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

VEVEE mmffc

VV mmffc

Ec

c

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

Strength

: flow stress of matrix at a given strain

Modulus

VV mmffc

VEVEE mmffc

VV mmffc

m

VEVdd

VEE ffmffc

m

m

strain given a atmatrix the of curve strain-stress the of slope : dd

m

m

3

Stage III : fiber & matrix – plastic

Strength

UTS

: ultimate tensile strength of fiber

: flow stress of matrix at the fracture strain of fiber

VV mmffc

Vdd

Vdd

E Modulus mfc

m

m

f

f

VV mmffucu

fu

m

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

matrix,ffiber,f

mmufu

mmuminV Vf

fu Vmin

mmu Vmin

)V1()V1(V fmufmffucu

5

mufmffucu )V1(V

fu Vcrit

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! (∵ ) VV mincrit

mfu

mmu

crit V Vf

mmu

0mu

Vcrit

6

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.

22

c ld

16E

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

)E or( GV) 1(2

EV

Gmm

m

m

m

mc

) 1(2E

G matrix, isostropic for mm

EEV3

EEVV2 fm

2/1

m

fmf2/1

fc

E ,G mm

Ef

Vf

m

m

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

matrix,ffiber,f

10

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

VVV mmmmuffu

m

VVV mmmmuffu

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

2l end fiber to plane crack from distance If c

2l end fiber to plane crack from distance If c

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

fr

fi

fi

13

Force Equilibrium

( lc : critical length of fiber )

1) Condition for fiber fracture,

2) Condition for fiber pullout,

lr2r iff2f

ciffu2f lr2r

dl

r2l

4c

f

c

i

fu

rl

2 f

c

i

fu

lr2r iff2f

fi

fuc r2

1dl

4

ll If

lr2r iff2f

fi

fuc r2

1dl

4

ll If

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

Wd

Wp

WWW pdfracture

Wp

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, )

2lk0 c

xi

)xk(di

dx)xk(ddx distanceapullouttoEnergy i

2dk

dx)xk(d W2

iip

2lk0 c

24

dldk

2dk

2l

1W

2ci

2i

cave,p

k

0

2cl

0

length debond : x x24dπ

E

σW

2

f

2fu

d

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)

plane, crack from ,2l

distance, a withinlocated is fiber a If c

ll l length, withfiber a of pullout fory Probabilit c

24dl

ll

W2cic

ave,p

l. length increasing withincreases distance pullout fiber , l l If c 2

pp lW l. length increasing withincreases, W l. length increasing withincreasestendency fracture fiber , l l If c

constant l l1

W l. length increasing withdecreases, W cpp

.ll whenmaximum, becomesW c p

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

diameter fiber :d VVd fracture ofEnergy

f

m2

i

d fracture ofEnergy

ppdfracture WWWW

yy

xx

18

If > interfacial tensile strength → delamination

→ crack deflection

Delamination Fracture in Laminate Composite

Fatigue → debonding at interface

Fracture → repeated crack initiation & propagation

xx

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

LexpLf 1

f

, d

20

Let, : kth moment of statistical distribution function

Mean Strength of Fibers

Standard Deviation for Strength of Fibers

Substituting

where : gamma function

Coefficient of Variation

f

11L /1

1

12

1LS 2

2/1

/1

dxx)xexp( n 1n

/11

/11/21S 2 2/1

)1

5.005.0 for only, )f(( 0.92-

0

d)(f M kk 0

Mk

Mdf 1 0

2/1212 MMS

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

11 ,1.0 8.57.2 ,4.02.0

plot L vs )1

1( )L( /1

/1

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

d )(f )(F

)1

(1 )L( /1

# of fibers

/1fufuB )eL()](F1[

0

23

Comparison of and

B

variation) of tcoefficien : ( . , As B

6.0 25.0

0.8 1.0

B

B

) (

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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 (>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

S or S or

X or X or

X or X

66

C22

T22

C11

T11

T

1XC

1XT

2XC

2X

25

ex) If uniaxial tensile stress is given in a direction at an angle with the fiber axis.

Failure occurs when,

Failure Criterion

x

0

0 ]T[ x

6

2

1

[ ]T

n mn

n mn

mn mn n

m

m

m

2

2

2

2

2

2

2

2

S mn or

X n or

X m

x6

T

2

2

x2

T

1

2

x1

failure shear planar mnS

or

failure tensile transverse nX

or

failure tensile allongitudin mX

x

2

T

1x

2

T

1x

Failure occurs by a criteria, which

is satisfied earlier.

x

1

2

26

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

S

66

S

66

C

22

T

22

C

11

T

11

or or

or or

or

T

1C

1T

2C

2S

6

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

1 S

X

X

X 2

2

12

2

2

22

1

212

1

2

1

x

nm

n

m

x6

2

x2

2

x1

substituting

1 X1

S1

nm Xn

Xm 2

x22

22

2

2

4

2

1

4

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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.

∴

1 F F )(f jij iii

j ii F ,F

1F2 F2 F2 F F F F F F 6226611621122666

2222

2111662211

0 = F =F = F Let 26 166

1F2 F F F F F 2112

2

666

2

222

2

1112211

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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,

1)X( FX F 1)(f ,X If 2T111

T11

T11

T

1XC

1X

1)(X FX F 1 f ,X If 2T222

T22

T22 σσ

266

2

666 S1

F 1SF 1)(f ,S If

221112 FF5.0F

1)X( FX F 1)(f ,X If 2C111

C11

C11

C1

T1

11XX

1F

C1

T1

1X

1

X

1F

1)(X FX F 1f ,X If 2C222

C22

C22 σσ

C2

T2

22XX

1F

C2

T2

2X

1

X

1F

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Boron/Epoxy composite

Intrinsic properties

MPa 5.6X ,MPa 4.52X

MPa 4.1S ,MPa 3.1X ,MPa 3.27XC

2

C

1

T

2

T

1

32

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.

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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)∵

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 CompositesCrack Density, Delamination, Modulus vs Cycles i) Ply cracking ii) Delamination iii) Fiber fatigue

38

Modulus Reduction during FatigueOgin et al.

Modulus Reduction Rate

where E : current modulus

E0 : initial modulus N : number of cycles

: peak fatigue stress A, n : constants

→ linear fitting

n

0

2

0

2

max

0 )E/E 1( E A

dNdE

E1

max

time

max

m

min

plot )E/E 1( E

log vs dN

dE

E

1 log

020

2max

0

39

Integrate the equation to obtain a diagram relating modulus reduction to numberof 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.

T