materials

Article
Hybrid Fiber Influence on the Crack Permeability of Cracked
Concrete Exposed to Freeze–Thaw Cycles
Wei Zeng 1,2 , Weiqi Wang 1,2 , Qiannan Wang 1,2 , Mengya Li 3 , Lining Zhang 4 and Yunyun Tong 1,2, *

1 School of Civil Engineering and Architecture, Zhejiang University of Science and Technology,
Hangzhou 310023, China; wei-zeng@zust.edu.cn (W.Z.); 121032@zust.edu.cn (W.W.);
wangqiannan@zust.edu.cn (Q.W.)
2 Zhejiang International Science and Technology Cooperation Base for Waste Resource Recycling and
Low-Carbon Building Materials Technology, Zhejiang University of Science and Technology,
Hangzhou 310023, China
3 Laboratory of Mechanics and Materials of Civil Engineering (L2MGC), EA 4114, CY Cergy Paris Université,
Cergy, 95000 Paris, France; mengya.li1@cyu.fr
4 Department of Mechanical and Mechatronics Engineering, The University of Auckland,
Auckland 1010, New Zealand; lzha977@aucklanduni.ac.nz
* Correspondence: 112013@zust.edu.cn

Abstract: This paper describes hybrid fiber’s influence on the crack permeability of cracked con-
crete exposed to freeze–thaw cycles. A permeability setup and a laser-scanning setup have been
designed to measure the crack permeability and the fractured surface roughness of cracked hybrid
fiber-reinforced concrete, containing polypropylene fiber and steel fiber, under a splitting tensile
load. The results show that, when the effective crack width of the specimens is less than 25 µm,
the rough crack surface significantly reduces the concrete’s crack permeability. As the crack width
increases, the effect of the concrete crack surface on crack permeability gradually decreases, and the
crack permeability of the concrete is closer to the Poiseuille flow model. The permeability parameter
α derived from the Poiseuille flow model is effective for assessing the crack permeability of concrete.
Compared to the modified factor ξ of crack permeability, the permeability parameter α can effec-
tively evaluate and quantify the development trend of crack permeability within a certain range of
crack widths. The permeability parameter α of SF20PP2.3, subjected to the same freeze–thaw cycles,
Citation: Zeng, W.; Wang, W.; Wang, decreases by 16.3–94.8% compared to PP4.6 and SF40, and SF20PP2.3 demonstrates a positive syner-
Q.; Li, M.; Zhang, L.; Tong, Y. Hybrid gistic effect on the crack impermeability of cracked concrete. The crack impermeability of SF40PP2.3,
Fiber Influence on the Crack subjected to the same freeze–thaw cycles, lies between that of PP6.9 and SF60. The roughness of crack
Permeability of Cracked Concrete
surface (X) and the crack permeability (Y) are highly correlated and follow an exponential curve
Exposed to Freeze–Thaw Cycles.
(Y = 1.0415 × 107 ·e−6.025·X ) in concrete. This demonstrates that hybrid fibers enhance crack imperme-
Materials 2024, 17, 1819. https://
ability by increasing the crack surface roughness. Furthermore, the combination of polypropylene
doi.org/10.3390/ma17081819
fiber and steel fiber effectively promotes the formation of micro-cracks and facilitates the propagation
Academic Editor: Carlos Leiva of multiple cracks in the concrete matrix. This combination increases the head loss of water flow
Received: 10 March 2024 through the concrete and decreases the crack permeability.
Revised: 8 April 2024
Accepted: 12 April 2024 Keywords: crack permeability; hybrid fiber; crack surface; positive synergistic effect; splitting
Published: 16 April 2024 tensile load

Copyright: © 2024 by the authors. 1. Introduction

Licensee MDPI, Basel, Switzerland.
In cold regions, concrete structures are usually exposed to freeze–thaw cycles in the
This article is an open access article
service stage. After freeze–thaw cycles, concrete suffers from freeze–thaw damage and its
distributed under the terms and
internal micro-cracks increase significantly [1,2]. This increases the permeability of concrete
conditions of the Creative Commons
materials. External harmful substances, such as chloride ion and sulfate ion, can easily
Attribution (CC BY) license (https://
enter the interior of the concrete structure through micro-cracks caused by freeze–thaw
creativecommons.org/licenses/by/
cycles, aggravating the corrosion of the concrete structure and reducing its load-bearing
4.0/).

