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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