Tensile strength of single fibers: test methods and data

analysis

J. Andersons

Institute of Polymer Mechanics, University of Latvia

Outline

• motivation

• fiber strength as a statistical variable

• inconsistency in Weibull distribution parameters

• modified Weibull distribution

• example 1: glass fibers

• example 2: flax fibers

• tensile strength of fiber-reinforced composite

Fiber tensile strength

determines the strength of a composite material

strength at ineffective length L

Weibull distribution (1939)

( )α

β

σσ

=

0

,l

llN

For Poisson distribution of flaws in a fiber with power-law intensity :

average number of flaws with strength

and fiber strength distribution

σ*

**

*

*

( )

−−=

α

β

σσ

0

exp1l

lP

flaw strength

σ≤

α

β

σ

=

0

1

ln

Fiber tension test (FTT)

fiber

glue

gauge

lenght, l

Output: fiber strength

+ straightforward method for long fibers

- labor intensive

- stress concentration at fiber ends

- complicated for short gauge length

Inconsistency of Weibull distribution parameters

lg l

lg <σ>

( )

−−=

α

β

σσ

0

exp1l

lP

( ) ( )αβσ α11

1

0 +Γ=−

ll

Weibull distribution

Mean strength according to Weibull distr.

P(σ)

σ

(α1, β1)

(α2 , β2)

αααα 2 ≥≥≥≥ αααα1

FTT

strength distribution

mean strength vs length

0.93.22.9Flax

EkoFlax

(Andersons et al., 2009)

0.555.22.8Flax

FinFlax

(Andersons et al., 2005)

0.695.4E-glass

Nito Boseki

(Andersons et al., 2002)

0.76.44.5Carbon

Torey T800

(Okabe et al., 2002)

from strength-

length data, α2

at a fixed fiber

length, α1

Ratio

α1/α2

Shape parameter of Weibull

strength distribution

Fiber type

Does the discrepancy of α1, α2 matter?

Mean strength as a function of length

α2=9

( ) ( )αβσ α11

1

0 +Γ=−

ll

E-glass fibers

Strength distribution at a fixed length

( )

−−=

α

β

σσ

0

exp1l

lP

Discrepancy in fracture

probability

α1=5.4

α1=5.4

α2=9

fiber length, mm

1 3 10 40 100

str

ength

, M

Pa

1500

2500

3000

4000

0

0.2

0.4

0.6

0.8

1

500 1500 2500

σσσσ, MPa

P

l =80 mm

Discrepancy in strength

interpolation

Gutans and Tamuzh, 1984

Watson and Smith, 1985( )

−−=

αγ

β

σσ

0

exp1l

lP

The modified Weibull distribution

Why classical Weibull distribution may not apply:

o Watson and Smith (1985): diameter variation between fibers

o Beyerlein and Phoenix (1996): variations in material texture from fiber to fiber

o Jeulin (1996): large-scale fluctuation of the density of defects in fibres

o Curtin (2000): each fiber has a Weibull strength distribution with a random scale parameter

( ) ( )αβσ αγ110 +Γ=

−ll

The modified Weibull distribution: interpretation

………

( )a

bln

=

10

1

1 σσ

( )a

bln

=

20

2

1 σσ

( )a

k

kbl

n

=

σσ

0

1

………

Defect density

Strength distribution

of fiber batch

……… ………

Strength distribution

of a given fiber

Probability of picking

an ith-type fiber

( ) ( ) ( )∑∞

=

=1i

ii bPPP σσ

( )

−−=

a

i

ibl

lP

σσ

0

exp1

Curtin (2000)

WoW model ( )

−−=

a

bl

lP

σσ

0

exp1 ( )

−−=

m

B

bbP exp1

Strength distribution of

an individual fiber

Scale parameter distribution

among fibers in a batch

( )

−−=

αγ

β

σσ

0

exp1l

lP

22 am

m

+=γ

22 am

ma

+=α

Strength distribution of fiber batch

Extensive numerical

simulations with various

a, m, B values

( )( )Bam75.0221

−+−=β

Curtin (2000) J Compos Mater

Fiber fragmentation test (FFT)

ε

Output: number of fiber breaks as a function of applied strain

+ yields strength distribution from a single test

+ characterizes an individual fiber

- residual strain estimate needed

- for non-linear elastic fibers, fiber stress-strain response

needed to convert limit strain to stress

SFF specimen preparation

Aluminium Frame

Tape

FibersResin

Mould Covered by Teflon Silicon Tube Sealing

Specimen

Matrix block

Clamping area

Fiber

25

-35

mm

3-4 mm

7-9

mm

2 mm

Specimen

MINIMAT

(Miniature Testing Machine)

Computer IBM PC

Motor

Load cell

Specimen

Grips

Electronic Unit

Video Recorder

Video Monitor

Microscope and

Video Camera

Strain Gage

Amplifier

SFF testing

strain, %

2 3 4 6 8

n, 1/m

m

0.02

0.05

0.1

0.3

1

a

b

E

ln

=

ε

0

1

( )