Materials 2024, 17, 1819. https://doi.org/10.3390/ma17081819 https://www.mdpi.com/journal/materials

Materials 2024, 17, 1819 2 of 22

capacity and service life. Generally, the application of macro fibers in civil engineering
can effectively reduce internal defects and decrease the concrete damage under harsh
environmental conditions such as freeze–thaw cycles [3,4]. The durability of concrete is
directly related to its permeability, and concrete structures in the service stage often work
with cracks [5–7], and the appearance of cracks significantly increases the permeability of
concrete [8]. The macro fibers can effectively limit the formation and development of macro-
cracks in concrete structures [9–11] and decrease the permeability of cracked concrete.
Because the effects of different types of macro fibers on concrete properties are signifi-
cantly different, the effect of the hybrid application of multiple fibers on concrete has been
explored to overcome the limitations of single-fiber reinforced concrete. Soner et al. [12]
found that using macro-PP fibers in a hybrid form with micro-PP fibers effectively reduced
abrasion loss and improved mechanical capacities more than using single macro-PP fibers.
Wang et al. [9] found that fibers with a high elastic modulus can increase the initial crack-
ing stress and ultimate strength, while fibers with a low elastic modulus can effectively
increase the toughness and strain capacity of the post-peak behavior of cracked concrete.
Athanasia et al. [13] found that hybrid cement–mortar composites, strengthened with
high-elastic-modulus carbon fibers and low-elastic-modulus polypropylene fibers, show
improved mechanical and electromechanical responses. Therefore, hybrid fiber-reinforced
concrete (HFRC) with a different elastic modulus and fibers can effectively show a better
performance than concrete reinforced with a single type of fiber. Banthia et al. [14] and
Caggiano et al. [15] found that the combination of polypropylene fiber and steel fiber
increases the bending–hardening performance of concrete and shows a positive syner-
gistic effect. Afroughsabet et al. [16] found that the combination of polypropylene fiber
and steel fiber with a certain ratio can significantly improve the mechanical properties
of concrete. Ding et al. [17] found that a hybrid fiber of polypropylene fiber and steel
fiber improves the flexural toughness of concrete more than steel fiber or polypropylene
fiber alone, with the same volume content. Furthermore, Rashid et al. [18] found that a
hybrid fiber of polypropylene fiber and steel fiber enhances the durability performance of
concrete structures. It is worth noting that hybrid fibers have a significant effect on concrete
properties (including mechanical properties, electromechanical properties and durability
properties). In cold regions, the effect of freeze–thaw cycles plays a key role in exacerbating
the durability deterioration of cracked concrete structures [19]. So, it is necessary to explore
the hybrid fiber (steel fiber and polypropylene fiber) influence on the crack permeability
of cracked concrete exposed to freeze–thaw cycles. However, this research work is still
very limited.
When the seepage medium and seepage pressure are constant, the crack morphology
of concrete significantly impacts the crack permeability of concrete [3]. Previous stud-
ies [20–22] have shown that the morphology of concrete cracks can often be represented
by two-dimensional parameters, such as the tortuosity of cracks on the surface of concrete
structures. However, since a concrete crack has three-dimensional characteristics, the two-
dimensional parameters do not accurately represent the morphology of concrete cracks.
Therefore, this study employs a self-developed 3D laser scanner to scan and construct the
crack surface. Based on the laser sensors in the scanner, the coordinates of each point on the
concrete crack surface can be effectively measured and the crack surface morphology can
be reconstructed, thus quantifying the crack surface roughness. Based on the data of crack
surface morphology, a three-dimensional parameter, such as a crack surface roughness
number (Rn ) [23–25], can be calculated, and the hybrid fiber’s influence on the morphology
of cracks in cracked concrete can be analyzed.
In engineering practice, the majority of concrete structures serve under loading. The
crack morphology under loading significantly differs from that of cracks after unloading.
Considering the loading condition, the study investigates the crack permeability of concrete
under loads in real time. As the cracks continue to expand, the crack permeability of the
concrete increases. Previous studies [26–29] typically calculated the crack permeability
of concrete by quantifying the mass of water flowing out from the water pipes of the
Materials 2024, 17, 1819 3 of 22

