1. 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

2. 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
: flow stress of matrix at a given strain

3. Stage III : fiber & matrix – plastic
Strength
UTS
: 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 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

5. 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! (∵ )

7. 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.

8. 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

9. 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

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 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

11. 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 l

12. 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

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 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
Load
Displacement WP Wd

15. 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, )

16. 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)

17. As Wd << 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

18. If > interfacial tensile strength → delamination
→ crack deflection
Delamination Fracture in Laminate Composite
Fatigue → debonding at interface
Fracture → repeated crack initiation & propagation

19. 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
: statistical parameters
L : fiber length

20. Let, : kth moment of statistical distribution function 
Let, : kth moment of statistical distribution function
Mean Strength of Fibers
Standard Deviation for Strength of Fibers
Substituting
where : gamma function
Coefficient of Variation  

21. 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

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-fracture
Let F(σ) : 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

23. Comparison of and

24. 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 (>0)
: ultimate uniaxial compressive strength in fiber direction (<0)
: ultimate uniaxial tensile strength in transverse direction
: ultimate uniaxial compressive strength in transverse direction
S : 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 Criterion x  1 2
Failure occurs by a criteria, which
is satisfied earlier.

27. 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

28. 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

29. 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. ∴

30. 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,

31. Boron/Epoxy composite
Intrinsic properties

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 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.

34. 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)

35. 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)

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 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.

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

38. 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
time

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 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.