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ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)
INGENIERO INDUSTRIAL
3D DETAILED FE PARAMETRIC MODELFOR A COMPOSITE BOLTED JOINED TEST
UNDER SHEAR-TENSION INTERACTION
Autor: Jorge Kraemer ÁvilaDirector: M.Sc. B.Eng. Eju Baruah (Airbus Operations GmbH)
MadridJunio 2012
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ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)
INGENIERO INDUSTRIAL
3D DETAILED FE PARAMETRIC MODELFOR A COMPOSITE BOLTED JOINED TEST
UNDER SHEAR-TENSION INTERACTION
Autor: Jorge Kraemer ÁvilaDirector: M.Sc. B.Eng. Eju Baruah (Airbus Operations GmbH)
MadridJunio 2012
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3D DETAILED FE PARAMETRIC MODEL FOR A COMPOSITEBOLTED JOINT TEST UNDER SHEAR-TENSION INTERACTION
Autor: Kraemer Ávila, Jorge
Director: Baruah, EjuEntidad Colaboradora: Airbus Operations GmbH ResumenHoy en día, el número de vuelos está creciendo exponencialmente, por ese motivo esindispensable encontrar nuevas formas de ahorro de combustible. Los materialescompuestos avanzados presentes en una era de cambio climático y rápida disminuciónde los recursos fósiles, no son sólo una oportunidad para crear robustas e innovadorasestructuras que ahorren peso para la industria aeronáutica, sino también un gran reto dela ingeniería, como se puede observar en este estudio sobre el comportamiento de TestARCAN. La fabricación, el mantenimiento y la necesidad de estructuras capaces desoportar carga de manera fiable hacen indispensable el uso de juntas mecánicas. Dada lafalta de experiencia con materiales compuestos en comparación con el metal en generaly con articulaciones en particular, el diseño de estructuras compuestas fiables es unacuestión de gran desarrollo así como los costes de experimentación.
Con el fin de reducir estos costes de experimentación y por lo tanto reducir el tiempo deentrega de diseño, avanzados métodos numéricos deben ser establecidos para validar losresultados de las pruebas con precisión. Con esta finalidad y mediante el uso del códigode elementos finitos Abaqus Explicit, se ha realizado una investigación sobre elcomportamiento de parámetros físicos y materiales en una junta mecánica de materialescompuestos bajo una carga cuasi-estática a través el Test ARCAN.
El funcionamiento interno del simulador consiste en una secuencia de comandos principal, llamada "Terminal", que está a cargo de la gestión de scripts y datos.
Terminal
PartCreator
Retainig Block
Bolts
Coupon
Nuts
Property
Materials
Sections
Assembly
Meshing
Interaction
Steps
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Este script recibe los parámetros de entrada del interfaz de usuario (abajomostrado) y gestiona todos los sub-scripts de comandos que envían de vuelta al
usuario la salida solicitada, es decir, el modelo 3D. Tal como se presenta en eldiagrama de flujo anterior, la herramienta se compone de la script de comandos principal y seis sub-scripts de comandos, cada uno de estos sub-scripts está acargo de una de las partes principales del modelo FE generados automáticamentecon el pre-procesador Abaqus CAE. Algunos de estos scripts de comandos sedividen en varias funciones con el fin de armonizar el código.
En una primera fase, el usuario llama al plugin mediante el pre-procesador, la siguienteinterfaz gráfica de usuario aparece (arriba mostrado), que permite la selección de variossub-menús. Dentro de cada sub-menú, una amplia gama de modelos y definiciones de
propiedad aparece, que permite la selección de las propiedades necesarias para crear elmodelo con las aportaciones correspondientes, y simular el modelo generado.
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Para tensión cortante pura, un método simpleutilizando elementos sólidos para las capas dematerial compuesto unidas por "tie constraint"ofrece un buen enfoque debido al hecho de que larigidez global del modelo está dominado
principalmente por rigidez del laminado, en particular, dominado por la rigidez en la dirección1 y 2, este comportamiento queda bienreproducido por la subrutina material.
Para tensión pura, la aproximación antesmencionada es buena para el cálculo de la rigidez,
pero ha de ser mejorada con el fin de capturar elfallo de delaminación. Por esta razón, laaproximación con "cohesive elements" se haimplementado, con resultados aún mejores para larigidez y la captura del fallo inter-laminar, elúnico problema que presenta este método es elenorme costo computacional en comparación conla aproximación anterior. Por esta razón, se haimplementado la aproximación de "cohesivecontact" se ha utilizado. Para obtener mejoresresultados con este enfoque, una versión másmadura del código de elementos finitos AbaqusExplicit ha de ser probado y utilizado, dónde lasincoherencias numéricas durante los análisis sehan eliminado.
La subrutina material utilizada para este estudiono tiene en cuenta la diferencia entre el fallo atensión y a compresión de la fibra ni de la matriz,
por esta razón se recomienda ejecutar los modelosutilizando subrutinas de material más complejas,ya que el "Enhanced Hashin" o el desarrollado
0 1 2 3 4 5
R F 1 [ k N ]
U1[mm]
Solids_wocoh
Experimental Results
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recientemente "Larc05". La aproximación con "cohesive elements" parece ser la másadecuada para Abaqus 6.10-1, pero presenta un enorme coste computacional. Se puedeutilizar, pero con el fin de minimizar este problema, es necesario establecer unaevolución tecnológica-computacional en términos de mejora de la paralelización de los
procesadores, y aumento del número CPUs, aunque el aumento del número de CPUsdespués de una cierta cantidad no siempre supone una reducción del costecomputacional. Por esta razón, el uso de Abaqus 6.12-1 es altamente recomendable pararealizar una aproximación de "cohesive contact" con éxito, evitando la incoherencia enel análisis numérico entre el comportamiento cohesivo y general de contacto.
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3D DETAILED FE PARAMETRIC MODEL FOR A COMPOSITEBOLTED JOINT TEST UNDER SHEAR-TENSION INTERACTION
Summary Nowadays the number of flights is growing exponentially because of which it isindispensable to find new ways of saving fuel. Advanced composite materials present inan era of climate change and rapidly decreasing fossil resources not only an opportunityfor robust and innovative weight saving structures for the aircraft industry, but alsogreat challenges in engineering as it can be observed in this study regarding the
behavior of the ARCAN Test. The manufacturing, maintenance and the need of reliableload carrying structures make indispensable the use of mechanically fastened joints.Given the lack of experience with composites in comparison to metal in general andwith joints in particular, the design of reliable composite structures is a matter of hugedevelopment as well as experimental costs.
In order to reduce these experimental costs and therefore reduce design laid time,advanced numerical methods must be established to validate test results accurately.Based on this assumption and by use of the finite element code Abaqus Explicit, anumerical and physical parameters investigation on the behavior of a composite bolted
joint under quasi-static loading have been performed through the ARCAN Test.
The internal performance of the simulator consists in a main script, so called“Terminal”, which is itself, in charge of script and data management. This scriptreceives the inputs from the User-Interface Modules and manages all the sub-scripts
sending back to the user the Output requested, that is the 3D model. As presented inflowchart above, the tool is composed of the main script and six sub-scripts; each of
Terminal
PartCreator
Retainig Block
Bolts
Coupon
Nuts
Property
Materials
Sections
Assembly
Meshing
Interaction
Steps
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these sub-scripts is in charge of one of the main parts of the FE Model automaticallygenerated with the Abaqus CAE Pre-processor. Some of these scripts are also divided
into several functions in order to harmonize the code.
At a first step, when the plugin-tool is called into the Pre-processor by the end-user, thefollowing graphical user-interface appears, that allows the selection of several sub-menus. Under each sub-menu, a range of model and property definitions appears, thatallows the selection of the required properties to create the model with the relevantinputs, and simulate the generated model.
0 1 2 3 4 5
R F 1 [ k N ]
U1[mm]
Solids_wocoh
Experimental Results
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For pure shear, a simple approach using solid elements for the layers of composite tiedtogether by tie contact offers a good approach due to the fact that the global stiffness ofthe model is dominated mainly by laminate stiffness, in particular dominated by thestiffness in direction 1 and 2, this behavior is well reproduced by the user subroutine.
For pure tension, the above-mentioned approachis good for the calculation of the stiffness, butneed to be enhanced with the purpose ofcapturing the delamination failure. For thisreason, the approach using cohesive elements has
been implemented, showing even better resultsfor the stiffness and capturing the inter-laminarfailure, the only problem that this method has isthe enormous computational cost in comparisonwith the previous approach. For this reason, theapproach of cohesive contact has been used. Toobtain better results with this approach, a morematured version of the FE code Abaqus Explicitis to be tested and used, where such numericalincoherences during the analyses are said to have
been removed.
