Fiber Diameter Optimization for Composite

8/13/2019 Fiber Diameter Optimization for Composite

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Fiber Diameter Optimization for FMLs

Bereket Berhane Tadese

Date 00-00-2013

Technical University of Delft

Faculty of Aerospace Engineering


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Fiber diameter optimization for Fiber Metal Laminates

Chapter 1Introduction to thesis work

1.1introducon 1.2 background 1.3 objecves 1.4 thesis outline

1.1 Introducon

This chapter provides an introduction regarding to a thesis research on iber diameteroptimization for Fiber Metal Laminates. It begins with a brief introduction to Fiber Metal Laminates(FMLs) history and background and the type of FML used in this thesis in section 1.2. Followed by theobjective and questions of the thesis discussed in section 1.3. Finally, Section 1.4 contains informationabout the structure of the thesis report.

1.2 BackgroundSince the irst light of the wright brothers in 1903 the aerospace industry has grown at very

fast pace, around 1960s it became major economy in the world. To achieve this milestone the aircraft

industry has gone through many obstacle and challenges in the span of the period between 1920 until1960s.

After the Second World War, research and development on aircraft structures started. The aimof the R&D was to advance the knowhow about the aircraft materials and structures; behavior underdifferent loading and environmental conditions as well as, the manufacturing processes and cost, howto reduce structural weight and how to reduce the maintenance time and cost.

But, prior to the application of advanced materials such as aluminum alloys and hybridmaterials, steel and wood were mainly used as construction materials in the aircraft industry [1]. Hybridmaterials are made from two or more materials bonded or consolidated in oven or autoclave havingdifferent mechanical and physical properties. Fiber reinforced plastic (composites) and FMLs are


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examples of hybrid materials. However, initially aluminum alloys replaced steel and wood becausealuminum alloys have lower speciic weight compared to steel, simpler production process and good -resistance against environmental effects [1]. The hybrid materials took few years to be used as aircraftconstruction materials.

Fiber reinforced plastics were also promising advanced materials with high strength, stiffnessand low speciic weight. Composite materials were mainly used as secondary aircraft structures butcomposite materials could not be used as primary structures until recent times. Because, compositematerials have high production cost, limited stress ranges of application due to lack of plasticity andinancial penalties incurred due to equipment downtime [2].

The need to develop advanced materials has it owns drive from different aspects of a designcriteria for aircraft. The design criteria are; reduction of fuel consumption, fail-safe construction,damage tolerant design, cheap manufacturing process and easy to maintain and repair.

Even though a fuel cost expense and manufacture cost of aircraft are vital concerns, the airframe

materials used to build the aircraft structures posed the biggest challenge to the industry. A tragichappening that took place in 1954 with the comet jet has revealed that metal fatigue was critical. Fatigueis an important factor in the design of aircraft [3]. Fatigue is a cyclic loading of structures for long periodof time below it failure stress and small cracks initiate and grow to a critical size. Then the material canno longer bear the applied load with the reduced cross sectional area and the material starts to fracturedrastically leading to material failure. The failure happens without any warning. For these reasons,Fatigue property of materials is an important design factor and needs a considerable amount ofattention when designing aircraft, automotive or machine components [4].

For which reason one can not only think the reduction of production process and fuel cost,without considering the major important factors such as the safety issues and maintenance. It ispractically impossible to design an aircraft component that will last for the given life cycle without any

damage. Since damages are inevitable on the structural component the so called Damage tolerancedesign concept was introduced in the aircraft industry [4].The concept is basically, cracks should notgrow to fast in order to detect the damage during regular inspection of the aircraft, the designer canintroduce structural element to obtain crack growth retardation or crack arrest.

A durability issue is also critical factor, in particular with damages like Fatigue, corrosion andenvironmental effect on any structural component of an airframe. Especially with the so called agingaircraft durability is a major issue [3]. As a result a regular maintenance is an obligation to guaranteethe safety of the leets. It brings extra maintenance cost to the airliners. Even with strict maintenanceroutine things can go wrong. One sad accident was the Aloha Airlines Boeing 737 in 1988, which haslost its upper fuselage cover due to accumulated fatigue cracks at the fuselage lap joint. Yet the cracks

didnt not reach the critical stage but interaction between the cracks may lead to total failure. It issometimes referred as multiple site damage [3].Again regular maintenance was not an answer tofatigue issues related to the multiple site damage. Another critical damage was corrosion damage.Corrosion damage that occurs in aircraft structures particularly at rivet holes a region where it is fatiguesensitive, is due to moisture ingress and fretting (sheet of materials sliding over each other). Thesecorrosion damages result in the initiate of fatigue crack. To inspect the damages it cost the airliners ahuge amount of money moreover it is hard to detect these corrosion damages. Undoubtedly to cut thecost, structural weight and to design safe durable structures, a damage tolerant material is required.

Broadly speaking the concept the damage tolerance design allows a designer to introducestructural elements to slowdown the damage growth (as in fatigue crack growth) to be detected duringthe regular interval inspection before reaching to critical stage that would cause failure to the

components of the airframes structures. A different philosophy is to develop a material which has a high


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crack growth resistance as an inherent material property and light weight without extra structuralelement. This philosophy was the fundamental for the development of (FML) Fiber Metal Laminates [4].However before the development of the Fiber Metal Laminates, almost two decades before in the 1950sa Delft university graduate (rob schliekekelmann) did some practical work at English aircraft

manufacture De Havilland a company that started to bond metal parts together. He brought the idea toFokker where he improved the process by treating the aluminum layer and by introducing the autoclave[5]. Later on, it was discovered at NLR by a team of professor Schijve who carried out a fatigue test onF-27 center wing, bonded metals appeared to have good fatigue properties.

The Delft University of Technology was involved in the development and research of bondedmetals almost from the start. In 1978 researchers by team of Prof L.B Vogelsang in Delft University ofTechnology at materials lab were experimenting by introducing iber to the bonded metals [5]. The iberintroduced at that time was Carbon iber unidirectional and Aramids in a weave form (Arall). This gaverise to a new material class as a combination of: iber, adhesive resin (polymer) and metallic alloys inshort Fiber Metal Laminates (FML). This new hybrid material showed even better fatigue property,impact properties and corrosion resistance than bonded metals. With its unique features iber bridging

and delamination mechanism [5]. More explanation on these mechanisms chapter 2.Understandingiber bridging and the delamination phenomena was necessary, Marissen [5] did research on iberbridging and the delamination mechanism which has led to the optimization of FMLs. As a result theinvention of the FMLs was solution for the damage tolerant design requirement and light weightstructural materials.

In the coming years after 1980 important milestones were achieved at Delft University oftechnology with patenting and application of FMLs.In1982/83 Arall (Aramid Reinforced AluminumLaminates) was commercialized with different variance depending on the production process andapplication [6]. In 1987 Glare (GLAss REinforced) was patented and in 1991 commercialized.

In this research assignment Glare is taken as the material to study.

Constituents of Glare:-

The Metallic layer sheet is Aluminum 2024-T3 thickness 0.4 mm The iber reinforced plastic made up of :Unidirectional S2-glass iber embedded in Cytec FM

94(matrix) epoxy resin, delivered as pre-impregnated (prepreg),

Figure 1.1 typical Glare layup [3]


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Figure 1.2 Cross section of Glare [3]The Department of Structural Integrity & Composites at the Delft University of Technology

Faculty of Aerospace Engineering is involved in research and innovation project. The research worksbeing conducted at the Department of Structural Integrity & Composites are on FMLs, metals andpolymer composite. A vast number of research work has been done on FMLs to study the mechanicalproperties, physical properties and production process. Hence this thesis assignment is considered as acontribution to the ongoing research work.

The thesis assignment is about iber diameter optimization for Fiber Metal Laminates. Theproject assignment is commissioned by the Department of Structural Integrity & Composites in a joint

collaboration with GTM-Advanced Structures. The project is carried out at TU Delft the Faculty ofAerospace Engineering. The stakeholders GTM-Advanced Structures, glass iber manufactures and theTU Delft Department of Structural Integrity & Composites are interested in inal outcome of the thesisindings and report that could optimize the FMLs properties necessary for certain applications.

To apply FMLs on aircraft structures the material has to be qualiied and tested to be safeaccording to the aviation regulation. To ensure FMLs applicability a profound property characterizationmethod is necessary. In This thesis the main concern is the characterization mechanical and fatigueproperties of FMLs in relation with the iber diameter. The characterization can be in the terms oftheoretical predictions, numerical analysis and experimental analysis. But in this thesis work theoreticalprediction and experimental analysis are only considered.

1.3ObjecveThe main objective of this thesis is to study the effect of iber diameter and its volume content

on the mechanical and fatigue properties of FMLs Glare (GLAss Reinforced aluminum). The objective isto study the effect of iber diameter and the iber volume fraction (FVF) of the pre-impregnated iberlayers on the mechanical and fatigue properties of the Glare. The research work is divided in to twoparts. The theoretical prediction of Glare properties by taking the FVF and iber diameter of prepregparameters as input. Secondly, the validation of the theoretical analysis with an experimentation.