−−=

a

bl

lP

σσ

0

exp1

Glass fiber fragmentation data

Linear elastic fibers, hence εσ E=

Number of breaks vs strain

a = 8.2

b= 3200 MPa

strain, %

2 3 4 6 8

n, 1

/mm

0.02

0.05

0.1

0.3

1

Glass fiber fragmentation data

Number of breaks vs strain for several fibers

Fragmentation diagrams:

� ~ the same slope (parameter a)

� location (parameter b) exhibits scatter

( )

−−=

a

bl

lP

σσ

0

exp1

a

i

ib

E

ln

=

ε

0

1

ln (b, MPa)

7.7 7.8 7.9 8.0 8.1 8.2 8.3

ln(-

ln(1

-P))

-4

-3

-2

-1

0

1

2

Individual glass fibers characterised, by FFT, by

( )

−−=

a

bl

lP

σσ

0

exp1

a = 8.4

B = 3150 MPa

m = 10.6

( )

−−=

m

B

bbP exp1

Fiber batch strength distribution

78.022

≈+

=am

mγ

7.622

≈+

=am

maα

( )( ) 3080175.022 ≈+−=

−Bamβ

( )

−−=

αγ

β

σσ

0

exp1l

lP

MPa

Curtin’s relations (2000)

0

0.2

0.4

0.6

0.8

1

500 1500 2500 3500

σσσσ, MPa

P

10 mm

20 mm

40 mm

SFT

( )

−−=

αγ

β

σσ

0

exp1l

lP

fiber length, mm

3 10 40 100<

σ>

, M

Pa

1500

2000

2500

3000

SFT

( ) ( )αβσαγ

110 +Γ=−

ll

Glass fiber strength: SFT and SFF

Distribution parameters determined by SFF

Elementary flax fiber

length ~ 3 cm

diameter ~ 20 µm

normalised distance y/rf

0 50 100 150 200

axia

l strain

, %

0.0

0.5

1.0

1.5

2.0

ÿÿÿÿ]

Eext=Em νext=νm

Eext=Ef/2 νext=νf

Eext=Ef νext=νf

Mechanical loading, ε =1 %

extension perturbation zone

Flax fiber

25-35 mm

50-150 mm Fiber extension

Fast drying glue

Flax fiber SFF

strain, %

2 3 4

n, 1

/mm

0.1

0.3

1

3

ε

ε

εa

bln

=

0

1

( )

−−=

ε

ε

εε

a

bl

lP

0

exp1

εn

strain

str

ess

69 20 GPa

0.32 0.42%n

E

ε

= ±

= ±

d, µm

0 10 20 30 40

E, G

Pa

0

50

100

150

10 mm

20 mm

( )nE εεσ −=

( )nE εεσ −=

Elementary flax fiber: response to axial tension

l, mm

5 10 20

<σ

>, M

Pa

600

700

800

900

1000

1100

SFT

SFF

The modified Weibull failure

strain distribution parameters

from SFF:0.79

4.97

2.5%

ε

ε

ε

γ

α

β

=

=

=

Flax fiber average strength: SFT and SFF

( )nE εεσ −=

+Γ

=

−

ε

α

γ

εα

βεε

ε

11

0l

l

x x

εFTT = εFFT

x x

εFTT = εFFT

FTT FTTFFT FFT

?

Man-made fibers Natural organic fibers

x

?

The effect of kink bands on bast fiber strength

Marginal effect:

o no correlation between the

amount of kink bands and fiber strength

Baley (2004); flax

Thygesen et al. (2007); hemp

Andersons et al. (2009); flax

Determining effect:

• absence of kink bands increases mean strength

of flax by ~20%

Bos (2002)

• failure in tension initiates within a kink band

in flax

Khalili et al. (2002)

Lamy, Pomel (2002)

Baley (2004)

( ) ( )[ ]σλσ dPP −−= exp1

( )

−−=

a

db

Pσ

σ exp1

s

l=λ

Derivation of bast fiber strength distribution

skink band spacing distribution of kink band strength

( )

−−=

a

bs

lP

σσ exp1

Probability of fracture initiated by defects:

Todinov (2000)

number of defects; in our case

defect failure probability; approximately

a

db

P

σ~

+Γ

=

al

sb

a 11

1

σ

Strength distribution Mean strength

Derivation of fiber batch strength distribution: random s

Curtin (2000)

WoW model

( )

−−=

a

bs

lP

σσ exp1 ( )

−−=

m

s

ssP exp1

Strength distribution of

an individual fiber

Kink spacing distribution

among fibers in a batch

( )

−−=

αγ

β

σσ

0

exp1l

lP

12 +=

m

mγ

12 +=

m

maα

Strength distribution of fiber batch

( )( )a

l

sbma

1

0

75.025.1 11

+−=

−−β

( )

−−=

αγ

β

σσ

0

exp1l

lP

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000

σσσσ, MPa

P( σσ σσ

)