The User Material Subroutine used for this studydoes not take into account the difference betweentension and compression failure for the fiberneither for the matrix, for this reason it isrecommended to run the models using morecomplex User Subroutine, as the EnhancedHashin or the recently developed Larc05.
The approach with cohesive elements seems to bethe most adequate for Abaqus 6.10-1, but presentsenormous computational cost. It can be used butin order to minimize this problem, it is needed
established a computational evolution in termsimprovement in parallelization of processors andnumber of CPU usage, as increasing the numberof CPUs after a certain amount that is not alwayslaid to reduce computational cost.
For this reason, the use of Abaqus 6.12-1 ishighly recommended to perform a successfullycohesive contact approach avoiding theincoherence in the numerical analysis between
cohesive behavior and general contact.
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Concluding with the way forward, it would be interesting to implement two differentapproaches into the model; one approach for the initial pretension in order to simulate
better the behavior of the joint under and after pretension and another approach formulti-scale approaches using multi-processors with the aim of distributing wiser theCPUs, using more for fine meshes and less for coarse meshes.
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iv
AgradecimientosMe gustaría dar las gracias a mis padres y amigos. Además me veo en la obligación dedar las gracias a Doña Eju Baruah, que ha supervisado este trabajo, y tantas horas de sutiempo me ha dedicado. Y me gustaría agradecer especialmente a todos los amigos queme han dado fuerzas en los momentos más complicados a lo largo de toda mitrayectoria universitaria, comenzando por aquellos compañeros que me siguieron en misinicios en la Universidad Pontificia Comillas y con los que forjé un lazo tan especial,continuando por aquellos que conocí y con los que tanto compartí durante miexperiencia en la Universidad Técnica de Múnich y por último, pero no menosimportante, aquellos que me han acompañado durante mi experiencia en AirbusOperations GmbH con los que tantos cafés y frustraciones he compartido. Muchasgracias a todos.
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AbstractThe following Thesis aims to study the complex failure phenomenon in composite
bolted joints in shear and tension under quasi-static loading, with the help of DetailedFinite Element modelling. With this intention a parametric plugin has been programmedto automate the development of the model. Different modelling methodologies areinvestigated in this study, with the emphasis being laid on simulating a pure tension anda pure shear test for the same joint configuration, and performing comparative analysesas well as correlations with physical tests. Different numerical approaches have beenimplemented and compared in order to obtain the better simulation accuracy-CPU usagerelationship possible.
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2.2.1.1.1 Microbuckling ........................................................................... 30
2.2.1.1.2 Fiber Breakage .......................................................................... 31
2.2.1.1.3 Matrix Cracking ........................................................................ 31
2.2.1.1.4 Delamination ............................................................................. 31
2.2.1.2 Failure Criteria .......................................................................... 31
2.2.2 Maximum Value Failure Theories ...................................................... 32
2.2.2.1 Maximum Stress Theory ........................................................... 32
2.2.2.2 Maximum Strain Theory ........................................................... 32
2.2.3 Interactive Failure Theories ................................................................ 32
2.2.3.1 Tsai-Wu Theory ........................................................................ 32
2.2.3.2 Tsai-Hill Theory ........................................................................ 34
2.2.4 Mechanical Failure Theories .............................................................. 35
2.2.4.1 Hashin-Rotem Criterion ............................................................ 35
2.2.4.2 Hashin Criterion ........................................................................ 36 2.2.5 Delamination Initiation Prediction ..................................................... 36
2.2.6 Damage Progression Models .............................................................. 37
2.2.6.1 Ply-Discounting Approach ........................................................ 38
2.2.6.2 Continuum-Damage-Mechanics Approach ............................... 38
2.3 Composite Bolted Joints in Aircraft Structures ........................................... 38
2.3.1 Mechanically Fastened Joints ............................................................. 38
2.3.1.1 Generalities ............................................................................... 38
2.3.1.2 Failure Modes of Mechanically Fastened Joints ....................... 39
2.3.1.2.1
Tension Failure ......................................................................... 40
2.3.1.2.2 Shear Failure ............................................................................. 40
2.3.1.2.3 Bearing Failure .......................................................................... 41
2.3.1.2.4 Cleavage Failure ....................................................................... 41
2.3.1.2.5 Pull-out Failure ......................................................................... 41
2.3.2 Design Considerations: Material Parameters and Joint Geometry ..... 42
2.3.2.1 Fiber Orientation ....................................................................... 42
2.3.2.2 Stacking Sequence .................................................................... 42
2.3.2.3 Joint Dimensions ....................................................................... 42
2.3.3 Design Considerations: Fastener Parameters ..................................... 42
2.3.3.1 Clamping Torque ...................................................................... 42 2.3.3.2 Friction ...................................................................................... 43
2.3.3.3 Bolt/Hole Clearance .................................................................. 43
2.4 Finite Element Modeling ............................................................................. 43
2.4.1 Finite Element Theory ........................................................................ 44
2.4.1.1 Hypothesis of the Finite Element Analysis ............................... 44
2.4.1.2 Explicit Finite Element Analysis .............................................. 44
2.4.2 Bolted Joints Finite Element Modeling .............................................. 46
2.4.2.1 Pin joints Modeling ................................................................... 46
2.4.2.2 Bolted Joints Modeling ............................................................. 46
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x
2.4.3 Interlaminar Damage Modeling ......................................................... 47
2.4.3.1 Fracture Mechanics Theory ...................................................... 47
2.4.3.1.1 Griffith Energy Criterion .......................................................... 47
2.4.3.1.2 Mixed-Mode Delamination ....................................................... 48
2.4.3.2 Virtual Crack Closure Technique ............................................. 49
2.4.3.3 Cohesive Zone Models .............................................................. 50
2.4.3.3.1 Principle .................................................................................... 50
2.4.3.3.2 Constitutive Equations .............................................................. 52
2.4.3.3.3 Mixed-mode Delamination ....................................................... 54
2.4.3.3.4 Comments on the Cohesive Zone Models ................................ 56
3 FE Modeling of the ARCAN Test 57 3.1 Description of the ARCAN – Physical Test ................................................ 57
3.1.1 Geometry ............................................................................................ 57
3.1.1.1 Specimen ................................................................................... 57
3.1.1.2 Test Fastener ............................................................................. 58
3.1.1.3 Test Fixture ............................................................................... 59
3.1.1.4 Retaining Blocks ....................................................................... 59
3.1.1.5 Fasteners .................................................................................... 60
3.1.1.6 Semi-Circumferential Parts ....................................................... 60
3.1.2 Test Description .................................................................................. 61
3.1.3 Test Procedure .................................................................................... 61
3.2 Description of the ARCAN – FE Virtual Test ............................................. 62
3.2.1 Development of a Tool for a Parametric Detailed FE Model ............ 62 3.2.1.1 Python-Script Flowchart ........................................................... 62
3.2.1.2 User-Interface Modules ............................................................. 63
3.2.2 Geometry ............................................................................................ 65
3.2.2.1 Composite Plate ........................................................................ 65
3.2.2.2 Fastener ..................................................................................... 66
3.2.2.3 Test Fixture ............................................................................... 66
3.2.2.3.1 Retaining Blocks ....................................................................... 66
3.2.2.3.2 Retaining Fasteners ................................................................... 67
3.2.3 Meshing and Element Properties ........................................................ 68 3.2.3.1 Test Fastener ............................................................................. 68
3.2.3.2 Core Vicinity ............................................................................. 69
3.2.3.3 Coupon Core ............................................................................. 70
3.2.3.4 Coupon Outer Cores .................................................................. 70
3.2.3.5 Cohesives (Approach of Cohesive Elements) ........................... 71
3.2.3.6 Retaining Block......................................................................... 72
3.2.3.7 Retaining Fasteners ................................................................... 72
3.2.3.8 Cohesive Elements .................................................................... 73
3.2.4 Constraints & Interactions .................................................................. 75
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xi
3.2.4.1 Tie Constraint ............................................................................ 75
3.2.4.2 General Contact ......................................................................... 75
3.2.4.3 Cohesive Behavior .................................................................... 77
3.2.5 User Subroutine .................................................................................. 78
3.2.6 Loading Steps ..................................................................................... 81
3.2.6.1 Initial Pre-tension ...................................................................... 81
3.2.6.2 Final Test Displacement ............................................................ 82
4 Analysis of Experimental Results 83 4.1 Protruding Head Bolt: Pure Tension ............................................................ 83
4.2 Protruding Head Bolt: Pure Shear ................................................................ 84
4.3 Comparison of Tests with Protruding Head Bolt ......................................... 85
4.4 Comparison of Tests with Countersunk Head Bolt ..................................... 87
4.5
Comparison in Terms of Maximum Peak Loads ......................................... 88
5 Analysis of Simulation Results 91 5.1 Solids without Cohesives ............................................................................. 91
5.1.1 Pure Tension ....................................................................................... 91
5.1.2 Pure Shear ........................................................................................... 93
5.2 Cohesive Elements ....................................................................................... 95
5.2.1 Pure Tension ....................................................................................... 95
5.3 Cohesive Contact ......................................................................................... 98
5.3.1 Pure Tension ....................................................................................... 98
5.4 Computational Cost ................................................................................... 100 6 Conclusion and Way Forward 103
6.1 Conclusion ................................................................................................. 103
6.2 Way Forward ............................................................................................. 103
A. Apendices 105
a. Data and Software-code 105
i. Terminal ¡Error! Marcador no definido.
