module 10 - nptelnptel.ac.in/courses/webcourse-contents/iisc-bang/composite... · learning units of...
TRANSCRIPT
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Module 10
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M10.3 Environmental Effects on Composites
M10.2 Joining of Composites
M10.1 Testing of Composites
Learning Units of Module 10
M10.4 Recycling of Composites
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Mechanical Testing of Composites
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Tensile Testing
Figure: Typical tensile composite test specimen (all
dimensions in mm)
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Compressive Testing
Figure: Celanese
compressive fixture and
specimen (all dimensions in
mm)
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Intra-laminar Shear Testing
Figure: Schematic
representation of the
asymmetric four-point
bend shear fixture
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Inter-laminar Shear Testing
Figure: The short beam shear test
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(a) Mode-I Fracture Toughness
Figure: An in situ Double Cantilever
Beam (DCB) test
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(b) Mode-II Fracture Toughness
Figure: An in situ End
Notch Flexure (ENF) test
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Standard Adhesion Testing Techniques For Non-fibrous Materials
• PEEL TEST
Quantitative
Reproducible
FE analyses exist
Mode I
Example: exposure of teflon to 500 eVAr+ sputtering
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• (NANO)SCRATCH TEST
Quantitative
Reproducible
Mode II
Diamond tip
(above a critical load)
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• FULL TEST
Quantitative
Not reproducible
Interpretation is problematic
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Fiber-matrix interfacial adhesion
• Two general methodologies:– Indirect (or macromechanical) testing - focuses on
the collective behavior of fibers in a polymer matrix [interfacial strength is interpreted via simplistic approximations; fast but questionable data collection]
– Direct (or micromechanical) testing – probes interfacial behavior of individual fibers interacting with a polymer matrix [more fundamental information; variability within and between techniques; issue of relevance to real-life composites (scaling-up)]
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Indirect (or macro-mechanical) testing
1. The transverse tensile test
2. The short-beam shear (or interlaminar shear strength -ILSS) test
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3. Asymetrical 4-point (Iosipescu) bend test
(And other macroscopic tests… )
CONCLUSION – They are simple to perform, but yield ambiguous, inaccurate data.
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Direct (or micromechanical) testing
1. The single-fiber pull-out test (several versions)
F F
~ 1 mm ~ 1 µm
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2. The single-fiber microcompression (or MIT) test
Tricky test, difficult to perform, fiber-matrix interfacial fracture is often only partial due to slight off-axis moments, loading must be exactly parallel to the fiber axis.
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3. The single-fiber fragmentation testSingle fiber embedded in matrix
Fragmented fiber
Pre-conditions for successful test: (a) failure strain of the fiber is much smaller than that of matrix; (b) Interfacial bonding must be fair to good.
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Importance of adhesive strength
Simple example: Unidirectional carbon/epoxy composite
The presence of interfaces, and their adhesive strength, is critical for various physical properties of composites
Weak interface leads to tough composites
Strong interface leads to brittle composites
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The Classification of Technological Features
Joints
Combined adhesive-mechanical joints
Adhesive-riveted
Adhesive-boltedAdhesive-threadedAdhesive-mechainal
Moulded-on
Spot mechanical joints
Riveted
Bolted
Theaded
Pin-bolted
Adhesive
Welded
Adhesive-rubber
Moulded-on
Continuous adhesive joints
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The Requirement of the Joint Design
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Advantage and disadvantages of bonded fastened joints
Limits to thickness that can be joined with simple joint configuration.Inspection difficulty.Prone to environmental degradation.Requires high level of process control.Sensitive to peel and through-thickness stresses.Residual stress problems when joining dissimilar metals.Disassembly is impossible without component damage.Requires surface preparation.
No stress concentrationin adherents.Stiff connection.Excellent fatigueproperties.No fretting problems.Sealed against corrosion.Damage tolerant.Small weight penalties.Fewer pieces, lower weight, good load distribution.
Bonded JointsDisadvantagesAdvantages
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Advantage & Disadvantages of Mechanically fastened joints
Considerable stress concentration.Relatively compliant connection.Relatively poor fatigue properties.Hole formation may cause damage to composite.Prone to fretting.Prone to corrosion.Large weight penalty.
Positive connection.No thickness limitations.Simple process.Simple inspection procedure.Simple joint configuration.Not environmentally sensitive.Provides through-thickness
reinforcement and not sensitive to peel stresses.
No residual stress problems.No surface preparation of
component required.Disassembly possible without
component damage.High tolerance to repeated loads.
