented for comparison

The accuracy of the effective stiffness approach and the Gob LOC approach is

validated comparing with FEM and they show a good prediction especially for low crack

density.

Figure 22. Bending stiffness of [𝟗𝟎 𝟎⁄ ]𝐬CF/EP cross-ply laminates according to different models

31

Conclusions

In the context of understanding of the influence of delamination on the

performance of the laminates, an attempt has been made to analyse the influence of local

delamination size on in-plane stiffness and to predict bending stiffness for damaged

cross ply laminates. The most important conclusions are the following:

The variation of the normalized average crack opening

displacement is linear with delamination length and the assumption of its

linearity as a function of ld/t90 shows a good agreement with FEM and GLOB-LOC

approach

The bending stiffness reduction with increasing crack density is

quasi- linear.

The results found in the tensile calculation could be used to

estimate the bending stiffness reduction: the stiffness reduction in the presence

of delamination is underestimated especially at high crack density.

Future work

Some recommendations for the future work are suggested:

• Numerical work

A model could develop expressions describing the dependency of crack opening

displacement for in-plane loading and the bending stiffness reduction on the

delamination length.

• Experimental work:

The bending stiffness for 4-point bending test could be obtained experimentally and

then compared with the results already found and with expressions (once they are

established).

32

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