permeability setup. However, since the permeability setup includes the specimens, vessels,
and water pipes, the process of water flowing out from the water pipes of the permeability
setup introduces a delay in the results, or water remaining in the permeability setup also
lead to errors in measuring the permeability of cracked concrete. Therefore, this study
designs a permeability setup, which makes water flow through the concrete by reducing
the air pressure at the downstream of the outflow water from the permeability setup. At
the same time, the water mass passing through the concrete is measured by an electronic
scale monitoring the water container at the upstream of the permeability setup. The water
quantity passing through the concrete is used to calculate the permeability of the concrete.
This method reduces the error in measuring the crack permeability and enhances the
precision of the results.
Hybrid fibers have been widely used in civil engineering. In order to estimate the
effect of the combination of polypropylene fiber and steel fiber on the crack impermeability
of concrete structures in engineering applications, a series of tests were carried out: (a) a
rapid freeze–thaw test was carried out to introduce freeze–thaw damage to the concrete,
(b) the relationships between the crack opening displacement (DCOD ) and the effective
crack area (Aeff ) of the fiber-reinforced concrete (FRC) specimens were estimated, (c) a
permeability test under a splitting tensile load was conducted to evaluate the permeability
of the concrete and (d) a morphological analysis of the concrete crack surface was conducted,
using the developed 3D laser-scanning setup. Furthermore, analyses of the relationship
between the crack surface roughness and the permeability parameter α, as well as the
positive synergistic effect of hybrid fiber on crack impermeability, were conducted. Based
on the relationship between the crack surface roughness and the permeability parameter α
obtained in this study, the permeability of the cracked concrete can be quickly evaluated by
the crack surface roughness in real engineering applications. In addition, the permeability–
deformation relationship of cracked concrete in this study can be combined with the
load–deformation relationship and finite-element analysis of cracked concrete in the service
stage. This can effectively estimate the permeability of fiber-reinforced concrete structures
under loading and provide a design basis for enhancing the permeability of hybrid fiber
(steel fiber and polypropylene fiber)-reinforced concrete under freeze–thaw conditions for
engineering applications.

2. Experiment
2.1. Materials
The mix design of the concrete was as follows: cement (P·O) 390 kg/m3 , coarse aggre-
gate (5–10 mm) 848 kg/m3 , fine aggregate (0–5 mm) 822 kg/m3 , fly ash 155 kg/m3 , water
272.5 kg/m3 and superplasticizer 4.0 kg/m3 . The macro fibers used include polypropylene
fiber and steel fiber. The geometries and performance parameters of the macro fibers are
shown in Figure 1. The hybrid fiber consists of polypropylene fiber and steel fiber, and
the fiber volume contents of the specimens were 0.50 vol.% and 0.75 vol.%, respectively.
The polypropylene fiber-reinforced concrete (PFRC) and steel fiber-reinforced concrete
(SFRC), with the same fiber volume content, and normal concrete (NC) were the reference
specimens. The specimens were named based on their macro fiber content, as shown in
Table 1.

Table 1. Fiber content of specimens.