Bonded JointsDisadvantagesAdvantages
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Design Considerations
The behaviour of mechanically fastened joints is influenced by:Material parameters.Configurational parameters.Fastener parameters, e.g.
• Fastener type (screw, bolt, rivet).• Fastener size.• Clamping force.• Washer size.• Hole size.• Tolerance.
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The Issues & Design Approaches to each Issue
Specially designed blind rivets.Verify joint strength with tests.
Damage induced by installation of blind fasteners and drive rivets
Avoid, if possible.Increased laminate thickness (locally).
Countersunk head
Larger fastener head.Washers (one or both sides).Limit on installation torque.
Preload relaxation
Larger fastener diameter.Insert (bushing).Increased laminate thickness (locally).
High local stresses
Closely controlled manufacturing operations.Inspection of drilled holes.
Drilling damage.ApproachIssue
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Failure Criteria: Allowable StressesThe allowable stresses in each of these modes are
a function of the following:Geometry of the joint including the hole size, plate width and distance of the hole from the edge of the plate.The clamping area and pressure.The fibre orientations ply sequence.The moisture content and exposure temperature.The nature of stressing, e.g. tension or compression, sustained or cyclic and any out-of-plane loads causing bending.
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Failure Modes
It is recommended that highly loaded structural joints be designed to fail in a bearing mode to avoid the catastrophic failures associated with net section failures.The failure stresses will depend on the degree of anisotropy at the hole and hence on the local fibre orientation.
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The Failure Modes
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Fastener Failure Modes
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Fastener Selection
Fastener requirements for joining composite structures differ from those joining metallic structures.Fastener selection considerations for joining composites include corrosion compatibility, fastener material, strength, stiffness, head configuration, importance of clamp-up, lightning protection, etc.
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Fastener Selection considerations
Corrosion Compatibility: Neither fibre glass nor aramid fibre reinforced composites cause corrosion problems when used with most fastener materials. Composites reinforced with carbon fibres are quite cathodic when used with materials such as aluminium or cadmium. Presence of galvanic corrosion between metallic fasteners and non-metallic composite laminates has eliminated several commonly used alloys from consideration. Conventional plating materials are also not being used because of compatibility problems. The choice of fastener materials for composite joints has been limited to those alloys, which do not produce galvanic reactions. The practice followed in aircraft industry is to coat the fasteners with anti-corrosion agent to alleviate galvanic corrosion.
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Continued…..Fastener material: The materials currently used in design include alloys of titanium and certain corrosion resistant stainless steels with aluminium being eliminated. The choice is obviously governed by the make up of the composite materials being joined, weight, cost, and operational environment. Titanium alloy Ti-6Al-4V is the most common alloy used with carbon fibre reinforced composite structures.Bolt bending: Due to increased inter-laminar shear between the composite plies, bending of the bolt occurs more easily. High modulus and high tensile strength fastener material is desired where bending may occur. Susceptibility of bolt bending in composite structure introduces higher reaction loads on the fastener head, which requires more careful consideration of head configuration. Bending should also be considered in multiple component fasteners such as blind fasteners. A threaded core bolt resists bending much better than a smooth bore pull-type blind fastener as shown in Figure (in next slide).
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Continued…..
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Continued…..Head configuration: Composites are sensitive to high bearing loads than are metals. This means fastener heads should be designed with as much bearing surface area as practicable. The larger area improves pull-through and delamination resistance in composites, while reducing over-turning forces from bolt bending. Countersunk or flush head fasteners are frequently used on exterior surfaces of the aircraft where aerodynamic smoothness is required. Countersunk fasteners for composites include tension head fasteners having the large head depths and shear head fasteners having smaller head depths with head angles ranging from 1000 to 1300 as shown in Figure (in next slide).
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Continued…..
Figure: Countersunk fasteners for composites with different head depths
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Continued…..Countersunk fasteners: tend to bear against the surrounding element more unevenly through the thickness than protruding head fasteners do. Tension head fasteners are generally preferred over shear head fasteners due to greater strength against head pull-through. However, if the joint element is so thin that the countersunk depth is greater than 70% of the element thickness, the tendency towards uneven bearing pressure in tension head fasteners is too great and shear head fasteners are recommended in this case. Caution should be observed in the use of 1300 countersunk head fasteners. Although this type of fastener increases the bearing area of the fastener and permits it to be used in thin laminates, pull-through strength can be adversely affected. Although; close tolerance fit fasteners are desirable for use with composites, interference fit fasteners cannot be used due to potential delamination of plies at the fastener hole.
