Fiber Bridging Characteristics of PVA-ECC Evaluated Based on the Sectional Image Analysis
B. Y. Lee1, Y. Y. Kim 2, J. S. Kim1, J. K. Kim1 Summary
A fiber bridging constitutive law was derived, which quantitatively considers the distribution of fiber orientation. In order to evaluate the fiber distribution characteristics an image processing technique was applied in the present study.
Fiber bridging curves were calculated taking the distribution of fiber orientation, which was measured by the use of the image analysis, into account. The bridging curves were then compared with those obtained from the assumption of two- and three-dimensional distribution of the fiber orientation.
Introduction
The fiber bridging curve governs composite tensile behavior. Therefore, to control and predict accurately it is very important to successfully design ECC material properties and tensile properties. Micromechanical parameters which affect of fiber bridging curve are properties of components such as fiber length, diameter, elastic modulus, strength, volume fraction, matrix strength, and elastic modulus and interfacial properties such as frictional bond strength, chemical bond strength, slip hardening effect. In addition to micromechanical parameters, fiber orientations affect the fiber bridging curve. For FRCC reinforced with three dimension randomly distributed fibers, the peak bridging stress could be 1/6 to 1/5 that of FRCC with 1 dimension aligned fibers[1]. Assumption of distribution of fiber orientation is primary important due to its effect to the number of fibers passing the crack plane, snubbing effect, and strength reduction. The distribution of fiber orientation is affected by geometry of specimen, rheological property of composite. However, most researchers assumed the distribution function of fiber orientation is 1/π for two dimension case or for three dimension case.
The purpose of this study is to present the impact of the fiber orientation on the fiber bridging curves. For this, the fiber bridging constitutive law considering quantitatively the distribution of fiber orientation is derived. And then, the distribution of fiber orientation was evaluated. The fiber bridging curves obtained on the basis of the distribution of fiber orientation measured using an image
1Dept. of Civil & Environmental Engineering, Korea Advanced Institute Science and Technology, Korea
2Dept. of Civil Engineering, Chungnam National University, Korea
analysis are compared with that obtained from the assumption of two dimension and three dimension distribution of the fiber orientation.
Fiber bridging constitutive law
Lin et al.[2] derived a theoretical single fiber debonding and pullout model based on simple stress analysis and energy balance principle. Eq. (1) and (2) represent the relation between load P and fiber pullout distance δ under debonding or pulling out and full debonding, respectively.
2
) 1 ( 2
) 1 ) (
(
3 2 0
3
2 π η
η δ τ
δ π + + +
= Efdf EfdfGD
P (1)
( )
[
0]
0 0 1 )
(δ π τ δ δ β − δ−δ
+ −
= e
f
f L
d d
P (2)
where Ef is elastic modulus of fibers, df is a diameter of fibers, τ0 is frictional stress, GD is chemical bond strength, η is
(
VfEf/
VmEm)
, Vf is volume fraction of fibers, Vm is volume fraction of matrix, β is slip-hardening coefficient, and Leis embedded length.The effect of fiber orientation which is called “snubbing effect” in the relation between P and δ is expressed as Eq. (3)[3,4].
θ P efθ
P
( )
=( 0 )
(3)The effect of fiber orientation in the relation between fiber strength σfu and δ is expressed as Eq. (3.14) (Lin et al. 1999).
( )
θ σ( )
θσfu = fu
0
e−f' (4)where f and f’ are snubbing effect coefficient and fiber strength reduction coefficient are determined by curve-fitting experimental data.
If resistance force of a single fiber at crack plane is P
(
θ,
Le,
δ)
composite bridging property can be expressed as Eq. (5).( ) (
θ δ) ( ) ( )
θ θ θ δ πσ π P L p dLd
d V
e L
e f
f B
f , , cos
4 /2
0 2 / 2
∫ ∫
0= (5)
If Nmis the number of fiber measured using image processing technique and Am is the measured area, Eq. (5) should be changed to the following Eq. (6).
( ) (
θ δ) ( ) ( )
θ θ θα π δ
σ π P L p dL d
d V
e L
e f
f nf B
f , , cos
4 /2 0
2 / 2
∫ ∫
0= (6)
where αnf is the fiber number coefficient which means the ratio of measured fiber numbers and assumed fiber numbers and can be expressed by Eq. (7).
