Presentation on theme: "Chap.8 Mechanical Behavior of Composite"— Presentation transcript:
1 Chap.8 Mechanical Behavior of Composite 8-1. Tensile Strength of Unidirectional Fiber Reinforced CompositeIsostrain Condition : loading parallel to fiber directionFiber & Matrix – elastic caseModulus : works reasonably wellStrength : does not work wellWhy?: intrinsic property (microstructure insensitive): extrinsic property (microstructure sensitive)Factors sensitive on strength of composite- Fabrication condition determining microstructure of matrix- Residual stress- Work hardening of matrix- Phase transformation of constituents
2 Analysis of Tensile Stress and Modulus of Unidirectional FRC Assumption : Fiber : elastic & plasticMatrix : elastic & plasticStress-Strain Curve of FRC - divided into 3 stagesStage I : fiber & matrix - elastic→ Rule of MixturesStrengthModulusStage II : fiber - elastic, matrix - plastic: flow stress of matrix at a given strain
3 Stage III : fiber & matrix – plastic StrengthUTS: ultimate tensile strength of fiber: flow stress of matrix at the fracture strain of fiber
4 Effect of Fiber Volume Fraction on Tensile Strength (Kelly and Davies, 1965)Assumption : Ductile matrix ( ) work hardens.All fibers are identical and uniform. → same UTSIf the fibers are fractured, a work hardenable matrix counterbalances the lossof load-carrying capacity.In order to have composite strengthening from the fibers,UTS of composite UTS of matrix after fiber fractureMinimum Fiber Volume FractionAs , .As , .degree of work hardening
5 In order to be the strength of composite higher than that of monolithic matrix, UTS of pure matrixCritical Fiber Volume FractionAs ↓, ↑.As ↑, ↑.degree of work hardeningNote that always! (∵ )
7 8-2. Compressive Strength of Unidirectional Fiber Reinforced Composites Compression of Fiber Reinforced CompositeFibers - respond as elastic columns in compression.Failure of composite occurs by the buckling of fibers.Buckling occurs when a slender column under compression becomes unstableagainst lateral movement of the central portion.Critical stress corresponding to failure by buckling,where d is diameter, l is length of column.
8 2 Types of Compressive Deformation 1) In-phase Buckling : involves shear deformation of matrix→ predominant at high fiber volume fraction2) Out-of-phase Buckling : involves transverse compression and tension ofmatrix and fiber→ pre-dominant at low fiber volume fractionFactors influencing the compressive strength :Interfacial Bond Strength : poor bonding → easy buckling
9 8-3. Fracture Modes in Composites 1. Single and Multiple FractureGenerally,When more brittle component fractured, the load carried by the brittlecomponent is thrown to the ductile component.If the ductile component cannot bear this additional load → Single FractureIf the ductile component can bear this additional load → Multiple Fracture
10 - predominant at high fiber volume fraction 1) Single Fracture- predominant at high fiber volume fraction- all fibers and matrix are fractured in same plane- condition for single fracturestress beared by fiber additional stress which can be supported by matrixwhere : matrix stress corresponding to the fiber fracture strain2) Multiple Fracture- predominant at low fiber volume fraction- fibers and matrix are fractured in different planes- condition for multiple fracture
11 2. Debonding, Fiber Pullout and Delamination Fracture Fracture Process : crack propagationDiscontinuous Fiber Reinforced Composite( lc : critical length )→ Debond & Pullout Good for toughness→ Fiber Fracture Good for strengthl
12 Fracture of Continuous Fiber Reinforced Composite Fracture of fibers at crack plane or other position depending on the positionof flaw↓Pullout of fibersFor max. fiber strengthening → fiber fracture is desired.For max. fiber toughening → fiber pullout is desired.Analysis of Fiber PulloutAssumption : Single fiber in matrix: fiber radiusl : fiber length in matrix: tensile stress on fiber: interfacial shear strength