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Continued…..
Clamp up: When tolerance fit holes are used, high clamp up appears to be beneficial for joint strength and fatigue life. The clamping forces, however, must be spread out over a sufficient area so that the compressive strength of the resin system is not exceeded and the composite crushed.
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Module 10.2
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Adhesive Bonding
Joining process whereby a filler material is used to hold two closely spaced parts together by surface attachment.Filler material (adhesive)
Usually non-metalUsually a polymer
CuringProcess (usually chemical) whereby the adhesive physical properties are changed from a liquid to a solid.
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The criteria for selection of adhesiveThe adhesive must be compatible with the adherends and able to retain its required strength when exposed to in-service stresses and environment.The joint should be designed to ensure failure in one of the adherends rather than failure within the adhesive bond line.Thermal expansion of dissimilar materials must be considered. Due to large thermal expansion difference between graphite composite and aluminium, adhesively bonded joints between these two materials are likely to fail during cool down from elevated temperature cures as a result of the thermal stresses induced by their differential expansion coefficients.Proper joint design should be used avoiding tension, peel or cleavage loading. If peel forces cannot be avoided, a lower modulus adhesive having high peel strength should be used.
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Continued……Surface preparation should be conducted carefully, avoiding contamination of the bond line with moisture, oil, etc.The adhesive should be stored at the recommended temperature.Use of adhesives that evolve volatiles during cure should be avoided.The recommended pressure and proper alignment fixtures should be used. The bonding pressure should be great enough to ensure that -the adherends are in intimate contact with each other.Traveller coupons should always be made for testing.The exposed edge of the bond joint should be protected with an appropriate sealing compound.
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General Properties of some adhesives
AcrylicThermoplastic, quick setting, tough bond at r.t.Tennis racquets, metal parts
EpoxyThermoset, strongest engineering adhesiveMetal, ceramic, rigid plastic parts
CyanoacrylateThermoplastic, touch“Crazy Glue”
Hot MeltThermoplastic, quick setting, easy to applyBonds most anything –Packaging, book binding, metal can joints
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PhenolicThermoset, strong, brittleBrake lining, clutch pads, honeycomb structures
SiliconeThermoset, slow curing, flexible, rubber likeGaskets sealants
Water base AnimalVegetableRubbers
Inexpensive, non-toxicWood, paper, fabricLeather, dry seal envelopes
Conitinued….
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Joint Design
Usually not as strong as welding or brazing jointsDesign principles
Maximize joint contact areaJoints are strongest in shear and or tension so joints should be designed to accommodate thisJoints are weakest in cleavage or peel. Avoid these stresses
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Adhesive bonding DisadvantagesJoints are not as strongAdhesive must be compatible with materials being joinedService temperatures are limitedCleanliness and surface preparation prior to adhesive application are importantCuring times can impose a limit on production ratesInspection of the bonded joint is limited.
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Joints
Bolted and Bonded
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Single Lap Joint
Bonded
Bolted
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Bonded
Bolted
Double Lap Joint
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Bonded
Single Doubler Joint
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Bonded
Double Doubler Joint
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Bonded
Stepped Lap Joint
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Bonded
Scarf Joint
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Bolted
Reinforced-Edge Joint
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Bolted
Shimmed Joint
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Bolted-Bonded Double Lap Joint
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More Examples
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ADHESIVE FAILURE
COHESIVE FAILURE
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• Thinner materials can be joined– Weight Savings– Cost Savings
• Fewer Production Parts• Less Machining• Unskilled Work Force• High strength to weight ratio
Advantages to Bonded Joints
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• Aerodynamic smoothness• Serves as a seal or corrosion barrier• Excellent electrical and thermal insulation• Superior fatigue resistance• Good damping and noise reduction• Can help in CTE mismatches
Continued…
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With Adherends of unequal thickness, the maximumshear stress will occur near the end of the overlap on the side of the thinner part.
Single Lap Joint
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• Adhesive stresses decrease with width, up to about a 4 in width. Then they remain constant.
• One in results can be used conservatively on wider specimens.
• max stresses do not decrease significantly with increasing bond area.
• Stresses decrease with increasing bond thickness
Single Lap Joint
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• Maximum stress increases almost linearly with shear modulus.
• As stiffness of adherends increase, the resistance of the joint to bending increases, and maximum stresses decrease.
• Maximum stresses in the adhesive are insensitive to the Poisson’s ratio of the adherends
Single Lap Joint
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Defined as the axial load divided by the nominal bonded area divided by the strength of the weaker adherend without the joint.