Macro Fiber
Mixture ID Steel Fiber Polypropylene Fiber
Volume Content
NC 0 vol.% 0 kg/m3 (0 vol.%) 0 kg/m3 (0 vol.%)
SF20PP2.3 20 kg/m3 (0.25 vol.%) 2.3 kg/m3 (0.25 vol.%)
SF40 0.50 vol.% 40 kg/m3 (0.5 vol.%) 0 kg/m3 (0 vol.%)
PP4.6 0 kg/m3 (0 vol.%) 4.6 kg/m3 (0.5 vol.%)
Materials 2024, 17, 1819 4 of 22

Table 1. Cont.
Materials 2024, 17, x FOR PEER REVIEW 4 of 23
Macro Fiber
Mixture ID Steel Fiber Polypropylene Fiber
Volume Content
SF40PP2.3 40 kg/m3 (0.5 vol.%) 2.3 kg/m3 (0.25 vol.%)
SF40PP2.3 40 kg/m3 (0.5 vol.%) 2.3 kg/m3 (0.25 vol.%)
SF60 SF60
0.75 vol.% 0.7560
vol.% kg/m3 (0.75 vol.%)0 kg/m3 (00 vol.%)
kg/m3 (0.7560vol.%) kg/m3 (0 vol.%)
Materials
PP6.92024, 17, x FOR PEER REVIEW 0 kg/m (0 vol.%) 4 of vol.%)
23 3
0 kg/m3 (0 vol.%)6.9 kg/m 6.9
(0.75
3 3
PP6.9 kg/m (0.75 vol.%)

SF40PP2.3 40 kg/m3 (0.5 vol.%) 2.3 kg/m3 (0.25 vol.%)

SF60 0.75 vol.% 60 kg/m3 (0.75 vol.%) 0 kg/m3 (0 vol.%)
PP6.9 0 kg/m3 (0 vol.%) 6.9 kg/m3 (0.75 vol.%)

(a) (b)
Figure
Figure1.1.Geometries and performance
Geometries parameters
and performance of macro fibers:
parameters (a) polypropylene
of macro fibers: (a) fiber; (b)
polypropylene fiber;
steel fiber.
(b) steel fiber. (a) (b)
2.2.
2.2.Specimens
Figure 1. Geometries and performance parameters of macro fibers: (a) polypropylene fiber; (b)
Specimens
steel fiber.
The
Thefresh
fresh concrete
concretewaswaspoured into steel
poured intomolds of 400 mm
steel molds × 400mm
of 400 mm×× 205 400 mm,
mmand× 205 mm,
it was
2.2. vibrated for 30 s to remove air bubbles on the vibrating table. A plastic wrap was
Specimens
and it was vibrated for 30 s to remove air bubbles on the vibrating table. A plastic wrap
usedTheto cover
fresh the surface to prevent cracking.
moldsAfter
of 400demolding, the× specimens were cured
was used toconcrete
cover thewas surface
poured into
to steel
prevent mm ×After
cracking. 400 mm 205 mm, and
demolding, the specimens were
for
it was vibrated for 30 s to remove air bubbles on the ◦vibrating table. A plastic wrap was drill was
28 days in a standard room at 20 °C and 95% humidity. After curing, a core
cured for 28 days in a standard room at 20 C and 95% humidity. After curing, a core drill
used
used totocover
extract cylindrical
the surface specimens
to prevent cracking.(size: diameter =the
After demolding, 100 mm, height
specimens = 205 mm), as
were cured
for 28used
was
shown days toa extract
in
in Figure standard cylindrical
room at 20 °Cspecimens
2a. Subsequently, and
the95% (size: specimens
humidity.
cylindrical diameter =were
After curing, a100 mm,
corecut height
drillinto
was = 205 mm), as
specimens
shown
used to
(size: in Figure 2a.
extract cylindrical
diameter Subsequently,
= 100 mm, specimens
height =(size:the cylindrical
diameter
50 mm), and=the specimens
100specimens
mm, heightwere were cut
= 205polished into specimens
mm), as to ensure (size:
diameter
shown in = 100
Figure mm,
2a. height =
Subsequently, 50
the mm), and
cylindrical the specimens
specimens
that their cut surfaces were parallel, as shown in Figure 2b. were were
cut intopolished
specimens to ensure that their
(size: diameter =were
cut surfaces 100 mm, height as
parallel, = 50 mm), and
shown the specimens
in Figure 2b. were polished to ensure
that their cut surfaces were parallel, as shown in Figure 2b.