ii. User Material Subroutine [21] 105
b. List of Figures 121
c. List of Tables 125
B. Bibliography 127
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Nomenclature
Formelzeichen Unit Kurzbezeichnung
t s Time
v m/s Velocity
ρ N/m 3 Density
S N/m 2 Tensile Strength
E N/m2
Young’s Modulus
V m 3 Volume
υ - Specific Volume
ν - Poisson Ratio
A m 2 Area
m kg Mass
F N Force
σi N/m 2 Tension Stress
εi - Longitudinal Strain
L m Lenght
G ij N/m 2 Shear Modulus
τ N/m 2 Shear Stress
γ - Shear Strain
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Abbreviations
FRP Fiber Reinforced PolymersFE Finite Element
ALLFD Friction Dissipated Energy of the Whole Model
ALLIE Internal Energy of the Whole Model
ALLKE Kinetic Energy of the Whole Model
ALLWK External Energy of the Whole Model
CAE Complete Abaqus Environment
COH3D8 Cohesive 3-Dimensional Element with 8 nodes
CPU Central Processing Unit
C3D8R Continuum 3-Dimensional reduced element with 8 nodes
ETOTAL Total Energy of the Whole Model
FEA Finite Element Analysis
RB Rigid Body
RP Reference Point
SLS Single Lap-Shear
VUMAT User Material for Abaqus Explicit
SDV State Definition Variables
CFRP Composite Fiber Reinforced Plastic
RAM Read Access Memory
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1 Introduction 1
1 Introduction
1.1 Motivation
1.2 Technological EnvironmentA composite is when two or more different materials are combined together to create asuperior and unique material. The first uses of composites date back to 1500 B.C. whenearly Egyptians and Mesopotamian settlers used a mixture of mud and straw to createstrong and durable buildings. Straw continued to provide reinforcement to ancientcomposite products including pottery and boats.
Later in 1200 A.D., the Mongols invented the first composite bow. Using a combination
of wood, bone, and “animal glue,” bows were pressed and wrapped with birch bark.These bows were extremely powerful and highly accurate. Mongolian composite bows
provided Genghis Khan with military dominance, and because of the compositetechnology, this weapon was the most powerful weapon on earth until the invention ofgunpowder. But we need to go forward 700 years to the future, to the early 1900s, when
plastics such as vinyl, polystyrene, phenolic and polyester were developed. And in1935, Owens Corning introduced the first glass fiber, fiberglass. Fiberglass, whencombined with a plastic polymer creates an incredibly strong structure that is alsolightweight. This is the beginning of the Fiber Reinforced Polymers (FRP) industry as
we know it today.After the First World War , improvements in chemical technology led to an explosion innew forms of plastics and with it the improvement of the composites, thanks to BrandtGoldsworthy, often referred to as the “grandfather of composites,” who developed newmanufacturing processes and products. Goldsworthy invented a manufacturing processknown as “pultrusion”. Today, products manufactured from this process include ladderrails, tool handles, pipes, arrow shafts, armor, train floors, medical devices, and more.
As we have seen, the use of composites is not something new, so the question is, whynow, and the easiest answer is because of the weight reduction that composites give usin the aviation industry. And there are two main reasons to get a weight reduction, toachieve a reduction of fuel consumption and more importantly, a reduction inenvironmental impact.
The Stern Review, 2006, identified that 1.6% of global greenhouse gas emissions comefrom aviation and that the demand for air travel will keep rising with our income. Tocombat the environmental threat that aviation poses, the Advisory Council forAeronautical Research in Europe in 2002 laid out targets to reduce the emission of CO2(an important greenhouse gas) from an aircraft by 50% by 2020. The reduction ofairframe weight through the extensive use of carbon composites is just one of a range oftechnologies that must be deployed to meet such a challenging target. [1]
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1 Introduction 2
Tab. 1: Evolution of final energy consumption in transport, by transport mode, 1990-2004, EU-25(in million toe) [2]
The increased criss-crossing of jet trails we see in the skies reflects the fastest rise inenergy consumption of all transport modes (including maritime transport). As illustratedin Table 1, between 1990 and 2004 energy consumption rose by 67 % in aviation, thiswas considerably greater than the 27 % growth recorded for road transport [2].
1.3 Project PresentationThe structures used in aircrafts to protect the passengers were firstly designed usingmetallic materials, the mechanical properties of which and especially the plasticcapabilities are well known. However, since some years now, other materials likereinforced composite materials have been used in aircraft key structure like the fuselage,and especially in the new generation airplane. Because of this, it is necessary to addressthe failure behavior of composite material structures.
1.3.1 Goals of this WorkThe global idea in this research project consists in the study of composite bolted jointsfailure, in order to be able to reproduce the test results with detailed FE models, to late
perform sensitivity studies. To try to get the better understanding of the failuremechanisms in the fuselage, a detailed modeling of a composite bolted joint will be
performed using the ARCAN test.
This study aims at:
Developing a friendly non-technical user oriented interface.
Developing a 3D parametric simulator.
Reproducing and capturing the behavior of the fastened joints.
Validation of the experimental results.
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1 Introduction 3
1.3.2 Methodology of this WorkIn order to develop this parametric 3D ARCAN test simulator, a highly operativescripting model was built up, using the following flow charts.
Tab. 2: Functional flowchart between the user and the tool.
This model consists in dividing each task of the script in separated sub-scripts with lessthan 200 code-lines obtaining the following advantages:
Reduction in computational cost.
Increment in the accuracy in scripting failure detection.
User-friendly further optimization of the tool.
Easy-to-use scripting interfaces and simulator for a new end-user.
In the following sections, the tests and the finite element models will be presented andcorrelation of the FE results with the test results will be performed.
User Terminal
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1 Introduction 4
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2 Theoretical Background 5
2 Theoretical Background
2.1 Composite Laminate Plate Theory
2.1.1 Composite MaterialsGenerally speaking, a composite material signifies that two or more components withdifferent properties are combined on a macroscopic scale to form a useful third material with
properties that are not attainable with the individual components, as has been used by man forthousands of years. The advantage of composite materials is that, they usually exhibit the bestqualities of their components and often some qualities that neither component possesses. Notall these material properties are improved at the same time. In fact, some of the properties are
in conflict with another.
2.1.1.1 Fiber MaterialsPolymers have low stiffness and are ductile, in contrast, ceramics and glasses are stiff andstrong but are catastrophically brittle. In fibrous composites, we exploit the great strength ofceramics while avoiding the brittle failure of fibers, leading to a progressive, not a sudden,failure. If the fibers of a composite are aligned along the loading direction, the stiffness andthe strength are roughly speaking, an average of those of the matrix and fibers, weighted bytheir volume fractions. But not all composite properties are just a linear combination of theseof the components. Their great attraction lies in the fact that, frequently, something extra isgained.
Tab. 3: Fiber and Wire Properties [3]
The fibers are subjected to special finished surface treatments to prevent fiber damage undercontact with processing equipment, to provide surface wetting when fibers are combined withmatrix materials and to improve the interface bond between fibers and matrices.