Joint Efficiency
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Small deflection theory not valid in analyzing the joint under load. Some authors suggest using maximum stress theory for adhesive and isotropic adherends and maximum strain theory for composite adherends.
Nonlinear Problem
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Eccentric Loading
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Stress Reduction-Maximum Peel and Shear
• Combination of flexible and stiff bonds• Stiffer lap plates• Tapered plates
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Ineffective Length
Defined as the lap length beyond which an increase in that dimension is ineffective in reducing peak values of peel and shear stresses.
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• More than twice as efficient as single-lap joints• Symmetric, so bending effects are minimized• Practical design rule: Lap length to adhesive
thickness ratios of 30
Double Lap Joints
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Important for repair applications where both sides of structure are not readily available
Single Doubler Joint
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• Smooth joint• Requires careful machining• Scales up to any load • Approaches the ideals of strain compatibility
in the adherends and uniform stress in the adhesive
• Adhesive ductility is less important than in other types of joints
• Provides efficient use of full overlap length• Not bounded by an “ineffective length”
Scarf Joints
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• Strength is not sensitive to number of steps• Light weight• Approximates the strain compatibility of scarf
joints• Scales up to any load • Provides efficient use of full overlap length• Not bounded by an “ineffective length”
Stepped Lap Joints
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Q1 Q2
N1
N2
M2
M1
σu(x) τu(x)
σl(x) τl(x)
General Adherend Element
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Net Tension Failure
Bolted Joint Failure Modes
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Cleavage Tension Failure
Bolted Joint Failure Modes
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Shear Out Failure
Bolted Joint Failure Modes
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Bearing Failure
Bolted Joint Failure Modes
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Bolt Pull Through
Bolted Joint Failure Modes
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Bolt Failure
Bolted Joint Failure Modes
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Environmental Effects on Composites
Composite usage has increased enormously mainly due to the advantages of lightweight, specific strength and stiffness, dimensional stability, tailor-ability of properties such as coefficient of thermal expansion and high thermal conductivity. Environmental effects on these properties may compromise a structure and must be considered during the design process.
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Biological AttackBiological attack on composite materials may consist of fungal growth or marine fouling. Fungal growth does not appear to be as damaging as the wet conditions that promote growth. Fungicide has been mixed in with resins to retard this growth. Even though marine organisms will grow on composite surfaces, mechanical properties do not appear to be affected and the fouling can be removed by scraping. Composites with graphite fibres have been used in medical applications for both internal and external purposes.
Internal composite structures such as artificial joints or plates for bone fracture support must be bio-compatible or the material may degrade over time. External composite designs (such as artificial limbs or orthotic braces) may experience impact damage, flexural and torsional loading during use.
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Fatigue
Fatigue, either through mechanical loads or acoustic vibrations, can cause crack growth or local defect formation. Fatigue designdepends not only on the load but also on the use temperature range and amount of moisture present. Very cold temperatures (below -50°C) may increase the stiffness of some composite materials thereby increasing the susceptibility to fatigue damage.
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Hygrothermal Behavior
Study of environmental effects (moisture and temperature) on composite material properties
and stress-strain behavior.
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ThermalThermaleffects are caused by
temperature
Hygroscopic Hygroscopic effects are caused by moisture.
Hygrothermal Hygrothermal effects are
caused by either temperature or
moisture.
Hygrothermal
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Topics to be Discussed
1. Matrix Dominated Property Degradation2. Stress-Strain Behavior3. Micromechanical Models
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Hygrothermal Degradationof Properties
1. Matrix dominated properties such as stiffness and strength under transverse, off-axis and shear loadings are altered.
2. Increased temperature causes a softening of the polymer matrix.
3. Glass transition effects.
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Glass Transition Temperature
At a certain value of the temperature the matrix
materials transitions from a glass-like behavior to a
rubbery behavior.
GlassyRegion
RubberyRegion
Wet
Dry
Temperature
Stiff
ness
0gTgwT
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Moisture is present in the operational environment in which a composite is manufactured and throughout its useful life.Water acts as a plasticizer when absorbed by the matrix, softening the material and reducing some properties of the laminate.Moisture may also migrate along the fibre-matrix interface thereby affecting the adhesion.Moisture in composites reduces matrix dominated properties such as transverse strength, fracture toughness and impact resistance.