For the proposed thesis assignment some critical questions could be asked. These thesis wouldprovide an understanding to the following questions:-


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What is the effect of changing iber diameter of a Glare on its mechanical property? What is the effect of changing the iber diameter on Glare fatigue property? Deriving a theoretical equation between prepreg parameters; FVF, iber diameter, iber wetted

area, iber spacing and thickness of the prepreg?

How to obtain the GLARE mechanical properties and fatigue behaviors using the theoreticalequation between prepreg parameters; FVF, iber diameter, iber wetted area and thickness of

the prepreg?

How to improve the mechanical and Fatigue properties of Glare? How to correlate the theoretical results to the Experiment data work for FMLs?

The thesis assignment is limited to the optimization of the iber diameter of the Glare. Due to the

complex nature of Glare not all the properties and other parameters will be covered in this thesis report,but some important characteristics shall be considered.

The selected characterizing properties for the Glare material are shown below.

The selected mechanical properties are:-

Tensile yield strength and tensile ultimate strength Compressive strength Blunt notch strength Tensile and shear elastic modulus

The fatigue case:-

Through the thickness cracks with Constant amplitude to measure crack growth rate1.4 Thesis outlineThe thesis is structured as follows

Part 1 literature review Chapter1: A general introduction of the topic and thesis work in detail.Chapter 2: Literature review will be given on FMLs Glare, Composite (Fiber layer) and aluminum. Thereview contains the general information about FMLs (Glare) - production method, static and fatigueproperties-application of Glare material in aerospace industry and the method of Analysis to predictproperties of Glare.

Chapter 3: The method to determine Fiber Volume Fraction (FVF) in composite structures as functionof iber diameter which includes Theoretical derivation and analysis. Current research indings byothers concerning the effect of iber diameter and iber volume fraction on composite materialsproperties.


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Part 2 analysis and pr operties prediction using FVFChapter 4: It comprises the derivation of FVF of the Glass iber prepreg using the equation andinformation formulated in the literature review. The variable for the equation cloud be iber diameter,nominal thickness, and the number of ibers in given cross section.

Chapter 5: With the data and equations obtained from Chapter 4, the Glare static and fatigue propertiesare predicted for a selected number of iber radius, FVF and wetted ibers in a given thickness.

Part 3 experimentation and data analysisChapter 6: The methods and techniques of specimen preparation; for image analysis, static propertiestest and fatigue property test are explained and the setup of the experiment is outlined properly. Acorrelation is established between the iber diameter and the specimen FVF using image analysis resultsand FVF equations.

Chapter 7: It includes experimental Results and data analysis. Calculating the averages, the standarddeviation within the same family samples etc.

Part 4 conclusion and discussionChapter 8: discussion of experimental work and analytical work results and comparisons.Chapter 9: conclusion.


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Chapter 2Fiber Metal Laminates

2.1 Introducon 2.2Glare orientaon and its constuent 2.3 The properes of Glare constuents

2.3.1 luminum alloy 2024-T3 2.3.2 S2-Glass Fiber 2.3.3 FM 94 Resin

2.4 Stac and Fague properes of Glare 2.4.1 Composite Materials 2.4.2 The Metal Volume Fracon method to predict Glare properes 2.4.3 The Fague properes of FMLs

2.1 Introducon

Fiber metal Laminates (FMLs) are a hybrid materials, made from thin metal sheets bondedtogether by an adhesive reinforced with ibers. FMLs are used as an aircraft structural material.

FML is laminated material of thin layer of metal sheet and unidirectional or weave iber layersembedded in polymeric adhesive system cured in autoclave.

The basic constituents of the FML are as follows: - [7]

Metal alloys: Aluminum 7075-T6, Aluminum 2024-T3, Aluminum 7475-T761, Titanium Ti-15-3 and AISI301 Stainless steel

Fiber: Aramid, S2-Glass, T300-Carbon, IM7-Carbon and Strail C-EP I-150 dhesive polymers: BSL-312-UL, AF163-2, FM94, FM906, epoxy and polyimide

The combination of these constituent yields a typical FML with trademark or commercial names.To mention some well-known FMLs with the Basic Constituents


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rallis made from Aluminum 7075-T6, Aramid ibers with Adhesive BSL-312-UL or AF163-2. Carall Aluminum 2024-T3 and T300-carbon iber with epoxy resin. GlareAluminum 2024-T3 and S2-Glass iber embedded in FM94 Adhesive polymer. TiGra metal sheet made from Titanium Ti-15-3 with IM7-Carbon iber embedded in polyimide

resin.The application of these FMLs are versatile and Depending on the constituent behaviors. As an

example, Glare was optimized for aircraft fuselage skin, because the irst class of FML (Arall) could notbe applied on aircraft fuselage due to its low strength in cross-ply orientation [5].

This chapter is divided in to three section. The Glare orientation and layup is discussed in section2.2. General information is provided about Glare constituents in section 2.3. The static and fatigueproperties and method of prediction of Glare is explained in section 2.4.

2.2 Glare orientaon and its constuent

Glare has six different standard grades. The ibers are unidirectional S2-glass ibers embeddedwith FM94 adhesive (lamina), with a thickness of 0.127 mm prepreg and a iber volume fraction of 59%.The term lamina will be used in this thesis as a reference to the (composite material prepreg) or iberlayer. The metal alloy is aluminum alloy 2024-T3 .The iber layer is laid-up in different orientationbetween the aluminum alloy sheets resulting in the different standard Glare grades [5]. When deiningthe orientation of the iber layer prepregs in the Glare, as a rule the 0 (L) or the rolling direction of thealuminum alloy is used as reference direction. Therefore the iber layer 0 direction, must be laid parallelor perpendicular to the L or 0 direction of the aluminum alloy. Additionally, the 0 (L) or the rollingdirection of the aluminum alloy the main loading axis when conducting on-axis mechanical and fatigueproperties tests or as a reference angle for off-axis situation.

Figure 2.1 orientations of luminum and iber [8]The LT transverse direction of the aluminum alloy is 90, as seen onFigure 2.1 the orientations

of the iber layer and aluminum alloy in Cartesian co-ordinate system.

The Glare grades have laminate coding system to comprehensively deine the laminate

formation.


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For example:-

Glare 4B-4/3-0.4Deined as

A Glare laminate with iber orientation according to the Glare 4B as deined inTable 2-1

4 layers of aluminum 3 [90/0/90]; 9 single layers of iber prepreg Each aluminum layer is 0.4 mm thick

For Glare 1, Glare 2 Glare 4, and Glare5 the prepregs in each iber layers are stackedsymmetrically. In the case of Glare 3 and Glare 6 the prepregs are laid in cross-ply manner. Therefore itneeds further explanation. For Glare 3 with even number of cross-ply prepreg the iber layer nearest tothe outer aluminum layer in the laminate the 0 direction of the iber layer is laid in the rolling directionof the aluminum alloy (0). Whereas for uneven number of cross-ply prepreg the irst prepreg in the

center the iber layer 0 is laid-up during the production in the 0 direction of the aluminum alloy. Forthe Glare6 the conditions are comparable to the deinition of Glare3.InTable 2-1 the different gradesare shown with the orientations associated to the application beneits [5].

Table 2-1 Glare Grades with the naming [5]Glare grade sub Metal sheet thickness[mm] and alloys Prepreg orientation

* ineach iber layers** Main beneicialcharacteristicsGlare 1 - [0.3-0.4] 7475-T761 0/0 Fatigue, strength,

yield stress

Glare 2

Glare 2A [0.2-0.5] 2024-T3 0/0 Fatigue, strength

Glare 2B [0.2-0.5] 2024-T3 90/90 Fatigue, strength

Glare3 - [0.2-0.5] 2024-T3 0/90 Fatigue, strength

Glare 4

Glare 4A [0.2-0.5] 2024-T3 0/90/0 Fatigue, strength in00direction

Glare 4B [0.2-0.5] 2024-T3 90/0/90 Fatigue, strength in900direction

Glare 5 - [0.2-0.5] 2024-T3 0/90/90/0 impact

Glare 6

Glare 6A [0.2-0.5] 2024-T3 +45/-45 Shear, off-axisproperties

Glare 6B [0.2-0.5] 2024-T3 -45/+45 Shear, off-axisproperties

*All aluminum rolling directions in standard laminates are in the same orientation; the rolling direction is deined as 0 0the transverse rollingdirection is 90.0

** The number of orientations in this column is equal to the number of prepregs (each nominally 0.127mm thick) in each iber layer.


