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 Strength E c  c  E f   f V f V f  E m   m V m V m : works reasonably well : does not work well Why?

E  c c : 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 1

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  c   f V f   m V m Modulus E c  E f V f  E m V m Stage II : fiber - elastic, matrix - plastic Strength  c   f V f   m V m Modulus    m E c  d d  m  m : flow stress of matrix at a given strain E f V f    d  m d  m    V m  E f    : slope of the stress V f strain curve of the matrix at a given strain 2

Stage III : fiber & matrix – plastic Strength  c   f V f   m V m Modulus E c    d  f d  f    V f    d  m d  m    V m UTS  cu   fu V f   m V m  fu : ultimate tensile strength of fiber  m : flow stress of matrix at the fracture strain of fiber 3

Effect of Fiber Volume Fraction on Tensile Strength (Kelly and Davies, 1965) Assumption : Ductile matrix (  f , fiber   f , matrix Al l 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,  cu   fu V f   m ( 1  V f )   mu ( 1  V f ) UTS of composite UTS of matrix after fiber fracture Minimum Fiber Volume Fraction V f  V min   fu  mu    mu  m   m As As   fu  mu   ,  m V min   ,  .

V min  .

degree of work hardening 4

In order to be the strength of composite higher than that of monolithic matrix,  cu   fu V f   m ( 1  V f )   mu UTS of pure matrix Critical Fiber Volume Fraction V f  V crit   mu  fu     m m As   fu  mu   m V crit  ↑, V crit ↑.

degree of work hardening Note that V crit  V min always! ( ∵  mu  5

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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,  c   2 E 16 d l 2 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  c  G V m m  2 ( 1  E m  m ) V m  G m ( or  for isostropic matrix, G m  2 ( E m 1 E  m  ) m ) → predominant at high fiber volume fraction 2) Out-of-phase Buckling : involves transverse compression and tension of matrix and fiber  c  2 V f V f E m E f 3 V m 1 / 2   E m  E f  1 / 2 → pre-dominant at low fiber volume fraction Factors influencing the compressive strength : G m , E m E f V f Interfacial Bond Strength : poor bonding → easy buckling 8

8-3. Fracture Modes in Composites 1. Single and Multiple Fracture Generally,  f , fiber   f , matrix 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 9

1) Single Fracture - predominant at high fiber volume fraction - all fibers and matrix are fractured in same plane - condition for single fracture  fu V f   mu V m   m V m stress beared by fiber additional stress which can be supported by matrix where  m : 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  fu V f   mu V m   m V m 10

2. Debonding, Fiber Pullout and Delamination Fracture Fracture Process : crack propagation l Discontinuous Fiber Reinforced Composite If distance from crack → Debond & Pullout plane to fiber end  l 2 c  Good for toughness If distance from crack → Fiber Fracture plane to fiber end  l c 2  Good for strength ( l c : critical length ) 11

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 r f : fiber radius  l : fiber length in matrix f : tensile stress on fiber  i : interfacial shear strength  i  f 12

Force Equilibrium  r f 2  f  2  r f  l i  r f 2  fu  fu  2  i l c r f  2  r f  i l c  fu 4  i  l c 2 r f  l d c ( l c : critical length of fiber ) 1) Condition for fiber fracture,  r f 2  f  2  r f  l i If l  l c   4 fu  i  l d    1 2 r f   2) Condition for fiber pullout,  r f 2  f  2  r f  l i If l  l c   fu 4  i  l d    1 2 r f   13

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 : W p W d W fracture  W d  W p Outwater and Murphy Load 14 W d W P Displacement

Energy Required for Fracture & Debonding W d    σ 2 fu E f      π d 2 24    x x : debond length elastic strain E. volume Energy Required for Pullout Let k : embedded distance of a broken fiber from crack plane x  i : pullout distance at a certain moment : interfacial shear strength 0  k  l 2 c Force to resist the pullout =  i  d ( k  x ) fiber contact area Energy to pullout a distance dx   i  d ( k  x ) dx Total energy(work) to pullout a fiber for distance k W p   k  i 0  d ( k  x ) dx   i  dk 2 2 c Average energy to pullout per fiber(considering all fibers with different k, ) 2  W p , ave  l c 1 2  l 2 c 0  i  dk 2 2 dk   i  dl 2 c 24 15

Fracture of Discontinuous Fiber Reinforced Composite If a fiber is located within a Probabilit y for pullout of a distance,  l c , from crack plane, fiber with 2 length, l  l Average energy to pullout per fiber with length, l l c → pullout W p , ave     l c l     i  dl 2 c 24 probability for pullout energy required for pullout Energy for Fiber Pullout vs Fiber Length(l) If l  l c , fiber pullout distance increases with increasing length l.

 W p increases, with increasing length l.

 W p  l 2  If W l  p l c , fiber fracture tendency increases with  W becomes p decreases, maximum, with increasing when l  l c .

length increasing l.

W p  1 l   length l.

 l c  constant 16

As W d << W p W fracture  W d  W p  W p 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 Energy of 2) Fiber pullout fracture  d   V V m f   d : fiber diameter fracture  d Energy of  i 3) Crack deflection (or Delamination) - ceramic matrix composite Cook and Gordon, Stresses distribution near crack tip  yy   xx 17 

→ crack deflection Delamination Fracture in Laminate Composite Fatigue → debonding at interface Fracture → repeated crack initiation & propagation 18

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 f  L    1 f  exp   L      : probability density function  ,  : statistical parameters L : fiber length 19

Let, M k   0   k f (  ) d  M k : kth moment of statistical distribution function Mean Strength of Fibers    0   f   d   M 1 Standard Deviation for Strength of Fibers S   M 2  M 2 1  1 / 2 Substituting f   S     1 /     1  1       1 /       1  2      2   1  1      1 / 2    0  n  1 Coefficient of Variation   S      1  2 /    1    1 / 1  1 /      1 / 2 (   f(  ) only, for 0 .