(a)(a) (b) (b)

(c) (d)
(c) (d)
Figure 2. Preparation of cylinder specimen: (a) cylindrical specimen (size: diameter = 100 mm,
height = 205 mm); (b) specimen (size: diameter = 100 mm, height = 50 mm); (c) specimen with white
color; (d) specimen with waterproof tape.
 Materials
Materials 2024, Materials 2024,
17, x FOR2024,
PEER 17,REVIEW
17, xx FOR
FOR PEER
PEER REVIEW
REVIEW 5 of 23 55 o
o

Figure
Figure 2.
Figure 2. Preparation Preparation
2.of of
of cylinder
cylinder specimen:
Preparation (a)specimen:
cylinder cylindrical(a)
specimen: (a) cylindrical
specimen specimen
(size:
cylindrical diameter
specimen (size: diameter
= 100
(size: mm, == 100
diameter 100 mm,
mm,
Materials 2024, 17, 1819 height == 205 mm); 5 of 22
height = 205 mm);
height(b) mm); (b)
specimen
205 specimen
(size:
(b) (size:
diameter
specimen diameter
= 100
(size: == 100
mm, height
diameter 100=mm, height
50 mm);
mm, (c)==specimen
height 50
50 mm);
mm); (c)
with
(c) specimen
specimen with
with
white
white color; (d) color; (d)
specimen
white color; (d) specimen
with withtape.
waterproof
specimen with waterproof tape.
waterproof tape.

AAfreeze–thaw
freeze–thawA
A freeze–thaw
testwas
freeze–thaw
test test
test was
wasconducted
conductedwason conducted
on on
on the
thespecimens,
conducted
the specimens, thetospecimens,
to to
to induce
inducefreeze–thaw
specimens,
induce induce freeze–thaw
freeze–thaw damage
freeze–thaw
damage dam
dam
according toaccording
ASTM
according toaccording
ASTM C666 to
C666 ASTM
[30].
to ASTM
[30]. The C666
The
C666 [30].
target The target
freeze–thaw
[30].freeze–thaw
target freeze–thaw
cycles
The target freeze–thaw were
cycles were 50, cycles
50,
cycles100 were
100wereand 50,
150,100 and
respec-
50, 100
and 150, respec- res
and 150, resp
150, res
tively. Twelve
tively. Twelve tively.
tively. Twelve
specimens
Twelve
specimens werespecimens
were prepared
specimens were
preparedwerefor eachprepared
for each
prepared for
specimen.each
for each
specimen. Before specimen.
Before the
specimen.
the splittingBefore
splitting
Before the splitting
splitting ten
tensile
the test,
tensile ten
test,two
the test,
thesurfaces
twotest, the
surfaces
the two
two
of the surfaces
ofsix
the
surfaces
specimensof the
six specimens
of the six
sixwere
were specimens
painted
specimens
painted were
were
white painted
white white
to record
topainted
record white
the to record
thetocrack
crack record the
patterns
patterns crack patte
theon
crack patt
patte
on specimen
the onsurfaces,
the specimen
on thesurfaces,
the specimen
specimen
as shownsurfaces,
as surfaces,
shown asFigure
inas
in Figure shown
shown inBefore
2c.in
2c. Before Figure
Figure 2c.permeability
the
2c. Before the
Before
the permeability the permeability
tests, tests, the
the other
permeability
tests, the other tests,
six the oo
six specimens
specimens six
were specimens
six were
wrapped
specimens onwere
wrapped theon
were wrapped
sides
wrapped sideson
thewith the
onwith
waterproofsides
the waterproofwith
sidestape. waterproof
tape.
withThis tape.
This prevented
prevented
waterproof This
water
tape. prevented
prevented w
water
leakage
This w
leakage from
from the sides leakage
the from
sides
of the from
leakage of the
the
specimens, sides of
specimens,
the sidesas shownthe specimens,
as shown in
in Figure 2d.
of the specimens, as shown
Figure 2d.in Figure
as shown in Figure 2d. 2d.