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The Glare grade used in this thesis is Glare 3-3/2-0.4 with 3 Aluminum layer sheets and 4 S2-Glass iber prepregs (lamina). SeeFigure 2.2 the graphical illustration of lay-up and orientation.inFigure2.2.The 90 direction indicated with S2-Glass iber prepreg refers to the orthogonal position of the 0 ofthe iber prepreg with respect to the rolling direction of aluminum alloy.

Figure 2.2 Glare 3-3/2-0.4 [8]2.3 The properes of Glare constuents

FML Glare is made of three main constituents. The constituents are aluminum alloy 2024-T3, S-glass iber and FM94 adhesive. A general information of these constituents is important for the input inthe theoretical prediction of the mechanical properties of Glare.in this section a mechanical and physicalproperties of the constituents is given.

2.3.1 Aluminum alloy 2024-T3

The aluminum alloy used to make Glare, as it was mention in the previous sections is 2024-T3alloy. It is proven by test that 2024-T3 alloy has a good fatigue property and moderate corrosionresistances compared to other aluminum alloys like 7075, 2024-T8 [4]. Aluminum alloys have isotropicproperty in general. It is beyond the scope of this thesis work to mention the processes involved and theapplications of these aluminum alloys to a Glare. Therefore the focus is on the 2024-T3 aluminum alloyby mentioning its: composition, physical properties and mechanical properties which are going to beused as material property input data for the calculation of Glare mechanical behavior. Brief explanationis also given concerning the surface pretreatment for bonding. The composition, physical and

mechanical properties of 2024-T3 is given onTable 2-2.

In order to obtain a good bonding strength between the composite and adhesive ilm to thealuminum layer, the aluminum surface should be treated before bonding it to the composite or adhesiveilm. Experience has revealed surface treatment prior to the bonding is critical for the long term servicedurability and enhancing the bonding process. A selected surface treatment tends to modify thesubstrate surface of aluminum layer by promoting wettability with primer and adhesive. The surfacetreatment for metals can be achieved with different techniques. The techniques used to treat thealuminum alloy are: chemical etching and later on electrochemical anodizing to improve the corrosioncharacter. The solution used to etch during the chemical treatment are chromic- sulfuric (CAE) acid andsulfo-ferric acid (P2) [9].


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Table 2-2 composition, mechanical and physical properties of aluminum alloy 2024-T3 [10] Composition units

Al (Aluminum) 94 %

Cu (Copper) 4.4 %

M Ma nesium 1.5 %

Mn Man anese 0.6 %

Mechanical properties Min MaxBulk Modulus 71 74.64 GPa

Compressive Strength 234 352 MPa

Elon ation 8 15 %

Elastic Limit 248 372 MPa

Endurance Limit 118 168 MPa

Fracture Toughness 37 41 MPa.m^1/2

Hardness - Vickers 108 148 HV

Modulus of Ru ture 248 372 MPa

Poisson's Ratio 0.33 0.3435

Shear Modulus 28 29.44 GPa

Tensile Strength 359 510 MPa

Youn 's Modulus 72 75.69 GPa

Physical propertiesDensity 2754 2782 kg/m^3

Maximum Service Temperature 110 170 C

Minimum Service Tem erature -273 C

S eciic Heat 942 980.4 k .K

Thermal Expansion 23.22 24.41 strain/C

2.3.2 S2-Glass Fiber

The S2-Glass iber reinforcement system in Glare is responsible for the strength, stiffness andfatigue properties. The glass iber is continuous long ibrous material with superior strength andstiffness in the axial direction (longitudinal axis). Glass iber is solid bar with cylindrical shape having a

circular cross section form. The smaller the diameter the greater the strength of the iber [2]. Thecurrent Glass iber used in Glare has almost 10 micron meters in diameter [5]. The composition of theS2-galss iber is 64%SiO2-24%Al2O3 -10%MgO. [11]. The mechanical and physical properties of S-glassiber is given onTable 2-32.3.3 FM 94 Resin

FM 94 matrix belongs to a thermosets polymer. It is a family of epoxy with outstandingproperties. FM 94 purpose is to bind the iber glass, keeps the iber in the proper orientation and tobond with the aluminum sheets. FM 94 is also responsible for the transfer of load to and between ibers,provides all the interlaminar shear strength of Glare and resistant to crack propagation and damage.However, FM94 has a service temperature up to 200C, which limits its use in high temperatureenvironment [2, 5]. The mechanical and physical properties of FM-94 is given below inTable 2-4


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Table 2-3 S2-Glass iber properties [11]Mechanical Property Min Max Units

Density 2485 2495 Kg/m3

Bulk Modulus 51 55 GPa



Endurance Limit 4050 4410 MPa

Fracture Toughness 0.5 1 MPa.m1/2

Modulus of Rupture 4500 4900 MPa


Shear Modulus 35 39 GPa


Strain to failure 5 10 %

Young's Modulus 88 93 GPa

Glass Temperature 920 950 K

Table 2-4 mechanical and physical properties of FM-94 [10]Mechanical properties Min Max Units

Bulk Modulus 4.5 4.8 GPa


Elongation 3 10 %


Endurance Limit 24.8 27 MPa

Fracture Toughness 0.4 0.7 MPa.m^1/2

Shear tensile 23 55 MPa


Shear Modulus 0.9 1.1 GPa


Young's Modulus 1.7 2.9 GPa

Density 1180 1240 kg/m^3

Maximum Service Temperature 160 200 C

Minimum Service Temperature -43 7 C


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2.4.1 Composite Materials

The composite material used in Glare is S2-glass iber embedded in FM94 epoxy. It is in the formof prepreg as a single layer which is called lamina. The ibers in composite materials can be woven,unidirectional, bidirectional or randomly dispersed. The composite material used in Glare isunidirectional lamina. A unidirectional lamina is where all the ibers are aligned in a single direction.The ibers in the unidirectional lamina are orientated along the longitudinal axis of the ibers, which is0 with respect to the axis 2(x-axis) as depicted inFigure 2.3 .

Figure 2.3 unidirectional lamina orientation [12]The strength and stiffness of unidirectional lamina composite is due to the stiff and strong

properties iber. However in a practical sense the strength and stiffness of the lamina depends on theorientation of the loading direction with respect to the iber direction and the amount of iber contentin the direction of the iber. The lamina properties are basically a function of the iber content (FVF) inthe direction of the iber. Furthermore, lamina properties change as the loading angle deviate from thelongitudinal axis of iber direction. Due to this directional sensitivity of unidirectional lamina has anorthotropic property. Orthotropic materials have different material properties with a differentdirections, in which the directions are orthogonal with each other [2, 13].

To obtain the orthotropic material properties of a unidirectional lamina a homogenizationtechnique shall be used. A homogenization techniques are methods to predict the elastic properties ofcomposite in terms of the elastic properties of constituents. Homogenization models are based onmodeling of the microstructure. These models are called micromechanics models, and the methods usedto acquire values of the laminas material properties are called micromechanics techniques. Themicrostructure model is used to modeling the heterogeneous composite constituent, in to homogenizedcomposite lamina model in a computational analysis of micromechanics [14].

As it was mentioned in the previous paragraph, micromechanics techniques are used topredicted the mechanical properties of lamina. The micromechanics techniques can be classiied intoempirical, semi empirical, analytical and numerical methods. But in this thesis work the analyticalapproach is used. There are several analytical homogenization micromechanics techniques. These areReuss model, Voigt Model, periodic Microstructure Model and Transversely isotropic Averaging. Todiscuss all these method is complex and beyond the scope of this thesis work. Hence, the most widelyused techniques such as Reuss model and Voigt model shall be discussed [14].


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Reuss Model

The Reuss model also known as Rule of Mixture (ROM) assumes there is equal strain in the iber,matrix and composite. In other words the iber, matrix and the composite deform or elongate togetherequally. This method is used to calculate the lamina materials property along the longitudinal iber axis

direction (0) of the lamina. The equation for ROM to ind the laminate property is given in equation(2.1) [14].

(1 )MP MP MPL F FVF M FVF (2.1)

Where

LMP is lamina mechanical property FMPis Fiber mechanical property

MMPis matrix mechanical property

FVF Fiber Volume FractionVoigt M odel

The Voigt model assumes that the stress in the iber, matrix and composite are all equal. It issometimes referred as the inverse of ROM. This method is used to compute the lamina property in thetransversal direction (90) to the longitudinal iber axis direction (0) of the lamina. The equation forthe inverse of ROM to ind the laminate property is shown with equation (2.2) [14].

1

1( ) ( )

MP

MP MP

LFVF FVF

F M

(2.2)

Where

LMP is lamina mechanical property FMPis Fiber mechanical property MMPis matrix mechanical property FVF Fiber Volume Fraction

The output of homogenized model (micromechanics model analysis) prediction lie betweenlower and upper bounds. These bounds are function of the FVF and physical properties of theconstituents. Equation (2.1)(2.2) demonstrate the inluence of FVF on the mechanical properties oflamina. Equation (2.1) (2.2) are used to predict basic engineering constants.