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2 Theoretical Background 6
The most commonly encountered surface treatments are chemical sizing performed during the basic fiber formation operation and resulting in a thin layer applied to the surface of the fiber,surface etching by acid, plasma or corona discharge, and coating of fiber surface with thinmetal or ceramic layers. With only a few exceptions (e.g., metal fibers), individual fibers,
being very thin and sensitive to damage, are not used in composite manufacturing directly, butin the form of tows (rovings), yarns and fabrics. A unidirectional tow (roving) is a looseassemblage of parallel fibers consisting usually of thousands of elementary fibers. Two maindesignations are used to indicate the size of the tow, namely the K-number that gives thenumber of fibers in the tow (e.g., 3K tow contains 3000 fibers) and the tex-number which isthe mass in grams of 1000 m of the tow. The tow tex-number depends not only on the numberof fibers but also on the fiber diameter and density. A yarn is a fine tow (usually it includeshundreds of fibers) slightly twisted (about 40 turns per meter) to provide the integrity of itsstructure necessary for textile processing. Yarn size is indicated in tex-numbers or in textile
denier-numbers (den) such that 1 9. Continuous yarns are used to make fabrics withvarious weave patterns. [4]
Fig. 2.1: Stress-strain curves for typical fibers used in reinforced composite
The processability is an important characteristic of fibers and can be evaluated as the ratio, of the strength demonstrated by fibers in the composite structure, to the strength offibers before they were processed. This ratio depends on the fibers’ ultimate elongation,sensitivity to damage, and manufacturing equipment causing the damage of fibers. Boron andhigh-modulus carbon fibers are the most sensitive to operational damage, possessingrelatively low ultimate elongation 5.
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Fig. 2.2: Fiber processability tests [5]: (a) straight tow, (b) tow with a loop and (c) tow with a knot
To evaluate fiber processability under real manufacturing conditions, three simple tests are
used -tension of a straight dry tow, tension of tows with loops, and tension of a tow with aknot (see Figure 2.2).
2.1.1.2 Matrix MaterialsTo utilize high strength and stiffness of fibers in a monolithic composite material suitable forengineering applications, fibers are bound with a matrix material whose strength and stiffnessare naturally, much lower than those of fibers. Matrix materials provide the final shape of thecomposite structure and govern the parameters of the manufacturing process. Optimalcombination of fiber and matrix properties should satisfy a set of operational and
manufacturing requirements that sometimes are of a contradictory nature and have not beencompletely met yet in existing composites. The most commonly used matrices are carbon,ceramic, glass, metal, and polymeric. Each has special appeal and usefulness, as well aslimitations. Richardson [6] presents a comprehensive discussion of matrices, which guidedthe following presentation.
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Tab. 4: Types of Matrix
Carbon Matrix . A carbon matrix has a high heat capacity per unit weight.
Ceramic Matrix . A ceramic matrix is usually brittle.
Glass Matrix . Glass and glass-ceramic composites usually have an elastic modulusmuch lower than that of the reinforcement.
Metal Matrix . A metal matrix is especially good for high-temperature use inoxidizing environments. There are three classes of metal matrix composites:
o Class I . The reinforcement and matrix are insoluble.
o Class II . The reinforcement and matrix exhibit some solubility and theinteraction will alter the physical properties of the composite.
o Class III . The most critical situations in terms of matrix and reinforcement arein this class.
Polymer Matrix . Polymeric matrices are the most common and least expensive. Theyare found in nature as amber, pitch, and resin. They are a low-density material.Because of low processing temperatures, many organic reinforcements can be used.There are two classes of polymer matrix composites:
o Thermoplastic . A thermoplastic can be remolded to a new shape whenit is heated to approximately the same temperature at which it wasformed.
o Thermoset . A thermoset cannot be remolded after it has been processed.
Matrix
Carbon
Ceramic
Glass
Metal
Class I
Class II
Class III
Polymer
Thermoplastic
Thermoset
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2.1.1.3 Laminate AnalysisAs it is shown in Figure 2.3, the analysis of a laminate is amulti-scale process, starting from the raw materials, the
matrix and the fibers, elements from which the lamina is built. It is possible to distinguish this category in threedivisions, depending on the allocation of the fibers: laminawith unidirectional fibers, lamina with woven fibers(biaxial weave) and lamina with woven fibers (triaxialweave), as shown in Figure 2.4. To determinate the
properties of the lamina, the use of micromechanics isneeded. It relies on the study of the interaction of theconstituent materials based on models that uses simple
assumptions regarding the lamina and/or experimentalmethods.
Fig. 2.3: The levels of analysis for a structure made of laminated composite. [7]
Fig. 2.4: Composite material systems. [7]
When the properties of the lamina are known, the analysis continues with the laminate, whichis built by stacking several laminas with different orientations. The properties of the laminateare calculated through macromechanics using the mechanical properties of the laminas. Thecomposite material is assumed homogeneous, therefore the contribution of the constituent
materials are detected as averaged properties of the composite material.
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2.1.2 Micromechanics
Fig. 2.5: Illustration of the matrix, fiber, and void volumes. [7]
Micromechanical approach is used to estimate the mechanical properties of compositematerials from the known values of the properties of the fiber and the matrix. There are threemajor categories of micromechanical approaches: (a) mechanics of material models based onsimplifying assumptions that make it unnecessary to specify in detail the stress–straindistributions, (b) elasticity models requiring that the stresses and strains be determined at themicromechanical level, and (c) empirical expressions resulting from curve-fitting elasticitysolutions or data.
2.1.2.1 The Rule of Mixture
We consider the volume V of an element shown in Figure 2.5. The volume of this element is
(2. 1)where the subscripts f, m and v refer to the fiber, the matrix, and the void. It is convenient tointroduce the volume fractions as follows:
(2. 2)With the equations A.1 and A.2, it can be deduced:
1 (2. 3)
When the void fraction is negligible ( 0), we have
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1 (2. 4)By taking in consideration the characteristics of the element 1, the following relationships can
be deduced, using the volume fraction and A, A f , A m, respectively the composite, the fiberand the matrix cross-sectional areas:
(2. 5)
(2. 6)
The mass of the element, by neglecting the void mass in Figure 2.6 is
· · (2. 7)where ρf and ρm are the fiber and the matrix densities, respectively. The density of thecomposite is
· · (2. 8)
Fig. 2.6: Representative elements: Element 1 (left) and Element 2 (right). [7]
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In the following, as is taken approach to consider unidirectional, fiber-reinforced compositeswithout voids. Deriving the properties using two types of elements, as shown in Figure 2.6,element 1 contains a single fiber bundle of circular cross section, element 2 consists of a fiberlayer sandwiched between two layers of matrix material. In element 2, the fiber and matrixvolumes are the same as in element 1.
2.1.2.2 Determination of the Mechanical Properties
2.1.2.2.1 Longitudinal Young‘s Modulus E1Element 1 is subjected to a force F1 in the fiber direction. The force, distributed over thesurface, is (Fig.); with σ1 the normal stress averaged on the cross-section with area A, we candecompose the force into two parts, one carried out for the matrix and the other one for thefiber:
F σ ·A σ ·A σ ·A (2. 9)When the Poisson effect is neglected, the normal stresses in the composite σ1, in the fiber
bundle σf1, and in the matrix σm1 are:
·
·
· (2. 10)
where E 1 and E f1 are the composite and the fiber longitudinal Young’s modulus and E m is thematrix Young’s modulus respectively. Using the hypothesis of identical deformation in thelongitudinal direction Equations (2.9) and (2.10) give:
· · (2. 11)Finally, using the Equation (2.6), the rule of mixture for the longitudinal Young’s modulus,depending on the fiber and matrix modulus, is obtained:
· · (2. 12)
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2.1.2.2.2 Transversal Young‘s Modulus E2In Element 2, the inner layer is a sheet of fiber with the same volume as the fiber bundle inElement 1. The length and the transverse Young’s modulus of this inner layer in x 2 direction
are L f and E f2 respectively, and the length and the Young’s modulus of the outer layers are
and E m, and the lateral surface of the element is .
Fig. 2.7: Element 1 subjected to a force in the fiber direction (left), and Element 2 subjected to a force inthe transverse direction (right). [7]
The normal stress in the transverse direction is:
· ∆ · (2. 13)where E 2 is the transverse Young’s modulus of the element and σ2 is the average transversenormal strain. The change in the length of the element is
∆ 2· · · (2. 14)When the Poisson effect is neglected, the transverse normal strains in the composite ε2, in the
fiber bundle εf2, and in the matrix ε m2 are:
(2. 15)By introducing equations (2.14) and (2.15) into equation (2.13), we obtain:
··· · · (2. 16)
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The transverse normal stresses in the composite σ 2, the matrix σ m2, and the fiber σ f2 layers areequal as follows:
(2. 17)The matrix and fiber volume fractions are:
(2. 18)By substituting equations (2.16) and (2.17) into equation (2.16), we obtain the transverseYoung’s modulus
(2. 19)
2.1.2.2.3 Transversal Young‘s Modulus E3By using the same theoretical procedure as for the longitudinal Young’s modulus but usingthe representative element 2. , we obtain:
· · (2. 20)2.1.2.2.4 Longitudinal Shear Modulus G12
We consider Element 2. The length and the longitudinal shear modulus of the inner layer areLf and G f12 respectively, and the length and the shear modulus of the outer layers are and G m. The element is subjected to a shear force F 12 (see Figure 2.8) distributeduniformly across the surface .
