Chap.8 Mechanical Behavior of Composite

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Chap.8 Mechanical Behavior of Composite

• 8-1. Tensile Strength of Unidirectional Fiber
  Reinforced Composite
• Isostrain Condition loading parallel to
  fiber direction
• Fiber Matrix elastic case
• Modulus
  works reasonably well
• Strength
  does not work well
• Why?
• 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

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• Analysis of Tensile Stress and Modulus of
  Unidirectional FRC
• Assumption Fiber elastic plastic
• Matrix elastic plastic
• Stress-Strain Curve of FRC - divided into 3
  stages
• Stage I fiber matrix - elastic
• ? Rule of Mixtures
• Strength
• Modulus
• Stage II fiber - elastic, matrix - plastic
• Strength
• flow stress of matrix at a given strain
• Modulus

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• Stage III fiber matrix plastic
• Strength
• UTS
• ultimate tensile strength of
  fiber
• flow stress of matrix
  at the fracture strain of fiber

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Effect of Fiber Volume Fraction on Tensile
Strength (Kelly and Davies, 1965)
Assumption Ductile matrix ( ) work
hardens. All fibers are identical and
uniform. ? same UTS If the fibers are
fractured, a work hardenable matrix
counterbalances the loss of load-carrying
capacity. In order to have composite
strengthening from the fibers,
UTS of composite UTS of matrix after
fiber fracture Minimum Fiber Volume
Fraction As ?, ?.
As ?, ?.
degree of work hardening

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In order to be the strength of composite
higher than that of monolithic matrix,
UTS of pure
matrix Critical Fiber Volume Fraction
As ?, ?. As
?, ?. degree of
work hardening Note that always!
(? )
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• 8-2. Compressive Strength of Unidirectional Fiber
  Reinforced Composites
• Compression of Fiber Reinforced Composite
• Fibers - respond as elastic columns in
  compression.
• Failure of composite occurs by the buckling
  of fibers.
• Buckling occurs when a slender column under
  compression becomes unstable
• against lateral movement of the central
  portion.
• Critical stress corresponding to failure by
  buckling,
• where d is diameter, l is length of column.

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2 Types of Compressive Deformation 1)
In-phase Buckling involves shear deformation of
matrix
? predominant at high fiber volume
fraction 2) Out-of-phase Buckling involves
transverse compression and tension of
matrix and fiber ? pre-dominant at low
fiber volume fraction Factors influencing the
compressive strength Interfacial
Bond Strength poor bonding ? easy buckling

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• 8-3. Fracture Modes in Composites
• 1. Single and Multiple Fracture
• Generally,
• When more brittle component fractured, the
  load carried by the brittle
• component is thrown to the ductile component.
• If the ductile component cannot bear this
  additional load ? Single Fracture
• If the ductile component can bear this
  additional load ? Multiple Fracture

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• 1) Single Fracture
• - predominant at high fiber volume fraction
• - all fibers and matrix are fractured in
  same plane
• - condition for single fracture
• stress beared by fiber additional
  stress which can be supported by matrix
• where matrix stress
  corresponding to the fiber fracture strain
• 2) Multiple Fracture
• - predominant at low fiber volume fraction
• - fibers and matrix are fractured in
  different planes
• - condition for multiple fracture

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• 2. Debonding, Fiber Pullout and Delamination
  Fracture
• Fracture Process crack propagation
• Discontinuous Fiber Reinforced Composite
• ( lc critical length )
• ? Debond Pullout
• ? Good for toughness
• ? Fiber Fracture
• ? Good for strength

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Fracture of Continuous Fiber Reinforced
Composite Fracture of fibers at crack plane
or other position depending on the position
of flaw ? Pullout of fibers For
max. fiber strengthening ? fiber fracture is
desired. For max. fiber toughening ? fiber
pullout is desired. Analysis of Fiber
Pullout Assumption Single fiber in
matrix fiber radius l fiber
length in matrix tensile stress on
fiber interfacial shear strength

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Force Equilibrium ( lc critical length
of fiber ) 1) Condition for fiber
fracture, 2) Condition for fiber
pullout,