In order to use this equation in the analysis lamina property some important assumption areconsidered these are: - [13]

Perfect bonding exist between iber and matrix Fibers are parallel, and uniformly distributed Both iber and matrix are isotropic and obey Hookes law

The applied loads are parallel of orthogonal to the iber direction


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2.4.2 The Metal Volume Fraction method to predict Glare properties

The glare type consider in this thesis work is Glare3 3/2-0.4. It shows both the constituentsproperties. It exhibits the elastic plastic behaviors of aluminum and elastic (orthotropic property) ofthe iber laminate. The mechanical property of Glare combines both iber and metal parts in which itbecomes a function of; iber orientation, loading direction, aluminum plasticity and residual stress aftercuring and a difference in thermal expansion of the constituents. Due to these factors on whichmechanical property depends, it is complex to explain the mechanical property of Glare using analyticaltools [15].

However, a simple powerful analytical tool was developed to predict the Glare materialproperties. The method is very similar to the ROM approach for the iber lamina. The method is calledMetal Volume Fraction (MVF). Before presenting the analytical formula of MVF method for determiningthe Glare property, at irst it will be shown how to determine the MVF values of any FMLs.

The MVF is expressed as a ratio between the sums of the individual aluminum sheet thickness to thetotal thickness of the FML see equation (2.3) [5]

1

P

al

lam

t

MVFt

(2.3)

Where:

t al is thickness of each separate aluminum layer tlamis thickness of the total laminate(FML)

p the number of aluminum layers

When the MVF ratio is 1 then it represents pure monolithic aluminum layer property and whenit is 0 it represents the pure prepreg lamina property. The MVF method assumes that there is a linearrelationship between the material properties at MVF is 1 metal layer contribution and the average oftested FML data. Then the line is extrapolated to MVF is 0 to a iber layer contribution. The analyticalequation to calculate Glare properties having any MVF values and the mechanical properties ofaluminum and iber layer laminate is given with equation(2.4) [5].

(MVF) (1 MVF)MP MP MP

FML MTL LAM (2.4)

Where FMLMP is FML mechanical property LAMMPis Fiber layer laminate mechanical property MTLMPis metal layer mechanical property MVF Fiber Volume Fraction

The MVF method will be used to predict the material properties of Glare in this thesisassignment. The static mechanical properties of Glare that are going to be used to evaluate the iberdiameter effect are; tensile strength, compressive strength, shear strength, blunt notch strength, andelastic and shear modulus. The MVF method equations for each material property will be shown with

unique symbols and term to distinguish from each property.


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The tensil e strength

The tensile strength of Glare can be divided in to two parts in the strain- stress curve. The twoparts are the ultimate strength and the yield strength. The ultimate strength of Glare depends on bothconstitutes behaviors, and failure limit stress and failure strain. On the other hand, the yield property of

Glare is related to the metal layer yield behavior, since the iber laminate does not yield. As a result thepredictions made by MVF method for yield property of Glare is quite accurate. Conversely, theprediction for the ultimate strength are rough estimations, since it assumes simultaneous failure of theconstituents, which in actual case is not correct [16]. The MVF equations for predicting tensile strengthultimate and yield of Glare are given respectively with equation(2.5) (2.6) [5].

, , ,( ) (1 )

GLR ult AL ult LAM ult MVF MVF (2.5)

Where

GLR ,ult is the ultimate tensile strength of Glare AL ,ultis the ultimate tensile strength of aluminum LAM ,ultis the ultimate tensile strength of the glass-iber laminate MVF is the Metal Volume Fraction

, , ,( ) (1 )GLR yld AL yld LAM yld MVF MVF (2.6)

Where

GLR ,yld is the yield tensile strength of Glare AL ,yldis the yield tensile strength of aluminum LAM ,yldis the yield tensile strength of the glass-iber laminate MVF is the Metal Volume Fraction

A stress quantity is used to measure the tensile strength. The unit for the stress is in mega Pascal(MPa). The tensile strength is assumed to be on-axis loading parallel to the iber 0 direction (x-axis) asshown inFigure 2.4 .

Figure 2.4Glare model under axial tension load parallel to the x axis [12]


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Compressive strength

Using the MVF method, it is possible to predict the compressive strength of Glare. The yieldcompressive strength will be considered to evaluate the effect of the iber diameter change on a Glarecompressive strength. The MVF method to predict the compressive strength of Glare is shown in

equation(2.7) [5].

,yld ,yld ,yld( ) (1 )CGLR CAL CLAM MVF MVF (2.7)

Where

CGLR ,yld is the ultimate tensile strength of Glare CAL ,yldis the ultimate tensile strength of aluminum CLAM ,yldis the ultimate tensile strength of the glass-iber laminate MVF is the Metal Volume Fraction

The unit of measure is in mega Pascal (MPa). The compressive strength is assumed to be on-axisloading parallel to the iber 0 direction (x-axis) opposite to the tension load direction as shown inFigure 2.5 .

Figure 2.5 Glare model under axial compression load parallel to the x-axis [12]Elastic modulus

The strain-stress curve of Glare under tension and compression is a combination of the iber-layer and metal sheet chart, which depends on the iber orientation and loading direction, and the curvehave semi bilinear shape [15] [5].

The elastic modulus of Glare is related to the elastic modulus of its constituents. The loadingdirection is very important, since it affects the iber layer stiffness. Elastic modulus is calculated alongthe iber direction (longitudinal axis). The elastic modulus of Glare could be used to investigate the iberdiameter effect on Glare mechanical property. The method of MVF will be used to predict the elasticmodulus of Glare as shown with equation(2.8) [5].

( ) (1 ) EGLR AL LAM

E E MVF MVF (2.8)


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Where

EGLR is the elastic modulus Glare EAL,is the elastic modulus of aluminum ELAM is the elastic modulus of the glass-iber laminate MVF is the Metal Volume Fraction

The blunt notch strength

The blunt notch strength is deined as the strength of a structure having a hole and it is importantdesign parameter because aircraft structures contain many holes at joints. Moreover, a notch causes astress concentration which in general results in a strength reduction of the structure under tensionloading case. The blunt notch strength can be used to demonstrate the effect of iber diameter effect onGlare mechanical property. The equation for blunt notch strength of Glare is shown with equation (2.9)[16] [5]. The unit of measure is in mega Pascal (MPa).

( ) (1 ) BGLR AL LAM

B B MVF MVF (2.9)

Where

BGLR is the blunt notch strength of Glare BAL is the blunt notch strength of aluminum BLAM is the blunt notch strength of the glass-iber laminate MVF is the Metal Volume Fraction

The term BLAM the blunt notch strength of lamina is computed using the Whitney and Nuismer(W-N) method. Whitney and Nuismer developed two criteria to account for the effect of hole on thetensile strength of composite laminates. The two criteria are called Average Stress Criterion (ASC) andthe Point Stress Criterion (PSC) respectively. In this thesis for the analysis of the iber diameter effecton the blunt notch strength of Glare the method of ASC will be used to predict the notch effect on thelamina.

The ASC require an expression for the stress distribution around the hole. For an ininiteorthotropic iber layer laminate having a circular hole with uniform far ield stress applied parallelto the x-axis then the normal stress x along the y-axis in the remaining ligament as depicted inFigure

2.6 and can be estimated by equation(2.10) [17].


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Figure 2.6 ininitely wide orthotropic plate with circular hole with Radius R loaded axial in y direction [12]

2 4 6 8

0, y 2 3 (K 3) 5 72

x T

R R R R

y y y y

(2.10)

Where:

R is the radius of the hole Y is the distance from edge of hole KTis orthotropic stress concentration factor for a plate

The stress concentration factor for orthotropic laminate plate is shown with equation(2.11) [17]in terms of laminate properties.

2 2

21

1 12

1 2T

E EK

E G

(2.11)

Where:

E1is the longitudinal (x-axis) elastic modulus E2is the transverse(y-axis) elastic modulus V21is in-plane Poisson ratio G12is in-plane shear modulus


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The ASC assumes that failure occurs in a notched laminate when the average normal stress, at acharacteristic distance (a0) ahead of the hole, reaches the unnotched strength(0). This phenomenacriterion is shown with diagram inFigure 2.7 and mathematically with equation(2.12) [17].

Figure 2.7 The SC area with the material parameter unnotched strength (0) & characteristic distance (a0) [12]0

0

0

1(0, )

y R a

xy R

xa

y d

(2.12)

Where:

0a is the ASC characteristic distance

0is the unnotched strength x(0,y) is the normal stress along the line through the center of the hole and perpendicular to

the loading direction (y-axis as shown inFigure 2.7)

The characteristic distance (a0) is determined by curve itting with strength Data from test oftwo or more hole sizes. The typical value for the a0 is 3.8 mm for a wide range of hole sizes and cross plyand quasi-isotropic glass-epoxy and graphite epoxy [18]. The unnotched strength of the laminate (0)is determined using the ROM method for composite lamina CLT.