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Fig. 2.8: Element 2 subjected to a shear force (left) and deformation of the “top” (ijkl) surface (right). [7]
The shear stress is: · (2. 21)
where G 12 is the longitudinal shear modulus of the element and γ12 is the average shear strain,
≅ tan ∆ (2. 22)The change in the length of the element due to the shear deformations of the matrix and fiberlayers is:
∆ 2· · · (2. 23)The shear strains in the composite γ2, in the fiber bundle γf2, and in the matrix γm2 are:
(2. 24)The surface where the shear stresses in the composite are applied, the matrix and the fibers arethe same, thus:
(2. 25)
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By introducing equations (2.22), (2.23), (2.24) and (2.25) into equation (2.21), we obtain thetransverse Young’s modulus:
(2. 26)2.1.2.2.5 Transversal Shear Modulus G23By using the same theoretical procedure as for the longitudinal Shear modulus but using therepresentative element 2, we obtain:
(2. 27)2.1.2.2.6 Transversal Shear Modulus G13By using the same reasoning as for the longitudinal Young’s modulus but using therepresentative element 2. Taking into account that the shear deformations are indeedequal
, the Shear force is distributed on the top surface, thus:
· · · (2. 28)where γm12 and γf12 are the shear strains in the matrix and fiber layers as follows:
· · (2. 29)It can be deduced from equations 2.28 and 2.29 the transverse Shear modulus G 23:
· · (2. 30)
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2.1.2.2.7 Longitudinal Poisson Ratio υ 12
Fig. 2.9: Deformation of Element 1 subjected to an axial force in the x
1 direction. [7]
As shown in Figure 2.9, element 1 is subjected to a force F1 in the longitudinal x1 direction.Because of this force the element deforms, as illustrated in Figure 2.9. The sides of thedeformed element are L ∆L, L ∆L and L ∆L. The change in the cross-sectionalarea A of the face ilkj is:
∆ ∆· ∆ ∆ ∆ (2. 31)
where ∆A and ∆A are respectively the change in the cross-sectional areas of the fiber andthe matrix. By definition, the normal strains in the x 2 and x 3 directions are: ∆ · ∆ · (2. 32)
By symmetry of element 1, it is assumable that ∆ ∆ and therefore: (2. 33)By introducing equations 2.32 and 2.33 in 2.31, the change in the cross-sectional area A
becomes:
∆ · 2· · (2. 34)
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By the same reasoning, change in the cross-sectional area of the fiber and the matrix are:
∆ 2· ·
∆ 2· · (2. 35)
By combining equations 2.31, 2.32 and 2.33, it is obtained that:
∆· · (2. 36)By combining equations 2.35 and 2.36, it is obtained that:
· · · · (2. 37)Taking into account that the deformation in the composite, the fiber and the matrix are thesame in direction 1:
(2. 38)Using the same procedure as for 2.36, the longitudinal Poisson ratio for the fiber and for thematrix can be expressed as:
(2. 39)Finally, using the equations 2.37, 2.38 and 2.39, the expression of the longitudinal Poissonratio is:
· · (2. 40)
2.1.2.2.8 Transversal Poisson Ratioυ
23For a transversely isotropic material, the transverse shear and the Young’s modulus arerelated by the following expression:
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· (2. 41)For the element 1, which is considered isotropic in the plane x 2, x3, it can be deduced:
· 1 (2. 42)2.1.2.2.9 Transversal Poisson Ratio υ 13
By using the same theoretical procedure as for the longitudinal Young’s modulus but usingthe representative element 2. We obtain:
· · (2. 43)2.1.3 Strain-Stress TheoryIn a composite material, the fibers may be oriented in an arbitrary manner. Depending on thearrangement of the fibers, the material may behave differently in different directions.According to their behavior, composites may be characterized as generally anisotropic,monoclinic, orthotropic, transversely isotropic, or isotropic. In the following, we present thestress–strain relationships for these types of materials under linearly elastic conditions.
2.1.3.1 Anisotropic MaterialA material is referred to as generally anisotropic, when there are nosymmetry planes with respect to the alignment of the fibers in the material.
For a linear elastic material, the following generalized Hooke´s lawexpressed in stiffness.
Fig. 2.10: Anisotropic fiber orientation for plane-strain condition.
In the local coordinate system it can be used in the coordinate system (x 1, x 2, x 3), the stress-strain relationships are:
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S S S S 0 0 SS S S 0 0 SS S S 0 0 S0 0 0 S S 00 0 0 S S 0S S S 0 0 S
(2. 48)
Using the Young’s and Shear modulus and Poisson ratios, it can be drawn:
S 0 0
0 0
0 0 0 0 0 00 0 0 0 0 0 (2. 49)
The number of independent terms is now reduced to 13.
2.1.3.3 Orthotropic MaterialA material is referred to as generally orthotropic, when havingthree planes of symmetry with respect to the alignment of thefibers; two-by-two are perpendicular. This configuration enablesnew simplifications in and . In addition to the previousterms equal to zero, the shear strain implied by normal stresses inthe remaining planes of symmetry is also equal to zero.
Fig. 2.12: Orthotropic fiber orientation for plane-strain condition.
0 0 0 0 0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0
(2. 50)
Using the Young’s and Shear modulus and Poisson ratios, it can be drawn:
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0 0 0
0 0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0 (2. 51)
The number of independent terms is now reduced to 9.
2.1.3.4 Transversely Isotropic MaterialA transversely isotropic material has three planes of symmetry and, as such, itis orthotropic. In one of the planes of symmetry the material is treated asisotropic. In this case, perpendicular-to-the-fibers planes can be treated asorthogonal. The compliance matrix is simplified as follows:
Fig. 2.13: Transversely Isotropic fiber orientation for plane-strain condition.
0 0 0 0 0 0 0 0 00 0 0 2 0 00 0 0 0 00 0 0 0 0
(2. 52)
Using the Young’s and Shear modulus and Poisson ratios, it can be drawn:
0 0 0 0 0 0 0 0 00 0 0 0 0
0 0 0 0
00 0 0 0 0
(2. 53)
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The following equation can also be deduced:
Q Q Q Q 0 0 0Q Q Q 0 0 0Q Q Q 0 0 00 0 0 0 00 0 0 0 Q 00 0 0 0 0 Q (2. 54)
The number of independent terms is now reduced to 5.
2.1.3.5 Isotropic MaterialThe isotropic material has no preferred directions, so every plane is a plane ofsymmetry. When a composite is built up of many plies equally oriented, this
behavior can be assumed. For an isotropic material, the coordinate system chosenis indifferent and and present their simplest form:
Fig. 2.14: Isotropic fiber orientation for plane-strain condition.
S S S S 0 0 0S S S 0 0 0S S S 0 0 00 0 0 2S S 0 00 0 0 0 2S S 00 0 0 0 0 2S S
(2. 55)
what can be simplified as:
S 0 0 0 0 0 0 0 0 00 0 0 0 00 0 0 0 0
0 0 0 0 0
(2. 56)
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In this case, there are still 12 non-zero terms, but only 2 independents, σ and ε. And also:
Q Q Q Q 0 0 0Q Q Q 0 0 0Q Q Q 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0 (2. 57)
2.1.4 Mechanics of a Ply
Ply or lamina is the simplest element of a composite material, an elementary layer ofunidirectional fibers in a matrix formed while a unidirectional tape impregnated with resin is
placed on the surface of the tool providing the shape of a composite part.
2.1.4.1 Plane Stress ConditionA ply can be considered as a thin plate, approximating the stresses in the lamina with a plain-stress state. It is convenient to work in the local coordinate system, to take advantage from the
possible planes of symmetry. In case of plain-stress condition, the normal-to-the-plane stressand the out-of-the-plane stresses are equal to zero:
0 0 0 (2. 58)Therefore, by using the equation A.44:
·
(2. 59)
The same kind of relationship with compliance matrix can be drawn:
· (2. 60)
Still with:
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(2. 61)
Then, being in the local coordinate system, according to the symmetries of the material, thematrices can be simplified referring to the expressions given in section 2.1.3. The stress andstrain tensors, σ and ε respectively, have been expressed in the local coordinate system. Thenext step is to transform these matrices and tensors in the global coordinate system.