13 Force Equilibrium( lc : critical length of fiber )1) Condition for fiber fracture,2) Condition for fiber pullout,
14 Fracture Process of Fiber Reinforced Composites Real fibers - non-uniform properties3 steps of fracture process1) Fracture of fibers at weak points near fracture plane :2) Debonding of fibers :3) Pullout of fibers :Outwater and MurphyLoadDisplacementWPWd
15 Energy Required for Fracture & Debonding elastic strain E. volumeEnergy Required for PulloutLet k : embedded distance of a broken fiber from crack plane: pullout distance at a certain moment: interfacial shear strengthForce to resist the pullout =fiber contact areaTotal energy(work) to pullout a fiber for distance kAverage energy to pullout per fiber(considering all fibers with different k, )
16 Fracture of Discontinuous Fiber Reinforced Composite → pullout Average energy to pullout per fiber with length, lprobability for pullout energy required for pulloutEnergy for Fiber Pullout vs Fiber Length(l)
17 As Wd << WpAdvantage of Composite Material:can obtain strengthening & toughening at the same timeToughening Mechanism in Fiber Reinforced Composite1) Plastic deformation of matrix - metal matrix composite2) Fiber pullout3) Crack deflection (or Delamination) - ceramic matrix compositeCook and Gordon, Stresses distribution near crack tip
18 If > interfacial tensile strength → delamination → crack deflectionDelamination Fracture in Laminate CompositeFatigue → debonding at interfaceFracture → repeated crack initiation & propagation
19 8-4. Statistical Analysis of Fiber Strength Real fiber : nonuniform properties → need statistical approachBrittle fiber (ex. ceramic fibers) - nonuniform strengthDuctile fiber (ex. metal fibers) - relatively uniform strengthStrength of Brittle Fiber→ dependent on the presence of flaws→ dependent on the fiber length : "Size Effect“Weibull Statistical Distribution Function: probability density functionProbability that the fiber strength is between and: statistical parametersL : fiber length
20 Let, : kth moment of statistical distribution function Let, : kth moment of statistical distribution functionMean Strength of FibersStandard Deviation for Strength of FibersSubstitutingwhere : gamma functionCoefficient of Variation
21 As L ↑, ↓. "Size Effect“As ↑, ↑. is less dependent on L.If , spike distribution function (dirac delta function)→ uniform strength independent on LGlass fiberBoron, SiC fibers
22 Strength of Fiber Bundle Bundle strength ≠ Average strength of fiber × n<Assumption : Fibers - same cross-sectional area- same stress-strain curve- different strain-to-fractureLet F(σ) : The probability that a fiber will break before a certain value of isattained.→ Cummulative Strength Distribution FunctionMean Fiber Strength of Bundle※ Mean Fiber Strength of Unit Fiber# of fibers
24 8-5. Failure Criteria of an Orthotropic Lamina Assumption : Fiber reinforced lamina - homogeneous, orthotropicFailure Criterion of Lamina1. Maximum Stress CriterionFailure occurs when any one of the stress components is equal to or greaterthan its ultimate strength.Interaction between stresses is not considered.Failure Conditionwhere : ultimate uniaxial tensile strength in fiber direction (>0): ultimate uniaxial compressive strength in fiber direction (<0): ultimate uniaxial tensile strength in transverse direction: ultimate uniaxial compressive strength in transverse directionS : ultimate planar shear strength
25 ex) If uniaxial tensile stress is given in a direction at an angle with the fiber axis. Failure occurs when,Failure Criterionx12Failure occurs by a criteria, whichis satisfied earlier.