A mathematical equation is obtained for predicting the notched strength of laminate plate by

substituting equation (2.10) in to equation (2.12) and performing the integration. The outcome of theintegration is a ratio of notched strength to the unnotched strength representing the effect of the bluntnotch size on the strength as illustrated by equation(2.13).

(2.13)

Where:

=R(R+0a )

2 4 6 8

0

2(1 )

2 ( 3)( )

N

TK


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2.4.3 The Fatigue properties of FMLs

In general, metallic Structures subjected to cyclic load, a fatigue crack can be initiated on amicroscopically scale, followed by crack grows to a macroscopic size and inally fail in the last cycle ofthe fatigue life. Essentially, fatigue life can be split in to two categories crack initiation period and crackgrowth period.Fatigue in the crack initiation period is a surface phenomenon, which is characterizedby several surface conditions. On the other hand, the crack growth period is characterized by theresistance of material bulk property. The fatigue property prediction method for the crack initiationperiod and crack growth period are fundamental different. For crack initiation the important parameteris the stress concentration factor (KT). Whereas, the stress intensity factor is used for prediction duringcrack growth period. A summary of fatigue life phases of material illustrated withFigure 2.8 [4].

Figure 2.8 The fatigue life phases and the related factor [4] In linear elastic fracture mechanics there are three different crack opening modes seeFigure 2.9

[4]. In many cases a crack grows perpendicular to the tensile stress that tries to open the crack which iscalled mode I and that is going to be consider in this thesis as a crack opening mode to evaluate thecrack growth rate [4].

Figure 2.9 The three modes of crack opening [4]A fatigue crack growth through the entire thickness of the material is referred as through the

thickness crack or simply through crack for most thin walled materials. On the other if the materialconsidered is thick crack grows not full though the entire thickness instead it grows partially throughthe thickness and it is labeled as part through crack. However in this thesis work only through crack isconsidered [4].The factor that affect the crack growth are many. Among all factors some important andrelevant factors to this thesis will be mentioned briely.one of the factors is the stress intensity factorwhich depends on the geometry of the structure and loading condition and it is shown withequation(2.14) [4]

K a

(2.14)


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Where:

is geometry correction factor is far ield stress away from the crack a is the crack length

The second factor is the application of the fatigue load on the structure. There are two types ofloading condition related to fatigue these are:

Constant amplitude loading(CA-loading) Variable Amplitude loading(VA-loading)

The constant amplitude loading condition is considered in this thesis work. The parametersrelated to the loading condition are; amplitude of loading, stress ratio and mean stress. Since the loadingcondition is in a cyclic manner it had maximum and minimum load limits. These maximum and

minimum limits are used to calculate the amplitude of the loading, stress ratio and mean stress.Amplitude loading, stress ratio and mean stress are represented respectively with equations(2.15),(2.16) and (2.17) [4].

max min

2A

(2.15)

max min

2M

(2.16)

min

max

R

(2.17)

Where:

A is a stress amplitude Mis a mean stress R is a stress ratio maxis the maximum stress limit minis the minimum stress limit

The concepts mention above about fatigue of structures apply to both monolithic metallic

materials and Glare material. But, Glares exhibit excellent fatigue and damage properties due to thepresence of fatigue insensitive iber Glass. The fatigue crack growth rate in Glare is considerably lowerthan that monolithic aluminum in equal loading case and thickness. Moreover the crack growth rate forGlare is constant for the majority of fatigue life. A characteristic difference between monolithicaluminum and Glare is the fatigue life phase. For a monolithic aluminum the majority of fatigue life ismainly crack initiation phase followed by fast crack propagating phase causing short life in the crackgrowth phase. On the other hand Glare has a shorter crack initiation phase but a longer life during thecrack growth phase [19]. The mechanism that contribute to the longer life during the crack growthphase and slower and constant crack propagation rate in Glare are Fiber bridging anddelamination oflayers.


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Fiber bri dging and Delamination of layers of Glare

The ibers in Glare transfer a signiicant amount of load over the cracked aluminum layer andrestrain the crack openings and this phenomenon is called iber bridging and the illustration is shownwithFigure 2.10 [4]. The amount of load transferred through the aluminum layer is small. As a result

the stress intensity factor at the crack tip of the aluminum layer of the Glare is smaller when comparedto the monolithic aluminum with equal crack length. Due to the load transfer between the aluminumlayer and iber layer the adhesive is subjected to a cyclic shear stress causing a delamination growthbetween aluminum layers and iber layer (lamina).

Some research work has revealed the amount of the bridging stress in the ibers is related to thecrack opening displacement and the length over which the ibers are elongated (delamination length).Large delamination length results in small bridging stress with small cyclic shear stress at interfaceinducing small delamination growth rate. In fact delamination growth rate and bridging stress are inbalance continuously inluencing each other. Since bridging stress affect stress intensity factor at thecrack tip of the aluminum layer, in turn that determines the crack growth rate. High bridging stressesresult in low stress intensities at the crack tip and thus small crack propagation rate. Therefore fatigue

crack growth in Glare is characterized by the process of crack growth in the aluminum layer and thedelamination growth at the interfaces, which continuously inluence each other [19].

Figure 2.10 load transfer through a crack, restraining crack opening and delamination [19] The crack growth rate of Glare will be used as a one of fatigue property to investigate the iber

diameter effect and the volume content of the iber in Glare. The model is refereed as Central CrackTension (CCT) loaded in tension. The model is provided with a central hole and with two saw cuts as astarter notch. See chapter 4

The analytical equations and model that is going to be used to calculate the crack growth rate ofthe Glare is a computer software package written in MATLAB code by Dr. Ren Alderliesten. Thesoftware plots the crack growth rate against the crack length. The software is called FML GROW V2.0.The main part of the code and a ile containing the parameter related to the thesis work are shown inAppendix A

. Figure 2.11 CCT coniguration


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Chapter 3.The Fiber Volume Fraction nalysis

3.1 Introducon 3.2 Fiber volume basic denion

3.2.1 FVF Based on the density and mass of the constuents 3.2.2 Determinaon of FVF using Image analysis and simulaon 3.2.3 nalycal model of FVF based on the geometry of the microstructure

3.3 FVF Model development analysis 3.3.1 nalysis of FVF of a Glare

3.4 The Shear lag method in relaon with the mechanical properes of the lamina. 3.5 Summary

3.1 IntroduconThe main objective of the thesis research is to investigate the effect of the iber diameter in

relation with a iber volume fraction of a composite lamina on the mechanical behavior of the Glarelaminate as a whole. In order to achieve the goals of the objective it is necessary to understand the micro-structure geometry of composite lamina and, a theoretical model of ibers coniguration in the matrix.Secondly, investigate and research, a micro-mechanics approach or homogenization techniquesmethods that relate the iber diameter and FVF to the mechanical property of lamina thus the Glare. Themain topics of this chapter are to research the existing theoretical microstructure geometry of laminaand a methodologies that relate the iber radius, and other lamina micro structure to the mechanicalproperty of Glare. Moreover, the works done by other researchers concerning the effect of iber radius,FVF and other parameters are also explored.

3.2 Fiber volume fracon basic denion

A composite lamina consisting of iber and matrix should have physical measuring quantity todescribe the content of the constituents. Particularly FVF is a quantity for measuring the content of the


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Andrei A. Guseva, et al [21] have studied the effect of the randomness, shape and the size of theinclusion (iber reinforcement) in an elastic medium (matrix) of composite materials properties. Thetechniques used by Andrei A. Guseva et al, to characterize the microstructure of the unidirectional glassiber/epoxy were experimentally with image analysis and with statistics of microstructures simulated

numerically with Mote Carlo method. The procedure and techniques how to apply the image analysiswith computer simulation is explained briely on the coming paragraphs.

The image analysis was conducted by Andrei A. Guseva, et al on a cross- sectional surface asshown inFigure 3.1taken from composite sample. The instrument used to accomplish the image analysiswas a transputer controlled image analyzer designed to measure the three-dimensional orientation ofibers. The sample is scanned in a raster manner. The image is analyzed with the foot print that eachiber makes with the cross section plan as seen inFigure 3.1.A different set of pixels were assigned tothe image, in order to distinguish the iber and matrix and for a further analysis on the microstructureof the composite. With the image analysis method it was possible to study the diameter variation, FVF,the variety of local packing arrangements across the cross-section and the spacing between the ibers.See igure an example of image analysis image sample.

Figure 3.1The ibers are shown in black and the matrix in grey FVF 54%. [21]A computer software analyzes the image based on the pixels difference and counts the number

of ibers in the give cross sectional area, then presents the average FVF percentage. To validate thepremise on the image analysis done by Andrei A. Guseva, et al a computer model of periodic unit cell ofibers was generated. Mote Carlo method was used to generate periodic computer models of unit cellcovering the randomness of the ibers, distribution of spacing and different diameters. The attempt wasto generate a FVF 54% that resembles the actually physically analyzed image. After several Mote Carlo

simulation runs the computer models had FVF of 54%. The Mote Carlo (MC) method has approximatedthe periodic unit cell arrangement of ibers to the realistic of randomness as seen inFigure 3.2.