Moisture Effects
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The relation between Moisture content and Glass transition temperature
Lowering of the glass transition temperature may also occur in epoxy and polyimide resins with an increase in absorbed moisture (as shown in Figure). Debonding can occur due to formation of discontinuous bubbles and cracking in the matrix. Mechanical properties can be reduced even further if heat is present or if the composite is under-cured or has a large amount of voids.
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Temperature EffectsTemperature effects on composite materials include cryogenic temperatures, elevated temperatures and thermal cycling between these extremes.Cryogenic temperatures do not appear to affect the mechanical properties of graphite/epoxies or graphite/polyimides significantly.Elevated temperatures for a prolonged period of time can seriously affect the properties of a composite, with even greater effect if moisture is present. Thermal cycling may induce micro-cracking in some composites thereby resulting in reduction of compressive and shear strength.
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93
Temperature Effects
Temperature effects are not limited to the matrix materials. Extended operation at 350°C (660°F) and 450°C (840°F) can cause oxidation of low modulus PAN-based fibres and high modulus PAN- or Pitch-based fibres, respectively. Oxidation resistance can be improved with higher purity fibres. Thermal cycling conditions are common for a number of applications, including aircraft and spacecraft.
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94
The effect of temperature and ageing on stiffness of composite Materials
Loss of stiffness with temperature and ageing is indicated in Figure; Susceptibility to matrix softening is not only dependent on the resin but also the lay-up.
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95
Protection against temperature effects
Protection against temperature effects can be achieved at the design stage itself by:Selection of resin system with high glass transition temperature.Potential degradation taken into account in the analysis and fatigue test.Protection against moisture exposure.
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96
Overheat Conditions
Heat generated by lightning strikes has been known to vaporize matrix resins and create large areas of delamination and fibre fracturing on composite rudders, ailerons, wing and stabilizer tips, nose domes and nacelle cowling. When exposed to hot gases over long periods, polymeric resin binders can become completely destroyed through a process of thermo-oxidation. Preventive methods may consist of application of heat resistant ablative coatings.
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97
Glass Transition Temperature
etemperatur transition glasswet"
etemperatur transition glass dry"
etemperatur transition glass
"T
"T
T
gw
0g
g
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98
Through-thickness Distributionof Temperature and Moisture
Plate
a
a
cT
a
a
cT
Thicknessh
Ta – Ambient Temperatureca – Ambient Moisture
Concentration
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99
Through-thickness Distributionof Temperature and Moisture
T
c
Thicknessh
z
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100
Temperature DistributionFourier Heat Conduction Equation
timedirection- in material of tyconductivi thermal
materialthe of heatspecific materialthe of density
==
==
∂∂
∂∂=
∂∂ρ
tzK
Cρ
zTK
ztTC
z
z
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101
Moisture DistributionFick’s Second Law
direction- in material of ydiffusivit mass zD
zcD
ztc
z
z
=
∂∂
∂∂=
∂∂
Temperature change is orders of magnitude faster than moisture change.
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102
Boundary and Initial Conditions
ionconcentratmoisture initiale temperatur ambient
compositethe of ionconcentratmoisture initialcompositethe ofe temperatur initial
a
a
i
i
ccTT
0thz,0z
ccTT
0tandha0
==
>==
==
≤<<
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103
Series Solution
( )( ) ( )
( )a
m
htDj
j
im
i
cc
eh
zjj
cccc
z
to relatedmaterial ofsurface at ionconcentratmoisture
⎟⎟⎠
⎞⎜⎜⎝
⎛ π+−∞
=⎟⎠⎞
⎜⎝⎛ π+
+π−
=−
−
∑2
2212
0
12sin12
141
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104
Average Concentration
( )( )
( )⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+π−−== ∑∫
∞
=
⎟⎟⎠
⎞⎜⎜⎝
⎛ π+−
1j2
h
tD1j2
im
h
0 1j2e81ccdz)t,z(c
h1c
2z
22
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105
Weight Percent Moisture
( )
( )
conditions ambient withsaturated fully henmoisture w percent weight
moisture percent weight
m
1j2
h
tD1j2
im
i
MM
1j2e81
MMMM
G2
z22
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+π−=
−−
= ∑∞
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ π+−
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106
Hygrothermal Degradation
21
00g
gw
0m TT
TTPPF ⎥
⎦
⎤⎢⎣
⎡
−−
==
Empirical formula for polymer resins
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107
ndegradatiobefore strength) or (stiffness property mechanicalmatrix Reference
ndegradatio after strength) or (stiffness property mechanicalMatrix
ratio retention property mechanicalMatrix
=
==
0P
PFm
Pressure Terms
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108
measured was whichate TemperaturP to ingcorrespond condition wet referenced fore temperatur transition Glass
condition dry referenced fore temperatur transition Glass
whichate Temperatur
00
0
PT
T
TT
gw
g
=
=
==
Temperature Terms
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109
Glass Transition Temperature
( ) 0gr2rgw T0.1M10.0M005.0T +−=
Empirical formula for polymer resins
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110
Properties
m0mmf1f1 EFEE vv +=etc.