2.1.4.2 Tensor and Matrix TransformationFor the purpose of getting the expression of [Q] and [S] expressed in the Cartesian coordinatesystem (x, y, z), base transformation matrices are used.
Fig. 2.15: Coordinate Base Transformation
For in-plane stress condition, this transformation is simple because the two coordinatesystems have usually the same normal and a rotation of ϴº. By noting with [T σ] and [T ε] thetransformation matrices for stresses and strains respectively from the stress and strain tensorsexpressed in the Cartesian coordinate system σ ’ and ε ’ to σ and ε , it can be expressed in theform:
′ ·
′ · (2. 62)
Where the transformation matrices are:
2 2 2 2 (2. 63)
with the notation:
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cos sin (2. 64)
Using the matrices of the equation 2.63, the expression of [Q] and [S] in the Cartesiancoordinate system (x, y, z) can be deduced ¡, where they will respectively be notedas and S:
(2. 65)
̅ (2. 66)2.1.5 Mechanics of LaminatesThe behavior of a laminate is characterized by three stiffness matrices [A], [B], and [D]. Theexpression of these matrices will be determined in the following pages by considering flatlaminates undergoing small deformations. The hypotheses used are hereafter presented.
2.1.5.1 HypothesesThe analysis will be made here in the frame of flat laminates subjected to small deformationsand relies on the laminate plate theory. Then, the following assumptions can be made [8]:
The strain varies linearly across the laminate.
The out-of-plane shear deformations are negligible.
The out-of -plane stress σz and the shear stresses τxz and τyz are small compared to thein-plane normal σx, σy and the shear stress τxy.
By using these assumptions, it is possible to consider plane-stress conditions. Thusrelationships which were determined in section 2.1.4 can be used here. Similarly to theclassical plate theory, a referred plane is considered, that corresponds to the plane (xy). Thenit is convenient to take the mid-plane of the laminate, shown in Figure 2.16 (left). In this planethe stresses are:
ε ε γ (2. 67)Where u and v are respectively the displacements along the x and y directions and thesuperscript º refers to the reference plane.
The Kirchhoff hypothesis, according to which normals to the reference plane remain straightunder deformations, is here used. This is illustrated in Figure 2.16. The angle of rotation ofthe normal to the reference plane χ and χcan be determined:
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χ χ (2. 68)where wº designates the out-of-plane displacement of the reference plane. The totaldisplacement of the reference plane along the x and y directions can be written as follows:
u u z·χ u z· v v z·χ v z· (2. 69)The strains of the laminate are defined as:
ε ε γ (2.70)By introducing the equation 2.69 into the equation 2.70, it is obtained:
ε z· (2. 71)ε z· (2. 72)
γ 2z· (2. 73)Thus, the expression in a matrix form can be written as follows:
· (2. 74)Where , and are the curvature of the reference plane of the plate.
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Fig. 2.16: Laminate under plane-stress conditions (left) and deformation of the plane in the x-z plane(right).
2.1.5.2 Stiffness Matrices of Thin LaminatesThe in-plane forces N and moments M acting on a very small element of the laminate aregiven by the following relationships:
σ (2. 75)
zσ (2. 76)Where the terms h b and h t refer to the distances from the reference plane to the surfaces of the
plate, as is shown in Figure. Furthermore, the following relation in (x, y, z) for a plane isalready known:
· (2. 77)With , the stiffness matrix in the Cartesian coordinate system (x, y, z), given in Equation2.65, where is expressed as a function of [Q] according to the section 2.1.5. By substitutingthe equations 2.74 and 2.78 into 2.75 and 2.76, it is obtained:
(2. 78)
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(2. 79)
We can simplify the previous equations using the stiffness matrices of the laminate, asfollows,
A B z D z (2. 80)where [A], [B] and [D] are the stiffness matrices of the laminate and the stiffness matrixof the ply located at the considered altitude. As is constant in a given lamina, theseintegrals can be written as a summation as follows:
A ∑ ·z z , 1,2,6 (2. 81)
B ∑ ·z z
, 1,2,6 (2. 82)
D ∑ ·z z , 1,2,6 (2. 83)Where N is the total number of plies in the laminate, z k and z k-1 are z-coordinates of the twosurfaces of the ply number k and are the elements of the stiffness matrix in the
Cartesian coordinate system (x, y, z) of the ply number k. Then, the equations 2.78 and 2.79can be merged in the following expression:
· (2. 84)
That represents the generic behavior law for a laminate. Being A ij the in-plane stiffnesses thatrelate the in-plane forces N x, N y, and N xy to the in-plane deformations , and . The D ij
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are the bending stiffnesses that relate the moments M x, M y, and M xy to the curvatures , and . Finally the B ij, are the out-of-plane coupling stiffnesses that relate the in-plane forcesto the curvatures and the moments to the in-plane deformations.2.2 Composite Failure TheoryThe failure in a mechanically fastened joint occurs as a combined and inter-dependent failureof each individual component that is not only the joined surface (plate, laminate…), but alsothe joining element (bolt or rivet). This makes prediction very difficult due to all the
parameters that have to be taken into account. To estimate the failure of joints in numericalmodels separate models are used for the fastener and the composite. As the fastener is usuallymade of metal the failure prediction is relatively easy and relies on the ultimate strength orultimate elongation of the fastener. But for laminates the failure prediction is very complex
due to the characteristics of the material. Failure can occur at different levels, in the fibers, inthe matrix or between the layers, and it occurs progressively.
The failure initiation prediction begins by predicting the failure of the elementary laminacomposing the laminate, based on failure criteria. Then the parameters are adjusted accordingto the results of the ply analysis and validated against physical material tests.
2.2.1 Failure of a Lamina
2.2.1.1 Failure Modes of a Laminate
As shown in Figure 2.18, there are four main failure modes of a laminate: microbuckling,fiber breakage, matrix cracking and delamination. The first three are microscopic failures, andthe last is a macroscopic one.
Fig. 2.17: Typical failure modes of composites.
2.2.1.1.1 MicrobucklingMicrobuckling or local fiber buckling reduces the compressive stiffness and strength of thelaminate. Microbuckling does not necessarily lead to immediate failure because thesurrounding matrix supports the fibers. The properties of the fibers and the matrix greatly
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affect the onset and magnitude of fiber buckling and the resulting losses in the compressive properties of the laminate.
2.2.1.1.2 Fiber BreakageFiber breakage refers to the failure of the fiber under a longitudinal tensile load. In order toconfront tensile loading composite materials have fibers. Advanced composites are commonlycomposed of highly elastic fibers with high tensile strength. Then if the fibers fail, the load isredistributed by the matrix to the surrounding fibers. The matrix acts as a bridge, transmittingthe load across the gap created by the breakage. Even with a certain amount of broken fibers,the laminate should be able to confront the loading.
2.2.1.1.3 Matrix CrackingMatrix cracking may occur under any loading. This can be a serious issue if several fibers arealready broken. The laminate could as well be weakened due to an increase in the moistureabsorption, reducing the stiffness of the laminate and the crack could be propagated into theinterface between adjacent layers, initiating delamination.
2.2.1.1.4 DelaminationThe delamination consists in the separation of two adjacent layers. It can occur because ofmanufacturing defects, edge defect, and impact by foreign objects. This failure is not easy to
predict and detect by a visual inspection because it often occurs within the laminate. Theresult is a reduction in the bending stiffness and strength, as well as the load carrying capacityof the laminate under compression.
To predict these phenomena theoretically, failure criteria are used.
2.2.1.2 Failure CriteriaFailure criteria are analytical functions aiming at predicting the failure onset of a lamina. They
are applied successfully to every ply constituting the laminate to determine the initiation ofthe failure. Failure criteria have usually the following form:
, , , , , , , ,⋯ 1 1 (2. 85)
In the expression, the stresses are the stresses in the ply and F 1, F 2… are the strength
parameter of the ply for different kind of loading.
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2.2.2 Maximum Value Failure TheoriesMaximum value failure theories are divided in two categories, one using stressconsiderations and other using strain considerations. These criteria belong to a structural
type and are based on the assumption that there can be three possible modes of failurecaused by stresses σ1, σ2, τ12 or strains ε1, ε2, γ12 when they reach the correspondingultimate values. This seems like a direct approach to evaluate the failure of a material.