27 2. Maximum Strain Criterion Failure occurs when any one of the strain components is equal to or greaterthan its corresponding allowable strain.Failure Conditionwhere : ultimate tensile strain in fiber direction: ultimate compressive strain in fiber direction: ultimate tensile strain in transverse direction: ultimate compressive strain in transverse direction: ultimate planar shear strain
28 3. Maximum Work Criterion Failure criterion under general stress stateTsai-Hillwhere X1 : ultimate tensile (or compressive) strength in fiber directionX2 : ultimate tensile (or compressive) strength in transverse directionS : ultimate planar shear strengthex) For uniaxial stress , having angle with the fiber axisFailure criterionsubstituting
29 4. Quadratic Interaction Criterion Consider stress interaction effect Tsai-HahnStress Functionstress term 1st interaction termThin Orthotropic Laminai, j = 1, 2, 6 (plane stress): strength parametersFailure occurs when,→ need to know 9 strength parametersFor the shear stress components, the reverse sign of shear stress shouldgive the same criterion.∴
30 Calculation of Strength Parameters by Simple Tests 1) Longitudinal uniaxial tensile and compressive tests,: longitudinal tensile strength: longitudinal compressive strength2) Transverse uniaxial tensile and compressive tests,3) Longitudinal shear test4) In the absence of other data,
33 8-6. Fatigue of Composite Materials Fatigue Failure in Homogeneous Monolithic Materials→ Initiation and growth of a single crack perpendicular to loading axis.Fatigue Failure in Fiber Reinforced Laminate CompositesPile-up of damages - matrix cracking, fiber fracture, fiber/matrix debonding,ply cracking, delamination Crack deflection (or Blunting) Reduction of stress concentrationA variety of subcritical damage mechanisms lead to a highly diffuse damagezone.
34 Constant-stress-amplitude Fatigue Test Damage Accumulation vs CyclesCrack length in homogeneous material - accelerate(∵ increase of stress concentration)Damage (crack density) in composites - accelerate and decelerate(∵ reduction of stress concentration)
35 S-N Curves of Unreinforced Plolysulfone vs Glassf/Polysulfone, Carbonf/Polysulfone Carbon Fibers : higher stiffness & thermal conductivity higher fatigue resistanceS-N Curves of Unidirectional Fiber Reinforced Composites (B/Al, Al2O3/Al, Al2O3/Mg)
36 Fatigue of Particle and Whisker Reinforced Composites For stress-controlled cyclic fatigue or high cycle fatigue, particle or whisker reinforced Al matrix composites show improved fatigue resistance compared toAl alloy, which is attributed to the higher stiffness of the composites.For strain-controlled cyclic fatigue or low cycle fatigue, the composites showlower fatigue resistance compared to Al alloy, which is attributed to the lower ductility of the composites.Particle or short fibers can provide easy crack initiation sites. The detailedbehavior can vary depending on the volume fraction, shape, size ofreinforcement and mostly on the reinforcement/matrix bond strength.
37 Fatigue of Laminated Composites Crack Density, Delamination, Modulus vs Cyclesi) Ply crackingii) Delaminationiii) Fiber fatigue
38 Modulus Reduction during Fatigue Ogin et al.Modulus Reduction Ratewhere E : current modulusE0 : initial modulus N : number of cycles: peak fatigue stress A, n : constants→ linear fittingtime
39 Integrate the equation to obtain a diagram relating modulus reduction to number of cycles for different stress levels.→ used for material design
40 8-7. Thermal Fatigue of Composite Materials Thermal StressThermal stresses arise in composite materials due to the generally largedifferences in thermal expansion coefficients() of the reinforcement and matrix.It should be emphasized that thermal stresses in composites will arise even ifthe temperature change is uniform throughout the volume of composite.Thermal FatigueWhen the temperature is repeatedly changed, the thermal stress results in thethermal fatigue, because the cyclic stress is thermal in origin. Thermal fatiguecan cause cracking of brittle matrix or plastic deformation of ductile matrix.Cavitation in the matrix and fiber/matrix debonding are the other forms ofdamage observed due to thermal fatigue of composites. Thermal fatigue inmatrix can be reduced by choosing a matrix that has a high yield strengthand a large strain-to-failure. The fiber/matrix debonding can only be avoidedby choosing the constituents such that the difference in the thermal expansioncoefficients of the reinforcement and the matrix is low.