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111
Lamina Stress-Strain Behavior
Including Hygrothermal EffectsIncluding Hygrothermal Effects
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112
Free Thermal Strains
(CTE) expansion thermal of tcoefficieni all fore where temperatur initial
etemperatur final)T-(Tchange e temperatur
0
0
===
==∆
=∆α==ε
α 0εT
TT
3,2,1iifT6,5,4iif0
Ti
Ti
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113
Hygroscopic Strains
0 0c
c
3,2,1iifc6,5,4iif0
Mi
Mi
=ε=
=β
=
=
=β==ε
when:conditionReference
(CHE) expansionc hygroscopi of tcoefficien
volume unit in material dry of massvolume unit inmoisture of mass
ionconcentratmoisture
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114
Hygrothermal Strains
⎪⎩
⎪⎨⎧
=
=β+∆α=ε
ε+ε=ε
4,5,6iif
1,2,3iif
0
cT iiHi
Mi
Ti
Hi
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115
Transverse Isotropy
32
32
β=βα=α
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116
Stress-Strain Behavior
cT
S
SS
SS
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
β
β
+∆
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
α
α
+
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
σ
σ
σ
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
γ
ε
ε
0000
0
0
2
1
2
1
12
2
1
66
2221
1211
12
2
1
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117
Stress-Strain Behavior
{ } [ ]{ } { } { }cTS β+∆α+σ=ε
Matrix Notation
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118
Stress-Strain Behavior
{ } [ ] { } { } { }( )cTS 1 β−∆α−ε=σ −
Invert
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119
Unrestrained Hygrothermal Exposure
{ } { } { }cT β+∆α=ε
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120
Completely Restrained Hygrothermal Exposure
{ } [ ]{ } { } { }
{ } [ ] { } { }( )cTS
cTS
β−∆α−=σ
β+∆α+σ==ε
−1
0
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121
Stress-Strain Behavior
cT
SSS
SSS
SSS
xy
y
x
xy
y
x
xy
y
x
662616
262212
161211
xy
y
x
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
β
β
β
+∆
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
α
α
α
+
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
σ
σ
σ
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
γ
ε
ε
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122
Transformation
CTE’s and CHE’stransform like tensor
strains.
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123
Tensor Strain Transformation
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
γε
ε
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−
=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
γε
ε
2
2
2
212
2
1
22
22
22
sccscs
cscs
cssc
xy
y
x
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124
Tensor Strain Transformation
[ ]⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
γε
ε
=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
γε
ε−
2212
2
1
1Txy
y
x
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125
Matrices
[ ]⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−
=−
22
22
22
1 2
2
sccscs
cscs
cssc
T
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126
Matrices
[ ]⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−=22
22
22
2
2
sccscs
cscs
cssc
T
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127
Transformation
[ ]⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
α
α
=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
αα
α−
0
T
2
2
1
1
xy
y
x
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128
( ) θθα−α=α
θα+θα=α
θα+θα=α
sincos2
cossin
sincos
21xy
22
21y
22
21x
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129
1 -by scaled by normalized sCTE' All 1
xy
2
1
21
αα
=αα
Off axis thermal expansion
0
0.5
1
1.5
2
2.5
0 30 60 90
Angle θ
CTE
's axayaxy
xαyαxyα
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130
1 -by scaled by normalized sCTE' All 1
xy
2
1
21
αα
−=αα
Off axis thermal expansion
-1.5-1
-0.50
0.51
1.52
2.53
3.5
0 30 60 90
Angle θ
CTE
's axayaxy
xαyαxyα
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131
1 -by scaled by normalized sCTE' All 1
xy
2
1
51
αα
=αα
Off axis thermal expansion
0
1
2
3
4
5
6
0 30 60 90
Angle θ
CTE
's axayaxy
xαyαxyα
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132
1 -by scaled by normalized sCTE' All 1
xy
2
1
51
αα
−=αα
Off axis thermal expansion
-2-101234567
0 30 60 90
Angle θ
CTE
's axayaxy
xαyαxyα