tensile strength of single fibers: test methods and data ... · tensile strength of single fibers:...
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![Page 1: Tensile strength of single fibers: test methods and data ... · Tensile strength of single fibers: test methods and data analysis J. Andersons Institute of Polymer Mechanics, University](https://reader036.vdocument.in/reader036/viewer/2022062414/5d16389588c993d4608b4c90/html5/thumbnails/1.jpg)
Tensile strength of single fibers: test methods and data
analysis
J. Andersons
Institute of Polymer Mechanics, University of Latvia
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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
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Fiber tensile strength
determines the strength of a composite material
strength at ineffective length L
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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Elementary flax fiber
length ~ 3 cm
diameter ~ 20 µm
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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
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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
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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
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x x
εFTT = εFFT
x x
εFTT = εFFT
FTT FTTFFT FFT
?
Man-made fibers Natural organic fibers
x
?
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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)
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( ) ( )[ ]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
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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
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( )
−−=
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
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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
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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
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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
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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
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( )( ) ( ) ( )
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
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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
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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
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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
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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
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Acknowledgements
Dr. Roberts Joffe
Prof. Masaki Hojo
Prof. Shojiro Ochiai