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Figure 3.2 the transition computer model of periodic unit cell to a realistic randomness model with Mote Carlomethod [21]The measured data from image analysis and numerically simulated data, distribution of iber

spacing is depicted withFigure 3.3 the y-axis represents the frequency of occurrences and the x-axis the

spacing in micro units. The graphs contain the measured data and the prediction of the simulation madeby MC method. As the graphs illustrate inFigure 3.3,that the measured data using image analysis hasalmost the same value as the MC solutions. As a result Andrei A. Guseva, et al work conirmed that thenumerical computer model can be used to approximate the randomness, the varieties of spacing andpacking arrangement in composites in order to predict the homogenous composite properties. Althoughthis method is acceptable, an ambiguity exist when talking about the MC method which Andrei A.Guseva, et al did not at all show the formula or either the method of MC. More over the approach tendsmore to numerical methods, which is more or less not the main objective of this thesis. In contrast tothe numerical approach method, the experimental technique (image analysis) will be used to determinethe actual FVF, iber diameter, iber spacing and packing arrangement of the FMLs.

Figure 3.3iber spacing frequency in a given cross section of composite [21]


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3.2.3 Model of FVF based on the geometry of the microstructure

The micromechanics approach is used to study the interaction of the iber with the surroundingmatrix. Micro-mechanics looks speciically to the individual constituents in terms of the geometricalshape of ibers, orientation of ibers, spacing between ibers and wetted areas of iber [22]. The thesisobjective is to drive an analytical equation of iber volume fraction relating the iber diameter, nominalthickness of lamina and wetted area. The iber diameter, wetted area of ibers and iber spacing are allmicro structure geometrical characteristics of the composite lamina. The FVF calculation based on themicro-structure geometrical entities of the lamina, is an analytical approach established on theoreticalassumption of micro-structure models of the iber inside a matrix. The orientation of the ibers in thecomposite lamina is important factor when formulating the FVF equation models. In order to drive theequation, lamina ibers orientation and geometry has to be deined.

Figure 3.4lamina with the passing plane - [12]The lamina considered in this thesis is unidirectional prepreg. As it was mentioned in section 2.

The 0 direction is the longitudinal direction or loading axis. If a plane A_A passes through the compositelamina perpendicular to the 0 as illustrated in Figure 3.4.A cross section will appear containing acircular ibers with certain diameter and spacing between the ibers occupied by a matrix. As seen inFigure 3.1the ibers (white circle) are distributed randomly in the prepreg. At the lamina level (meso-analysis) the randomness could be replaced by periodic microstructure model [14]. In the periodicmicrostructure model, the ibers are modeled with a constant circular cross section and regular spacingbetween the ibers seeFigure 3.5.

Figure 3.5a periodic arrangement of ibers with regular spacing [12]


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The periodic microstructure model approach is not only used to simulate the micromechanicsanalysis of composite materials, it is also used for other purposes. For instance it can be used to modeland simulate the production process of composites, such as the impregnation process. TheImpregnation process is impregnating of ibers with resin to produces prepreg, or to wet the ibers with

resin during resin transfer molding (RTM). During impregnation process the ibers are scattered andrandom therefore it is very dificult to predict the impregnation process parameters (iber spacing,impregnation time pressure drop required to drive the low) and low characteristics (velocity, planerelongation and shear rates) [23]. In order, to predict these process parameters and low characteristicsS.S. BAFNA and D.G.BAIRD [23] have modeled the ibers in the matrix as periodic micro-structure. Theperiodic micro-structure wasmodeled with varying space distance and angles between the ibers.Theiber packing geometries assumed by S.S. BAFNA and D.G.BAIRD [23] were the symmetric squarepacking and symmetric equilateral packing arrangement.

Figure 3.6symmetric square iber packing (=900S1> S2) [23]The symmetric square packing is shown inFigure 3.6,Where S1 is the diagonal spacing distance

between two ibers and S2represents the horizontal and vertical spacing distance between two ibers.The angle alpha () is the angle between the vertical and horizontal line. The D represents the diameterof the ibers and R is the radius of the ibers. In the symmetric square packing geometry the distance S2is equal is every direction (horizontal and vertical). The angle alpha () is 900. Moreover the distance S1is greater than the distance S2.

The symmetric equilateral packing is shown in Error! Reference source not found.where S1 is thediagonal spacing distance between two ibers and S2 represents the horizontal and vertical spacingdistance between two ibers. The angle alpha () is the angle between the vertical and horizontal line.The D represents the diameter of the ibers. In the symmetric equilateral packing geometry the distanceS2is equal is every direction (horizontal and vertical). The angle alpha () is 600. Moreover the distanceS1is equal to the distance S2which is also equal to s.

These iber packing patterns greatly simplify the complexity of the randomness of micro-structure of composite lamina. The Figure 3.6 and Error! Reference source not found.show how theperiodic microstructure of ibers is arranged in the composite prepreg in to manageable iber packingpattern in order to perform quantitative analysis concerning the strength performance of the compositeor to predict important processing parameters.


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Figure 3.7symmetric equilateral iber packing (=600S1= S2=S) [23]The equilateral packing arrangement sometimes it is called hexagonal packing. The square packing andhexagonal packing arrangement are found in many composites mechanics textbooks, to analysis the FVFof a lamina. The main parameter for the square packing and hexagonal are iber spacing(h) and iberradius (r), with theses parameter it is possible to calculate the FVF of a composite lamina. SeeFigure 3.8

for the geometric arrangement of square packing and hexagonal packing [24].

Figure 3.8(a) hexagonal and (b) square packing arrangement [24]Where: his iber spacing, 2Ris the distance between the centers of two neighboring ibers and ris theradius of the iber. By using the R, h, and,rvariables the FVF equation is formed [24].The FVF equation for the hexagonal packing geometry is shown with equation (3.4)

2

2 3

rFVF

R

(3.4)

Where:-

f is FVF iber volume fraction r is radius of iber R half of the center to center distance

The FVF equation for the square packing geometry is shown with equation(3.5)

2

4

rFVF

R

(3.5)

Where:-

f is FVF iber volume fraction r is radius of iber R half of the center to center distance


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The FVF equation for hexagonal and square packing arrangements are based on the area andvolume ratios. The area ratios area the area of the ibers divided to the area of the circumscribingpolygons, then the volume ratio is the volume of the iber divided to the volume of the surroundingpolygon as seen inFigure 3.9 .To simplify the analysis simple geometry and trigonometric analysis are

used to deine the area of the circumscribing polygons and the iber cross sections.

Figure 3.9 the polygons circumscribing the ibers for the given packing arrangements [24]It is depicted inFigure 3.8 the length 2R is the distance between the centers of two neighboring

ibers, h being the spacing between the neighboring ibers and r is the radius of the ibers.Equation(3.6) could be used to calculate the given spacing between the ibers.

2 2R r h (3.6)

An alternative method to calculate a FVF of lamina, which does not depend on the packinggeometry of the iber, but on the periodic micro-structure is called FVF or Fiber scaling method. The

main parameters of this method are the number of ibers, thickness of the lamia and the radius of theibers. An example of this method geometry is illustrated inFigure 3.10.The width and the length of thelamina could be assumed as a unit value (1 unit).

Figure 3.10 lamina with a thickness ts [12]The formula to compute the FVF of a lamina using the iber scaling method is given with

equation(3.7).

2

s

n rFVF

t l

(3.7)


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Where:

n is the number of ibers r is the radius of the circle tsis the thickness of the lamina l is the length of the lamia

Note that the length of the lamina is taken as a unit (1 unit)

3.3 FVF Model development analysisIn this section the FVF of a lamina will be analyzed with analytical method using parameters

like: iber spacing, iber radius, geometric packing arrangement, thickness of lamina and number ofibers. Up to now, equations and literature studies have been conducted on the methodologies and

techniques to calculate the mechanical properties of Glare and FVF of a lamina. The equations given insection 3.2 will be used to perform the FVF analysis. The analysis is aided by micro mechanics theoriesthat will set boundaries and limitations to the parameters in order to predict an acceptable FVF.

In many literatures it is mentioned that the FVF parameters namely the spacing between ibersand the number iber in a given area have a limited values for a given FVF and iber diameter. Thelimiting criteria are based on the micro-mechanics theories and assumptions. One of the basic governingmicro-mechanics criteria is the interfacial bond strength between iber and matrix. Composite lamina(FRP) the iber-matrix interface is considered as a critical factor that affect the overall mechanicalproperties of the composite. The iber-matrix interface is a region where the load transfer occursbetween the iber and the matrix and between the ibers.