2.2.2.1 Maximum Stress TheoryThis theory predicts the failure of the ply when any component of the stress tensor exceedsthe strength of the material for this solicitation, that means in a compressive, tensile or shearstate. The direction 1 is thus along the fiber direction, the direction 2 in the lamina plate,normal to the first one and the direction 3 normal to the lamina plane. The failure thus occurs
when the following relation is verified
, , , , , , , , 1 (2. 86)where X t, Y t, and Z t are the tensile strengths and X c, Y c, and Z c are the compressive strengthsof the ply in the directions 1, 2 and 3 respectively and S ij are the shear failure strengths in the
plane (ij).
2.2.2.2 Maximum Strain TheoryUsing the same scheme with E 1t, E 2t, and E 3t are the tensile strengths and E 1c, E 2c, and E 3c arethe compressive strengths of the ply in the directions 1, 2 and 3 respectively and E ij are theshear failure strengths in the plane (ij), the following equation can be drawn:
, , , , , , , , 1 (2. 87)2.2.3 Interactive Failure TheoriesInteractive criteria are criteria predicting the failure load by using polynomial equationsinvolving stress components. As all the loads are taken into account at the same time, loads inany direction can influence the failure in every other direction. The two common interactivefailure theories are the Tsai-Wu and Tsai-Hill.
2.2.3.1 Tsai-Wu Theory
The Tsai-Wu theory is a tensor polynomial theory predicting the failure onset of a ply usingthe following second-order-tensor polynomial equation:
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∑ ∑ ∑ 1 , 1,2,⋯,6 (2. 88)The coefficient F i and F ij are the strength parameters, functions of the tensile, compressive orshear strengths of the material.
For example, in case of plane stress state, the equation 2.74 can be simplified and the failureis initiated when:
2 1 (2. 89)In which the strengths are given by:
| | | | | | | | | | | | (2. 90)
Where like previously, X t, Y t and Z t are respectively the tensile strengths of the ply in the
directions 1 and 2, X c and Y c are the compressive strengths in the directions 1 and 2, and S 12 is the shear failure strength in the plane (12). One can notice that the parameter F 12 is notconsidered. This parameter is usually determined experimentally by a biaxial test althoughsome formulas exist to try to predict it.
Tsai and Wu originally insisted on the need of maintaining closed, ellipsoidal failure surfacesfor all stress states. They proposed stability conditions on that purpose like the inequality:
0 (2. 91)The summation is not implied by repeated indices. In the case of plane-stress conditions in the
plane (12), thus:
≥0 (2. 92)
The equation 2.78 is the equation of an ellipse. The failure surface is illustrated in Figure 2.19for F 0. One can notice in equation D.8 the influence of F 12 which entails the rotation ofthe failure surface.
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2.2.3.2 Tsai-Hill TheoryTsai-Hill theory is the application of Hill´s anisotropic plasticity theory to the failure ofheterogeneous, anisotropic materials. In plane-stress state in the plane (12) the failure occurs
when:
′ 2 ′ ′ ′ 1 (2. 93)Where the strength parameters are:
′
′
′
′ (2. 94)
With X, Y and Z equal to X t, Y t and Z t or X c and Y c depending on the sign of σ1and σ2 andS12 of the shear strength in the plane (12).
Note that in the case of plane stress in 12, for a transversely isotropic laminate ( ), so′ can be considered. This is the quadratic criterion.
All these interactive theories brought an enhancement compared with the maximum value
theories by considering the interaction between the stresses. Composite materials areheterogeneous and furthermore anisotropic materials so failure does not occur the same wayin the same direction, something that is not accounted for these theories.
Fig. 2.18: Failure surfaces for some failure criteria. [9]
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2.2.4 Mechanical Failure TheoriesThe mechanical failure criteria are nowadays widely used because they enable to predict thefailure of the lamina by identifying the failure mode as well as which loading state is
responsible for this failure.
2.2.4.1 Hashin-Rotem CriterionHashin-Rotem failure criterion is a 2D criterion, one of the firsts to predict the failure mode incomposite materials. This criterion is applied assuming plane stress conditions in the plane(12). The equations are as follows:
Tensile fiber failure mode:
1 (2. 95)Compressible fiber failure mode:
1 (2. 96)Tensile matrix failure mode:
1 (2. 97)Compressible matrix failure mode:
1 (2. 98)Where X t and Y t are respectively the tensile strengths in the fiber direction and in thetransversal direction, X c and Y c are the compressive strengths in the fiber direction and in thetransversal direction and S 12 is the shear failure strength in the plane (12). It can be seen thatthe Hashin-Rotem criterion is a further development to the Tsai-Hill criterion as an enhancedfailure criteria.
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2.2.4.2 Hashin CriterionThe Hashin criterion comes from the Hashin-Rotem criterion. Indeed, this is an extension ofthe previous criterion to three dimensional problems. Now the equations are as follows:
Tensile fiber failure mode:
1or 1 (2. 99)Compressive fiber failure mode:
1 (2. 100)Tensile matrix failure mode 0:
1 (2. 101)Compressive matrix failure mode 0:
| | 1 1 (2. 102)
where it is assumed that and where S23 is the shear strength in the plane 23. For the
2D case, the failure criteria can be obtained easily by getting rid of every values accountingfor the direction 3.
2.2.5 Delamination Initiation PredictionMost of the delamination onset criteria used are stress based criteria, some hereafter
presented. Relying on the same principle as the other criteria, Pearce [10] lists in his paperssome stress based criteria; it is based on the comparison to allowable stress values.
The first natural criterion is:
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Damage progression models are used in order to take into account the progressive material properties degradations. These models alter the stiffness or compliance coefficients ruling theelastic constituting relations in a laminate.
There are two main types of degradation models. On one hand, the models are based on ply-discounting material degradation approach. On the other hand, the models are based on acontinuum-damage-mechanics approach using internal state variables.
2.2.6.1 Ply-Discounting ApproachThe ply-discounting approach is a common approach for the failure analysis. It is taken theconsideration that the laminate behaves as a homogeneous material. The laminate plate theoryis used to calculate the stresses and strains in each ply. Then it is analyzed with a laminafailure criterion in order to determine which ply fails first and possibly the mode of failure. A
stiffness reduction model is then applied to decrease the stiffness of the laminate due to this ply failure.
The stiffness reduction model relies on the application of decreasing coefficient to the elasticmaterial stiffness coefficients affected by the ply failure. There are several stiffness-reductionmodels like the parallel springs, distinct for the fibers and the matrix or the incrementalstiffness reduction mode [17].
Due to its easy implementation, this approach is widely used. Using this method, the predictions can however either under- or over- predict the stiffness decrease because of matrixdamage depending on the stacking sequence.
2.2.6.2 Continuum-Damage-Mechanics ApproachContinuum-damage-mechanics (CDM) models generally describe the internal damage in thematerial by defining one or more internal state variables [17]. The CMD models use similardamage law relationships as the elastic law relationships to describe the behavior of thelaminate, except for the coefficients of the stiffness or compliance matrices, which depend ona variable describing the damage state of the material (noted usually as ω). These models canalso be used to predict the delamination propagation and can be implemented through thefinite element analysis.
2.3 Composite Bolted Joints in Aircraft Structures
2.3.1 Mechanically Fastened Joints
2.3.1.1 GeneralitiesFastening consists of connecting two distinct parts together in order to create an assembly or astructure capable of sustaining a certain loading. The joint created has to be able to transferthe loads between the different parts. In a structure, a joint is the weaker area and mostfailures initiate around the joints. Because of this, joints are sometimes eliminated by
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integrating the structure. As used by K. Mazumdar [12], joints have the followingdisadvantages:
A joint is a source of stress concentration. It creates discontinuity in the load transfer.
The creation of a joint is a labor-intensive process; a special procedure is followed tomake the joint.
Joints add manufacturing time and cost to the structure.
There are several techniques for joining components of fiber-reinforced composites,mechanical fastening and adhesive bonding and fusion or melting.
Mechanical joining is most widely used in joining metal components. Examples ofmechanical joints are bolting, riveting, screw and pin joints. Similar to the mechanical jointsof metal components, composite components are also joined using metallic bolts, pins, and
screws; except in a few cases where RFI shielding and electrical insulation are required,composite fasteners are used.
The main advantages of mechanical fastening are:
No abnormal inspection problems
Disassembly possible without component damage
No surface preparation of components required
The main disadvantages of mechanical fastening are:
Holes cause unavoidable stress concentration
Can incur a large weight penalty
For most mechanical joints, an overlap is required in two mating members and a hole iscreated at the overlap so that bolts or rivets can be inserted. When screws are used forfastening purposes, mostly metal inserts are used in the composites, the reason being that thethreads created in the composites are not strong in shear and therefore metal inserts are used.In bolted joints, nuts, bolts and washers are used to create the joint. In riveting, metal rivetsare used. Bolted joints can be single lap joints, double lap joints or butt joints.