05    0 .

5     0.92

 1  ) 20

  (  L )  1 /   ( 1  1  )     1 /  vs L plot As L As  ↑, ↓.  ↑, ↑.  "Size Effect“  is less dependent on L.

Glass fiber → uniform strength independent on L Boron, SiC fibers     0 .

1 0 .

2 ,   0 .

4  11 ,   2 .

7  5 .

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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 F (  )   0  f (  ) d  Mean Fiber Strength of Bundle  B   fu [ 1  F (  fu )]  ( L  e )  1 /  ※ Mean Fiber Strength of Unit Fiber   (  L )  1 /   (1  1  ) 22

Comparison of and B  As   (   ) ,   B  .

(  : coefficien t of variation)   0 .

1   B    0 .

25   B   0.8

 0 .

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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  1  X T 1 or  1  X C 1 or  2  X T 2 or  2  X C 2 or  6  S or  6  S where X 1 T : ultimate uniaxial tensile strength in fiber direction (>0) X 1 C : ultimate uniaxial compressive strength in fiber direction (<0) X T 2 : ultimate uniaxial tensile strength in transverse direction X C 2 : ultimate uniaxial compressive strength in transverse direction S : ultimate planar shear strength 24

      1 2  6      [ T ]      0 0 x     [ ]        m 2 n 2 n m 2 2 mn mn Failure occurs when,  1   x m 2  X 1 T or  2   x n 2  X T 2 or  6   x mn  S  m 2 2 mn 2 mn  n 2     Failure Criterion  x  X m T 1 2 longitudin al tensile failure or  x  X 1 T n 2 transverse tensile failure or  x  S mn planar shear failure 25  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  1   1 T or  1   C 1 or  2   T 2 or  2   C 2 or  6   S 6 or  6   S 6 T where : ultimate tensile strain in fiber direction  1 C  T 2  C 2  S 6 : ultimate compressive strain in fiber direction : ultimate tensile strain in transverse direction : ultimate compressive strain in transverse direction : ultimate planar shear strain 27

3. Maximum Work Criterion Failure criterion under general stress state Tsai-Hill  X 1 2 2 1   1  2 X 2 1   2 2 X 2 2   2 1 S 2  1 where X 1 X 2 : ultimate tensile (or compressive) strength in fiber direction : ultimate tensile (or compressive) strength in transverse direction S : ultimate planar shear strength  x  1   x m 2  2   x n 2  6   x m n substituting Failure criterion   m 4 X 2 1  n 4 X 2 2  m 2 n 2   1 S 2  1 X 2     2 x  1 28 

4. Quadratic Interaction Criterion Consider stress interaction effect Tsai-Hahn Stress Function f (  )  F i  i  F i j  i  j  1 stress term 1st interaction term Thin Orthotropic Lamina i, j = 1, 2, 6 (plane stress) F i , F i j : strength parameters Failure occurs when, F 1  1  F 2  2  F 6  6  F 11  2 1  F 22  2 2  F 66  2 6  2 F 12  1  2  2 F 16  1  6  2 F 26  2  6 → need to know 9 strength parameters  1 For the shear stress components, the reverse sign of shear stress should give the same criterion.

Let F 6 = F 16 = F 26 = 0 ∴ F 1  1  F 2  2  F 11  1 2  F 22  2 2  F 66  2 6  2 F 12  1  2  1 29

Calculation of Strength Parameters by Simple Tests 1) Longitudinal uniaxial tensile and compressive tests, If If  1  1 X 1 T   X T 1 , f (  ) X C 1 , f (  )   1  F 1 X T 1 1  F 1 X C 1   F 11 ( F 11 ( X T 1 X C 1 : longitudinal tensile strength ) 2 ) 2   1 1 F 11 F 1   1 X X T 1 1 T 1 X C 1  1 X C 1 X 1 C : longitudinal compressive strength 2) Transverse uniaxial tensile and compressive tests, If If σ 2 σ 2   X T 2 , f X C 2 , f     1  F 2 1  F 2 X T 2 X C 2   F 22 (X T 2 F 22 (X C 2 ) 2 ) 2   1 1 F 22 F 2   1 X X T 2 1 T 2 X  C 2 1 X C 2 3) Longitudinal shear test If  6  S , f (  )  1  F 66 S 2  1  F 66  1 S 2 4) In the absence of other data, F 12   0 .

5 F 11  F 22 30

Boron/Epoxy composite Intrinsic properties X X T 1 C 1  27 .

3 MPa , X T 2  52 .

4 MPa , X C 2  1 .

3 MPa , S  1 .

4 MPa  6 .

5 MPa 31

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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) 34

S-N Curves of Unreinforced Plolysulfone vs Glass f /Polysulfone, Carbon f /Polysulfone Carbon Fibers : higher stiffness & thermal conductivity  higher fatigue resistance S-N Curves of Unidirectional Fiber Reinforced Composites (B/Al, Al 2 O 3 /Al, Al 2 O 3 /Mg) 35

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 37

Modulus Reduction during Fatigue Ogin et al.

Modulus Reduction Rate  1 E 0 dE dN  A   E 0 2 ( 1  2  max E / E 0 )   n  max  m where E : current modulus E 0  max : initial modulus : peak fatigue stress  min  N : number of cycles A, n : constants log    1 E 0 dE dN   vs log   E 0 2  2 max ( 1  E / E 0 )   plot → linear fitting time 38

Integrate the equation to obtain a diagram relating modulus reduction to number of cycles for different stress levels.

→ used for material design 39

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.

      T 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.

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