=
∫ ∫
/20 2 / 0
2
) cos(
) (
4 π θ θ θ
α π
Lf
e m
f
m f nf
d dL p
A V
N
d (7)
This equation is derived from ( ) ( ) ( )
m m e L
e f
f
nf A
d N dL p
L d P
V f
∫ ∫
θ δ θ θ θ =α π4 π/2 , , cos
0 2 /
2 0 .
On the basis of Eq. (1) – (7), composite bridging property can be expressed as Eq. (8) using normalized length divided by
Lf/ 2 .
( )
( )
[ ]
∫ ∫
∫ ∫
−
−
−
+
+ + + +
=
4 3
4 3
2 1
2 21
) cos(
) ( 4 1
) cos(
) 2 (
) 1 ( 2
) 1 ( 4
0 0
2 0
3 2 0
3 2
2
θ θ
θ θ θ
θ
θ θ θ δ δ δ δ β
τ δ π π
α
θ θ η θ
δ π η τ π α π
δ σ
e e
e
L
L e
f e
f f
f f nf
e L f
L
D f f f
f f
f nf B
dL d p
e d L
d d V
d dL p
G e d E d
E d
V
(8) Distribution of fiber orientation and fiber bridging curves
This study used the PVA-ECC produced by Kim et al.[5]. Table 1 presents the fiber distribution. Although the fiber dispersion coefficients are little difference among specimens, ECC specimens with slag have higher values. This means that ECC with slag has more homogeneously distributed fibers in the composite. On the other hand, Fn values among specimens are regardless of addition of slag. The wc60wos specimens have smallest Fn which affects bridging curves. Figure 1 shows probability density function according to specimens. Although there is little difference according to specimens, probability
density is much difference comparing with those of 2 and three dimension random distribution.
Bridging curves according to the distribution of fiber orientation and specimens with different mix proportions are shown from Figure 2. Bridging curves predicted on the basis of PDFs measured using image analysis are similar with those predicted on the basis of PDFs assuming two dimension randomly distribution of fiber orientation. If the number of fibers is not considered to calculate bridging curve, bridging curves predicted using image analysis exhibits more about 25% higher stress. Therefore, crack openings at peak bridging stress of bridging curves predicted using image analysis are larger than those of predicted on the basis of PDFs assuming two dimension randomly distribution of fiber orientation. These differences in crack openings at peak bridging stress will cause the difference in prediction of the ultimate tensile strain capacity of tensile behavior. Furthermore, there is little difference in bridging curves in case of calculation of bridging curves on the basis of 2 or three dimension randomly distribution of fiber orientation.
Table 1 αf values calculated by proto-type and enhanced algorithms PDF wc60wos wc60ws wc48wos wc48ws αf
0.311 (0.0118)
0.321 (0.00836)
0.315 (0.00550)
0.317 (0.000106)
Fn
(number/mm2)
Measured 8.95 (0.391)
9.94 (0.291)
10.6 (0.968)
9.91 (0.144)
1D 16.7
2D 10.7
3D 8.37
(a) wc60wos (b) wc60ws
(c) wc48wos (d) wc48ws Figure 1 Probability density function according to specimens
(a) wc60wos (b) wc60ws
(a) wc48wos (b) wc48ws
Figure 2 Bridging curve according to the distribution of fiber orientation Conclusion
This paper presents the impact of the fiber distribution on the fiber bridging curves of PVA-ECC. A fiber distribution analysis was carried out. The fiber bridging constitutive law and crack spacing that quantitatively considers the distribution of fiber orientation and matrix spalling effect are derived. Then, in order to evaluate the fiber distribution characteristics, an image processing
technique was applied. During deriving these, the fiber number coefficient, which means the ratio of measured fiber numbers and assumed fiber numbers, is adopted. The bridging curves predicted on the basis of PDFs measured using image analysis are similar with those predicted on the basis of PDFs assuming two-dimensional random distribution of fiber orientation. If the number of fibers is considered not to calculate bridging curve, bridging curves predicted using image analysis exhibits more about 25% higher stress. Therefore, crack openings at the peak bridging stress of bridging curves predicted using image analysis are larger than those predicted on the basis of PDFs assuming two-dimensional random distribution of fiber orientation.
Acknowledgement
This study was supported by the Korea Ministry of Education, Science and Technology via the research group for control of crack in concrete and the Korea Science and Engineering Foundation(KOSEF) grant funded by the Korea government (MEST) (No.R01-2008-000-11539-0)..
Reference
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