2.3.1.2 Failure Modes of Mechanically Fastened JointsExperimentally determined behavior of both small and full-size joints is additionally requiredto complement the theoretical work. Knowledge of failure modes at macroscopic andmicroscopic levels is essential, if theoretical models are to be refined. One of the problemswith composites is the very large number of material that can be evolved by changing fiber,matrix, lay-up, stacking sequence, etc. Taking into account the huge cost of testing all
possible combinations, as far as a general predictive method, which does not exist until date,the detailed three-dimensional analyses show encouraging correlation between stress
distribution and experimentally observed failure modes.
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Fig. 2.19: Modes of failure for bolted joints in advanced composites.
2.3.1.2.1 Tension FailureIt is also known as Net section failure, the tensile load required to fail a laminate through asection at which holes occur (net section) is less than at a section at which there are no holes(gross section), as is shown in Figure 2.16. The failure stresses at these sections can berepresented by
(2. 108)And
(2. 109)where P is the critical load, W the width at the net section of the joint, N the number of holesof D diameter at that section, and t the joint thickness. This failure is one of the mostdangerous modes because it is sudden and occurs without any early warning.
2.3.1.2.2 Shear FailureThe shear strength is the interlaminar shear strength obtained using the well-known short
beam shear test technique [13]. The shear strength, as for isotropic materials, is given as
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(2. 110)where E is the distance (parallel to the load) between the center of the hole and the free edge,usually known as end distance. The joint shear strength of unidirectional composites, whenloaded in the direction of the fibers, is low in comparison with the in-plane shear strength.This mode is common for highly orthotropic laminates.
2.3.1.2.3 Bearing FailureThe bearing failure is a more progressive failure mode that does not entail a maximumindicating the beginning of the bearing failure, but which is then still able to carry the loaduntil the complete failure of the joint in another failure mode like pull-through or shear-out.
For practical purposes, the bearing strength of a composite material is usually expressed as anaverage stress acting uniformly over the cross-sectional area of the hole, so that
(2. 111)The point of failure will depend upon the fiber orientations and the proportion of eachorientation contained in the laminate. It has also been shown that laminates containing some
±45º and/or 90º plies perform well under bolt bearing conditions although this is contrary towhat might be expected from knowledge of the compressive strength behavior of ±45º and90º laminates, due to the lateral constraint provided by clamping action of the bolt.
2.3.1.2.4 Cleavage FailureCleavage failure occurs only in 0 ° lay-ups which contain low proportions of non-axialfibers. Failure initiates in a single shear mode, this is then followed by the failure of the netsection on one side of the laminate. For most applications, lay-ups containing fiber orientationthat are susceptible to this type of failure would not be used. [14]
2.3.1.2.5 Pull-out FailureA composite bolted joint in single lap-shear configuration undergoes a very complex failureand damage mechanism. In most cases, the first failure load is reached with the onset of inter-laminar cracks, leading to bearing failure and elongation of the hole, and finally a pull-outfailure of the bolt.
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2.3.2 Design Considerations: Material Parameters andJoint Geometry
2.3.2.1 Fiber OrientationThe influence of the fiber orientation and the proportion of fibers at the various orientationscan best be demonstrated by looking at single bolted joint in a particular family of laminateorientations. As has been shown by Collings [15], the best overall joint performance forreinforced composite structures is achieved by using a combination of 0º and ±45º plies,which strongly influences both the material property and the joint behavior.
2.3.2.2 Stacking SequenceThe stacking sequence has a significant impact on the joint behavior. Most of the time, the
preferred configuration for composite plies is the quasi-isotropic configuration composed of0º, ±45º, 90º fibers [16]. This kind of composite material is indeed more likely to be able tosustain the complex loading in a joint. The non-zero degree fibers carry the load around thehole to prevent shear-type failures, whereas the 0º fibers carry out the primary bearing loadand tension. The stacking sequence has to be chosen according to the desired failure mode.Park [17] studied the influence of the stacking sequence for different quasi-isotropiccomposite laminates 90/ 45/0 , 0/ 45/90 ,90//0 45 with andwithout clamping, pointing out that in both cases, the trend of variations for the ultimate anddelamination bearing strength of bolted joints for the same stacking sequence were the same,
but clamping allowed to reach significantly higher strength of the laminate.
2.3.2.3 Joint DimensionsThe geometry has also a critical influence on the failure of the joint. The effects of specimengeometry on joint life were eva1uated for the 50/40/10 and 19/76/5 lay-ups using specimenswith different edge distances and widths. Fatigue strength was not changed for the 50/40/10lay-ups. For the matrix-dominant 19/76/5 lay-up, fatigue strength was found to increase whenthe w/d ratio was increased from 3 to 4. Further, a change of failure mode from bearing to netsection occurred when w/d ratio was reduced from 4 to 3.
2.3.3 Design Considerations: Fastener Parameters
2.3.3.1 Clamping TorqueThe procedure for installation of fasteners in composite structures can produce unacceptabledamage. Before the installation of fasteners, holes must be cleanly drilled through thecomposite structure. This is not an easy task because composites are prone to fraying withinthe hole and to splintering on the exit side of the hole as a result of the drillingoperation; consequently, special drill bits and procedure different from those used in
metal drilling, are required in order to avoid introducing any local weakening of thecomposite structure. While the fastener is being installed into the hole, its rotation isnot recommended because this would lead to breaking and lifting of fibers from the
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surface. Also, axial misalignment of fastener in the hole could lead to damage. For example,misalignment of a blind fastener during installation could cause the blind head to dig into thesurface of the hidden side, crushing or delaminating the composite skin. Severalmanufacturers have introduced fastener designs which have significantly facilitated theirinstallation in composite structures.
2.3.3.2 FrictionThe effect of friction is commonly ignored in the analytical work published in the literature.Friction between plate surfaces can, however, significantly affect joint bolt load distribution.Experimental work by Wittmeyer and Smode [18] indicates that the clamp-up force resultingfrom bolt tightening relieves the joint load transmitted by fastener shear. Significant frettingwas observed in the region of failure. This observation led to the belief that failure was theresult of the propagation of cracks which were initiated by a fretting mechanism. Incomposites, this contact problem in the faying plate surfaces is further complicated by the factthat the behavior of friction and wear is a function of varying fiber orientations with respect tothe sliding direction.
2.3.3.3 Bolt/Hole ClearanceThe drilling is a critical task in composite materials because, drilling defects can produce asignificant effect on the strength of the joint, inducing additional damage to the material,actually, a poor manufactured hole can produce the 60% of all part rejections.
Fig. 2.20: Schematics of delaminations caused by drilling (a) upon entry, and (b) upon exit. [19]
Drilling entails the weakening of the material by triggering a discontinuity in it but moreover,
delamination in the outer layers can occur during manufacturing process, as it is shown inFigure 2.17.
2.4 Finite Element ModelingFinite element methods are taking an increasing role in the analysis and design of complexstructures. With the development of many finite element codes and the increase ofcomputational power, it becomes indeed easier to create a good model of an ARCAN test andto simulate it. It also appears as a way of saving time by reducing or even sometimes, forreally accurate model, replacing to a great extent experiments. It enables, as a result, to savetime and cost of developments. This was precisely the purpose of the already mentionedEuropean program BOJCAS [20] for bolted joints. In the frame of this project, researchers
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tried to get a better understanding of the behavior and the mechanisms occurring in bolted joints by using of course experiments and also using finite elements analysis. From theirwork, they have been able to develop a complete tool for the design of bolted joint finiteelements models called Abaqus [21].
In this section, after having given the main equations of the finite elements method usedhereafter, we will give a short overview of some works made in bolted joints modeling andwe will finally describe the main damage modeling method in finite element analysis.
2.4.1 Finite Element Theory
2.4.1.1 Hypothesis of the Finite Element AnalysisMechanically fastened joints made of reinforced composite material have many specificities
that make its analysis complicated. Three-dimensional and singular phenomena have to bemodeled or determined to obtain a realistic model of the joint. Among these, the boltclamping force and thus the bolt pretention, the bearing stresses, the stress repartition at the
bolt hole interface, the delamination or also the secondary bending need to be mentioned, allthese mechanisms interacting.
Other parameters like contact and friction oblige to take into account non-linearity effects. Asmentioned section 2.2, the bolt-hole contact, the washer-composite contact or the contact
between the composite plates can greatly influence the stress repartition.
The properties of these contact zones, like the friction definition or note, have also to be
carefully addressed. It will be specified as it is pa