5 mm

20 mm

Elementary flax fibers, Finflax

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1s, mm

P(s

)Kink spacing distribution (5 mm fibers) Fiber strength distribution

approximation

prediction

21.0

4.1

=

=

s

m

1700

9.2

=

=

β

α

MPa

( )

−−=

m

s

ssP exp1

8.012

≈+

=m

mγ

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500

σσσσ, MPa

P( σσ σσ

)

5 mm

20 mm

Elementary flax fibers, Ekotex

0

0.2

0.4

0.6

0.8

1

0 0.03 0.06 0.09 0.12s, mm

P(s

)

Kink spacing distribution Fiber strength distribution

( )

−−=

m

s

ssP exp1

prediction

07.0

2.5

=

=

s

m

98.012

≈+

=m

mγ

1400

1.3

=

=

β

α

MPa( )

−−=

αγ

β

σσ

0

exp1l

lP

approximation

Flax-fiber reinforced composites

Extruded flax / PP and MAPPRTM flax mat / thermoset polymer

Fiber length ~ few cm

Fiber orientation ~ in-plane random Fiber length ~ 1 mm

Fiber orientation ~ 3D random

Composite stiffness: model

oE lE f f m mE E Eη η ν ν= +

( )( )

( )max

min

tanh 211

2

l

lE

l

ll h l dl

l l

ξη

ξ

= ⋅ − ⋅

∫

3 8oEη =

1 5oEη =extruded composite (3D distribution):

FFM composite (2D distribution):

( )20 0

2

ln

m

f

G

r E R rξ =

Cox - Krenchel model:

fiber length distribution

FFM / AR FFM / VE1 FFM / VE2

E, M

Pa

0

2

4

6

8

10

12

matrix

experimental

predicted

νf

0.0 0.1 0.2 0.3

E,

MP

a

0

1

2

3

4

5

6

flax / PP

flax / PPM

Extruded flax / thermoplastic matrix

composites

Flax fiber mat / thermoset matrix

composites

Composite stiffness: results

Predicted

νf = 0.3

( )( ) ( ) ( )

0 40

0

cos1

n

uc f f m

l

l lh l g d dl

l

θσ θ

σ ν θ θ ν σ∞

= + −∫ ∫

Modified Fukuda-Chou critical zone strength theory

( )( )

0

1 2

2

c uf c

c uf c

l l l ll

l l l l

σσ

σ

− ≥=

<

( )0 arccos nl lθ =

Composite strength: model

critical zone

1

3

2

θ0 loading axis

θ

nl nl

bridging fibers

� all fibers bridging the critical zoneand the matrix fail simultaneously� fiber stress at failure proportionalstrength, reduced by ineffective length

fiber orientation distribution

fiber length distribution

( )mffufsc σννσησ −+= 1

Composite strength: theoretical vs experimental

0

20

40

60

80

100

0 20 40 60 80 100 120 140

theoretical strength, MPa

exp

eri

men

tal str

eng

th, M

Pa

Flax fiber mat / thermosetmatrix composites

Extruded flax / thermoplastic matrixcomposites

n cl l=The critical zone width is chosen equal to the critical fibre length

Composite strength: extruded flax / PP

( )( ) ( )

mf

l

ccuff

sc

c

dllhl

l

l

l

l

llσν

σνησ −+

−

−= ∫

∞

12

115

5

*

( ) ( ) ( )αβσ αγσ 110 +Γ=

−llluf

( ) ( )αγα

αγ

σ

τ

αβ+

−

+Γ=

0

11

l

rl

f

c

f

muf

mE

Eσσ =

20

25

30

35

40

45

0 0.1 0.2 0.3

νf

σc,

MP

a

flax/PP

flax/PPM

PP

0

20

40

60

80

100

120

FFM/AR FFM/VE1 FFM/VE2

σ, M

Pa

matrix

experimental

predicted

( )( ) ( )

mf

l

cccccuff

sc

c

dllhl

l

l

l

l

l

l

l

l

l

l

llσν

σ

π

νησ −+

−

−

++= ∫

∞− 1

21123cos3

4

22

1*

( ) ( ) ( )αβσ αγσ 110 +Γ=

−llluf

( ) ( )αγα

αγ

σ

τ

αβ+

−

+Γ=

0

11

l

rl

f

c

f

muf

mE

Eσσ =

Composite strength: Flax fiber mat / thermoset

νf = 0.3

Conclusions

• The modified Weibull distribution for fiber strength should be used if

the variation of strength with length is of interest

• To evaluate distribution parameters by fiber tension tests alone,

strength data for at least two gauge lengths are needed

• Alternatively, fiber fragmentation tests can be applied to obtain

strength distribution of brittle man-made fibers (glass) and average

strength-length relation for bast fibers (flax)

• The modified Weibull distribution enables accurate scaling of fiber

strength from tests at length of cm range to mm range, needed for

composite strength analysis

Acknowledgements

Dr. Roberts Joffe

Prof. Masaki Hojo

Prof. Shojiro Ochiai