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• Fracture Process of Fiber Reinforced Composites
• Real fibers - non-uniform properties
• 3 steps of fracture process
• 1) Fracture of fibers at weak points near
  fracture plane
• 2) Debonding of fibers
• 3) Pullout of fibers
• Outwater and Murphy

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Energy Required for Fracture Debonding
elastic strain E. volume Energy
Required for Pullout Let k embedded
distance of a broken fiber from crack plane
pullout distance at a certain
moment interfacial shear strength Force
to resist the pullout fiber
contact area Total energy(work) to pullout
a fiber for distance k Average energy to
pullout per fiber(considering all fibers with
different k, )

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Fracture of Discontinuous Fiber Reinforced
Composite ? pullout Average energy
to pullout per fiber with length, l
probability for pullout energy required for
pullout Energy for Fiber Pullout vs Fiber
Length(l)

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• As Wd ltlt Wp
• Advantage of Composite Material
• can obtain strengthening toughening at the
  same time
• Toughening Mechanism in Fiber Reinforced
  Composite
• 1) Plastic deformation of matrix - metal matrix
  composite
• 2) Fiber pullout
• 3) Crack deflection (or Delamination) - ceramic
  matrix composite
• Cook and Gordon, Stresses distribution near
  crack tip

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• If gt interfacial tensile strength ?
  delamination
• 
  ? crack deflection
• Delamination Fracture in Laminate Composite
• Fatigue ? debonding at interface
• Fracture ? repeated crack initiation
  propagation

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• 8-4. Statistical Analysis of Fiber Strength
• Real fiber nonuniform properties ? need
  statistical approach
• Brittle fiber (ex. ceramic fibers) -
  nonuniform strength
• Ductile fiber (ex. metal fibers) - relatively
  uniform strength
• Strength of Brittle Fiber
• ? dependent on the presence of flaws
• ? dependent on the fiber length "Size
  Effect
• Weibull Statistical Distribution Function
• probability density function
• Probability that the fiber strength is
  between and .

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• Let, kth moment of statistical
  distribution function
• Mean Strength of Fibers
• Standard Deviation for Strength of Fibers
• Substituting
• where
  gamma function
• Coefficient of Variation

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• As L ?, ?. ? "Size Effect
• As ?, ?. ? is less dependent on L.
• If , spike distribution function
  (dirac delta function)
• ? uniform strength independent on L
• Glass fiber
• Boron, SiC fibers

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• Strength of Fiber Bundle
• Bundle strength ? Average strength of fiber
  n
• lt
  
• Assumption Fibers - same
  cross-sectional area
• - same stress-strain
  curve
• - different
  strain-to-fracture
• Let F(s) The probability that a fiber will
  break before a certain value of is
• attained.
• ? Cummulative Strength
  Distribution Function
• Mean Fiber Strength of Bundle
• ? Mean Fiber Strength of Unit Fiber

of fibers
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• Comparison of and

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• 8-5. Failure Criteria of an Orthotropic Lamina
• Assumption Fiber reinforced lamina -
  homogeneous, orthotropic
• Failure Criterion of Lamina
• 1. Maximum Stress Criterion
• Failure occurs when any one of the stress
  components is equal to or greater
• than its ultimate strength.
• Interaction between stresses is not
  considered.
• Failure Condition
• where ultimate uniaxial tensile
  strength in fiber direction (gt0)
• ultimate uniaxial compressive
  strength in fiber direction (lt0)
• ultimate uniaxial tensile strength
  in transverse direction
• ultimate uniaxial compressive
  strength in transverse direction
• S ultimate planar shear strength

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• ex) If uniaxial tensile stress is given
  in a direction at an angle with the fiber axis.
• Failure occurs when,
• Failure Criterion

?x
1
?
2
Failure occurs by a criteria, which is satisfied
earlier.
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• 2. Maximum Strain Criterion
• Failure occurs when any one of the strain
  components is equal to or greater
• than its corresponding allowable strain.
• Failure Condition
• where 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