The strength and stiffness of the interfacial bond are essential factors affecting the lamina

mechanical properties and fatigue behavior. The interfacial bonded areas are the surface areas of theibers. If the total bonded area of a lamina is increased the load may transfer eficiently from iber toiber and iber to matrix. Moreover, the increased interfacial surface area could compensate for damagedibers and imperfectly bonded regions. Another way of improving the mechanical property of the laminais by increasing the FVF of lamina [25]. Based on this interfacial bond strength concept criterion andimproved mechanical property due to FVF increase, the goal will be to increase the number of ibers ina given lamina and the FVF. The objectives could be achieved: [26].

by changing the iber diameter and spacing between the ibers with constant FVF by changing the iber diameter and spacing between the iber with varying FVF by changing the FVF and spacing but with constant iber diameter

These methods will be analyzed with equations(3.4)(3.5)(3.7) in the coming analysis. To realizea larger interfacial surface area and an optimal higher FVF in relation with iber diameter is a sub-objective of the thesis work as in optimizing the current Glare.

Increasing the number of ibers or FVF could improve the interface strength and the mechanicalproperty of lamina. However, the improvements of the mechanical property and interfacial the strengthof lamina due to the increase of FVF of lamina have certain limiting point. Beyond the limiting point thatis beyond the maximum value of FVF the bonding between the ibers and the matrix will start to degrade.The de-bonding of ibers from the matrix will take place leading to micro-crack in the matrix inally to afailure of the lamina [25].The loss of lamina strength at higher FVF could be caused by the ibers packingproblem creating large amount of voids and the incomplete penetration of matrix between the ibers.

These two factors affect the interfacial surface areas of the lamina [27].


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Another approach that explains the limiting factors is, due to the iber spacing between the ibersbecoming too small as the FVF increase and number of ibers. This phenomena is dealt with a shearstrength criterion within the composite lamina. The criterion is called shear-lag theory [28] . The shear- lag analysis it was proposed by Cox [28] to analyze the stress transfer between ibers and matrix by

means of interfacial shear stress for broken and damaged ibers in lamina. The shear lag theoryassumes the following ideas to perform the analysis [28]:

A damage to a iber or a broken ibers The model is linear elastic both the iber and matrix behave elastically The FVF across the cross section is constant A prefect bonded between iber and matrix The strain in the matrix and iber are equal

The shear-lag equation that predicts the maximum interfacial shear stress and axial stress has an

important parameter called the share-lag parameter, shown with equation (3.8)

1 2

ln

m

f

G

Rr E

r

(3.8)

Where:

Gmis shear modulus of the matrix Efis the elastic modulus of iber R is the half of the center to center distance r is the radius the iber is the shear- lug parameter

The matrix volume effectively involved in load transferring which the region is covered by R. Inunidirectional composites this volume fraction(R) is presumed to be the distance between the ibers.

The shear-lag parameter could be measure experimentally. The value of obtained fromexperimental work, coupled with iber and matrix property and equation (3.8) is rearranged to

determine R/r (iber spacing ratio) or R for a given iber radius. The computed iber spacing ratio at agiven iber radius and shear-lag parameter could be used to determine the maximum FVF of the laminaand the minimum iber spacing distance between neighboring ibers. Measuring shear-lag parameterexperimental is not the scope of this thesis work and requires ibers with different diameters. Theequation to predict the maximum shear stress around the iber is shown with equation (3.9)

max coth4

2 ln( )

mbf

f

G l

RE

r

(3.9)


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Where:

mG is shear modulus of the matrix

fE is the elastic modulus of iber

R is the half of the center to center distance r is the radius the iber is the shear- lug parameter

max is the maximum shear stress in the system

bf

is the iber breaking stress

ll is the length of the iberEquation(3.9) provides the relation between the shear-lag parameter, iber spacing ratio(R/r)

or R and the maximum shear stress. When (R/r) decreases the value of max

will increase as shown

inFigure 3.11.The minimum iber spacing R is determined when max

is co-related to the maximum

interfacial shear stress or shear strength of the matrix near the iber s

[28].If the iber length is

relatively long enough the term coth ( 4l ) approaches 1 and, this is the case considered in this thesis

work to compute the minimum iber spacing related to the maximum interfacial shear stress.

Figure 3.11 Shear stress verse Fiber spacing ratio0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0

1

2

3

4

5

6

Shear Stress as function of Fiber Spacing Ratio

Fiber Spacing Ratio(R/r)

Sh

earStress(Gpa)

intersection point


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The strength ratio of iber breaking stress to the interfacial shear stress (bf s ) and stiffness

ratio of the matrix shear modulus iber elastic modulus to the iber elastic modulus (m f

G E ) are given

onTable 3-1 the data are found on ref [28].

Table 3-1 the iber matrix properties ratio used in the shear-lag calculation [28]Item Range

Strength Ratio(bf s ) 1.5-5.0

Stiffness Ratio(m fG E ) 0.02-0.3

The data given inTable 3-1 is used to compute the maximum interfacial shear stress (s

), at a

given iber breaking stress and selected strength ratio range. The selected strength ratio is 2.4 anaverage value of the range. When the strength ratio is 2.4 and the iber ultimate strength is 4.8Gpa then

the interfacial shear stress (s

) is 1.41Gpa.

As seen inFigure 3.11aline is drawn from the y-axis of the interfacial shear stress that intersectthe graph. Then from the intersection point a perpendicular line is drawn toward the x-axis to ind thecorresponding iber spacing ratio. This corresponding iber spacing ratio is the minimum allowable iberspacing ratio for a given strength ratio and stiffness ratio. Now with the interfacial shear stress (

s ) it

is possible to ind the minimum iber spacing using the graph inFigure 3.11.

The maximum allowable FVF for the two types of packing (hexagonal and square) conigurationand the related minimum iber spacing ratio is shown in Table 3-2. The FVFs are computed withequation (3.4) and equation(3.5). The minimum iber spacing ratio is obtained fromFigure 3.11graphdata. InTable 3-2 across the row each packing coniguration of FVF is shown and each column showsthe related iber and matrix properties, parametric ratios and FVF.

Table 3-2stress ratio and maximum allowed FVF

s

min( )R r

By rearranging equation (3.6) and by substituting R from the iber spacing ratio in terms of rand a constant representing the ratio of minimum spacing( ) equation (3.10) is obtained to predict h.

The spacing between two neighboring ibers can be calculated with equation(3.10), for a given radius

and minimum iber spacing Ratio.

2 ( 1)h r (3.10)

min

R

r


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Where:

h is iber space between two neighboring ibers r is iber diameter R is the half of ibers center to center distance is the iber spacing ratio constant

In order to perform analysis on the iber radius, iber spacing, the number of ibers and FVF oflamina the nominal or the current lamina should be analyzed. The current lamina has a iber diameterof 10m, a FVF of 57% and nominal thickness of 0.125mm a 1mm length. The remaining Fiber spacingand number of ibers for a given thickness are calculated using equation(3.4), equation(3.5) andequation(3.7). The results of the analysis for the nominal lamina is depicted onTable 3-3.

Table 3-3the current lamina parametersr (m) h (m) n FVF ts(m) l (m)

Square 5 1.785 796 .57 125 1000Hexagonal 5 2.69 796 .57 125 1000

In the next sub section the methods to analyze FVF and iber radius are discussed.

3.3.1 Analysis of FVF of a Glare

In this section analysis is conducted on a lamina at a given nominal thickness, with varying iberdiameter, iber spacing, number of ibers and FVF. This analysis is conducted to explore the number ofibers with changing iber diameter and spacing. Moreover, to set a basis for the investigation of amechanical and fatigue property of Glare for changing number of ibers and iber radius and FVF. Thelimiting factor and other physically imposed assumption are considered in this analysis. The analysis isperformed with equations(3.4)(3.5)(3.7). The iber radius range is between 1m up to 30m. The iberspacing range is between 1m up to 5m. Also the maximum allowed FVF is included in the graphs. Agraph will be plotted for the two different iber packing coniguration. The results and data arepresented using graph and tables.

The selected FVF are as follows

FVF 1 with 40 % FVF FVF 2 with 50% FVF FVFS max with 57% FVF FVF3 with 60% FVF FVF H max with 66%FVF

From Error! Reference source not found.andTable 3-2 it is observed that the maximum allowed FVF forthe square packing is 0.57.In addition the current FVF value is also 0.57. To obtain a higher FVF bychanging either by the number of ibers, iber diameter or iber spacing for square packing is theoreticalmeaning less. Therefore in the coming analysis the square packing coniguration is not analyzed, insteadthe hexagonal packing coniguration is assumed as a model of the microstructure of the lamina.


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Figure 3.12 FVF for square packing as function of radius

Figure 3.13 FVF as function of radius iber for hexagonal packing

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

FVF as function of fiber radius for a square packing

Fiber radius( micrometer)

FVF

h=0 micrometer

h=1 micrometer

h=2 micrometer

h=3 micrometer

h=4 micrometer

h= 5 micrometer

FVF 1

FVF 2

FVF S

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FVF as function of radius for Hexagonal packing

Fiber radius (micrometer)

FVF

h=0 micrometer

h=1 micrometer

h=2 micrometer

h=3 micrometer

h=4 micrometer

h=5 micrometer

FVF 1

FVF 2

FVF 3

FVF H

FVF S


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TheFigure 3.13 depicts the FVF as a function of a iber radius. It is illustrated in theFigure 3.13the FVF increase as the iber radius increase for a given iber spacing. Moreover for a given iber radiusthe FVF can be increased or decreased by changing the iber spacing. To increase the FVF the iberspacing is reduced from the previous value. For example inFigure 3.13 to change the FVF from 55% to

60% of a iber radius of 10m, the spacing is reduced from 5m by moving up in the graph to 4m.Additionally, to increase the FVF without increasing the iber radius or keeping it constant, isaccomplished by tracing to the left of the graph onFigure 3.13.When tracing back to the left there issimultaneous change of iber radius and iber spacing, in which the size of the iber radius and iberspacing is diminishing.

In Figure 3.13 for a given FVF and a ixed iber spacing path the iber radius vary betweenspeciic intervals in order to attain the given maximum FVF. For example, inFigure 3.13 for a given FVFof 57% and iber spacing of5m the range of the iber radius to reach the maximum FVF 57% is between1m to 10m.

For the changes that were discussed in the preceding two paragraphs, the ibers number presentin a given area will be analyzed. The given area is (125m125m=15625m 2). Before discussing the

changes, a visual graph is needed to demonstrate the changes and the data. Equation(3.7) is rearrangedto plot the number of ibers present in a given area for a given FVF as a function of iber radius. Numberof iber for a given FVF's as function of iber radius

Figure 3.14Number of iber for a given FVF's as function of iber radiusThe Graph onFigure 3.14 demonstrates the variation of number of ibers in a given area for a

different FVF and iber radius. These number of ibers could vary depending up on any change on theiber radius, ibber spacing and FVF. Next the three approaches to vary the number of ibers arediscussed.

0 4 5 6 7 8 9 10 11 12 13 14 150

200

400

600

800

1000

1200

1400

1600

1800Number of fiber for a given FVF's as function of fiber radius

Fiber radius(micrometer)

Numberoffibers

FVF=0.4

FVF=0.5

FVF=0.57

FVF=0.6

FVF=0.66


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Constant FVF and varying f iber diameter and spacing

For instance take the constant line of FVF (0.5 the green line) from the graph on Figure 3.14 asa target FVF. Any simultaneous change of the iber radius and spacing (h) of the graph onFigure 3.13has an immediate effect on the number of ibers in a given area for the target FVF 0.5. If the simultaneous

change is tracing back to the left of the plot onFigure 3.13,the dimensions of the iber radius and spacingwill reduce. While the spacing and iber radius is decreasing the number of ibers in the given area willincrease inversely proportional to the square of the iber radius as depicted onFigure 3.14.

On the other hand, the number of ibers decreases as the iber radius and spacing increases, onmoving from left to the right onFigure 3.13.OnTable 3-4 it is illustrated the number of ibers and therelated iber radius and spacing for a given FVF.

Constant radius with varying fi ber spacing and FVF

At a constant iber radius if the iber spacing is varied automatically the FVF is altered. The iberspace is varied by going vertically up or down on the graph onFigure 3.13.During the variation of theiber spacing the matching FVF is sought out horizontally from the corresponding iber spacing on

Figure 3.13.If the direction taken to change the iber spacing is vertically upward meaning reducing thedimension of the iber spacing ,will induce a higher FVF. The consequence of the higher FVF withconstant iber radius is the increase of number of ibers in a given area. As the iber spacing is changingonFigure 3.13 the FVF get higher or lower. These changes are correlated to the number of ibers onFigure 3.14 by assigning the corresponding FVF with the selected iber radius. OnTable 3-5 a data ispresented on the number of ibers at a constant iber radius with changing iber spacing and FVF.

Constant fi ber spacing with changing f iber diameter and FVF

Another way of changing the FVF and the number of ibers in a given cross sectional area is bychange the iber diameter with a selected iber spacing. OnFigure 3.13 for example let the iber spacing(h=2m) be the selected constant iber spacing. If the iber radius is traced to the right of the graph, theFVF will increase together with the increasing size of the iber radius. The changing iber radius and FVF

are directly co related toFigure 3.14.

Interestingly, the number of ibers will not increase as the FVF increase together with the iber diameter.The number of iber could stay constant or it might decrease. The changing FVF is marked by theincreasing constant line on theFigure 3.14.A summary of the changing FVF and iber radius and ibernumber is given onTable 3-6.

Table 3-4 number of iber constant FVF and varying iber diameter and spacing( r, h ) [m] Number of ibers FVF

(8,5) 342 0.55

(6,4) 553 0.55(5,3) 796 0.55

Table 3-5number of ibers for constant iber Radius and changing FVF and spacing( r, h ) [m] Number of ibers FVF

(8,5) 342 0.55

8,4 360 0.58

(8,3) 398 0.64


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Table 3-6number of ibers for constant spacing and changing iber radius and FVF( r, h ) [m] Number of ibers FVF

(8,5) 342 0.55

9,5 285 0.58

(10,5) 238 0.60

3.4 The Shear Lag Method in relaon with the mechanical properes of the

lamina.So far methods were discussed how to calculate the mechanical properties of lamina as a

function of FVF and the mechanical properties of Glare as function of the MVF. However, methods that

relate the iber diameter or radius directly to the mechanical properties of lamina has not beenmentioned yet. At this time, few analytical technique are known or documented about the effect of iberdiameter on the mechanical properties of lamina or composite. More over the micro mechanical method(homogenization techniques) are all based on the volume fraction. Therefore, in this section a selectedtheoretical or analytical method that relate the iber radius to some mechanical properties and fatiguebehaviors of lamina or composite is explored. This method is based on the micromechanics approachwhich cover the geometry of the iber and the spatial orientation of the ibers in a matrix medium. Themethods is called Shearlag analysis

Shear lag methodThe shear lag model analysis was irst proposed by Cox [28] to determine the stress in a iber

embedded in an elastic matrix. For this shear lag analysis the composite is loaded in tension parallel tothe longitudinal axis. As it was mentioned in the assumptions in section3.3, the strain in matrix and iberis equal. This means no shear stress exists at the interface of iber and matrix. On the other hand it isalso mentioned in the assumptions in section 3.3 the iber ends are broken. Due to this broken iber endsthe matrix momentarily transmits a large portion of the load. As a consequence strain gradient will ariseat the iber-matrix interface until the load is redistributed back to the broken iber or to the ibersaround the broken iber. And this redistribution of the load occurs via shear stress at the iber-matrixinterface region. This phenomena could be analyzed using shear lag method. A shear lag model is usuallymodeled by a single broken iber surrounded by a matrix material. The model for the shear lag is shownonFigure 3.15 [26] [29].

Figure 3.15 the iber model in hexagonal packing for the shear lag analysisThe load is transferred over a inite length of the iber. On igure 44566 the iber break is mark with the

latter A . at the vicinity of the break the interfacial stress rise


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References

[1] J. Schijve, Development of Fiber Metal Laminates ARALL and GLARE new fatgiue resistantmaterials, 1993.

[2] M. C. Niu, Composite Airframe Structures practical design information and data, 1st ed.

[3] A. Vlot, Glare history of the develpoment of new Aircraft materials, Kluwer Academic .

[4] J. Schijve, Fatigue of Structures and materials, 2nd ed.

[5] A. Vlot and W. Gunnink, Fiber Metals Laminates an introduction, Kluwer Academic.

[6] C. A. J. R. VERMEEREN, An Historic Overview of the Development of Fibre Metal Laminates, Delft:kluwer Academic, 2003.

[7] R. Alderliesten, On the Development of Hybrid Material Concepts for Aircraft Structures, Delft,2008.

[8] "FMLC Fiber Metal Laminates Center of Competence," [Online]. Available:http://www.fmlc.nl/research-development/glare-types-conigurations/.

[9] E. A. ,. M. . B. ,. O. . Tamer Sinmazelik, "A review: Fibre metal laminates, background, bonding

types and applied test methods," Materials and Design, 2011.

[10] E. Pack, Edu Pack 2009 materials selection software.

[11] Azom.com, "The A to Z Materials," [Online]. Available:http://www.azom.com/properties.aspx?ArticleID=769.

[12] B. Berhane, "Pictures".

[13] J. REDDY, MECHANICS of LAMINATED COMPOSITE PLATES and SHELLS theory and analysis, 2nded., CRC PRESS, 2004.

[14] E. J.

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