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• 3. Maximum Work Criterion
• Failure criterion under general stress state
• Tsai-Hill
• where X1 ultimate tensile (or
  compressive) strength in fiber direction
• X2 ultimate tensile (or
  compressive) strength in transverse direction
• S ultimate planar shear strength
  
• ex) For uniaxial stress , having angle
  with the fiber axis
• Failure criterion

substituting
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• 4. Quadratic Interaction Criterion
• Consider stress interaction effect
• Tsai-Hahn
• Stress Function
• stress term 1st interaction term
• Thin Orthotropic Lamina
• i, j 1, 2, 6 (plane stress)
• strength parameters
• Failure occurs when,
• ? need to know 9 strength parameters
• For the shear stress components, the reverse
  sign of shear stress should
• give the same criterion.
• ?

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• Calculation of Strength Parameters by Simple
  Tests
• 1) Longitudinal uniaxial tensile and
  compressive tests,
• longitudinal tensile strength
• longitudinal compressive strength
• 2) Transverse uniaxial tensile and compressive
  tests,
• 3) Longitudinal shear test
• 4) In the absence of other data,

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• Boron/Epoxy composite
• Intrinsic properties

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• 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
  Composites
• Pile-up of damages - matrix cracking, fiber
  fracture, fiber/matrix debonding,
• ply cracking, delamination
• ? Crack deflection (or Blunting)
• ? Reduction of stress concentration
• A variety of subcritical damage mechanisms
  lead to a highly diffuse damage
• zone.

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• Constant-stress-amplitude Fatigue Test
• Damage Accumulation vs Cycles
• Crack length in homogeneous material -
  accelerate
• (? increase of stress concentration)
• Damage (crack density) in composites -
  accelerate and decelerate
• (? reduction of stress concentration)

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• S-N Curves of Unreinforced Plolysulfone vs
  Glassf/Polysulfone, Carbonf/Polysulfone
• Carbon Fibers higher stiffness thermal
  conductivity
• ? higher fatigue resistance
• S-N Curves of Unidirectional Fiber Reinforced
  Composites (B/Al, Al2O3/Al, Al2O3/Mg)

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• 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 to
• Al alloy, which is attributed to the higher
  stiffness of the composites.
• For strain-controlled cyclic fatigue or low
  cycle fatigue, the composites show
• lower 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 detailed
• behavior can vary depending on the volume
  fraction, shape, size of
• reinforcement and mostly on the
  reinforcement/matrix bond strength.

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• Fatigue of Laminated Composites
• Crack Density, Delamination, Modulus vs Cycles
• i) Ply cracking
• ii) Delamination
• iii) Fiber fatigue

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• Modulus Reduction during Fatigue
• Ogin et al.
• Modulus Reduction Rate
• where E current modulus
• E0 initial modulus N number of cycles
• peak fatigue stress A, n
  constants
• ? linear fitting

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• Integrate the equation to obtain a diagram
  relating modulus reduction to number
• of cycles for different stress levels.
• ? used for material design

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• 8-7. Thermal Fatigue of Composite Materials
• Thermal Stress
• Thermal stresses arise in composite materials
  due to the generally large
• differences in thermal expansion
  coefficients(?) of the reinforcement and matrix.
• It should be emphasized that thermal stresses
  in composites will arise even if
• the temperature change is uniform throughout
  the volume of composite.
• Thermal Fatigue
• When the temperature is repeatedly changed,
  the thermal stress results in the
• thermal fatigue, because the cyclic stress is
  thermal in origin. Thermal fatigue
• can cause cracking of brittle matrix or
  plastic deformation of ductile matrix.
• Cavitation in the matrix and fiber/matrix
  debonding are the other forms of
• damage observed due to thermal fatigue of
  composites. Thermal fatigue in
• matrix can be reduced by choosing a matrix
  that has a high yield strength
• and a large strain-to-failure. The
  fiber/matrix debonding can only be avoided
• by choosing the constituents such that the
  difference in the thermal expansion
• coefficients of the reinforcement and the
  matrix is low.