2.3.
2.3.Acquisition
2.3.ofof
2.3.
Acquisition Crack
CrackData
Acquisition
Acquisition ofofofCrack
Dataof Concrete
DataSurface
Data
Concrete
Crack of
Surface under
of Concrete
under
Concrete Splitting
Surface Tensile
under
Splitting
Surface Load
Splitting
Tensile
under Load Tensile
Splitting Tensile Load
Load
Splitting
Splittingtensile
tensile tests
Splitting
tests were
were conducted
conducted on the
conducted
on the specimens,
on the
specimens, following
Splitting tensile tests were conducted on the specimens, following ASTM
tensile tests were specimens,
following ASTM C496
following
ASTM C496 [31]
[31] C496
ASTM C496
and GB/T 50081-2002
and
and GB/T 50081-2002 GB/T
and GB/T [32].[32]. An
50081-2002
An MTS
50081-2002 MTS 250
[32].
250An
[32]. kN
An
kNMTS hydraulic
MTS 250
hydraulic kNservo machine
hydraulic
250 kNservo machine
hydraulic (Manufacturer:
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servo machine
(Manufacturer:(Manufacturer:
MTS
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machine (Manufacturer: M
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Systems Corporation
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(China), Shenzhen,
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with aa closed-loop
a closed-loop
China) used,
control wascontrol
closed-loop used, was
control was uu
under
underaadeformation
under rate ofof0.012 mm/min. Two cameras recorded crack patterns in the
under a deformation rate of 0.012 mm/min. Two cameras recorded crack patterns in
deformation a deformation
rate 0.012 rate of
mm/min. 0.012Twomm/min.
camerasTwo cameras
recorded recorded
crack crack
patterns in patterns
the in
specimens
specimensunder
specimensloading, as
underasshown
loading, in Figure 3.3. in Figure 3.
under
specimens loading,
under loading,inas
shown shown
Figure
as shown in Figure 3.

Figure
Figure3.3.Setup
Setup 3.
3. Setup
forsplitting
Figure splitting for
for splitting
tensile
Setuptensile test. tensile
splitting tensile test.
test.
Figure for test.

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The The
surfacecrack
crack
The surface
patterns
surface
patterns crack
crack
of patterns
ofthe
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concrete of
of the
the concrete
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specimens specimens
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specimens were
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the concrete
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[33].the concrete
[33].
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crack gradu
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on the on
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formed the
on the concrete
surface. surface.
concrete surface.

Table2.2.Crack
Table Table
Crack 2. Crack
Crack
evolution
evolution
Table 2. evolution process.
process.
process.
evolution process.

DCOD DCOD
D COD
COD DCOD Morphology
Morphology of
Morphology
Morphology of
Crack onof Crack
the on the
Specimen
ofCrack
Crack on
on theSurface
the Specimen
Specimen
Specimen Surface
Surface
Surface
10 µm 10
10 µm
µm 10 µm

50 µm 50
50 µm
µm 50 µm

100 µm 100
100 µm
µm 100 µm
Materials 2024, 17, x FOR PEER REVIEW 6 o

200 µm 200
200 µm
µm 200 µm

300 µm 300 µm

Wang et al. [34] found that concrete was a multiphase heterogeneous material,
cracks typically first appear on one side of the concrete specimen and then extend to
other side. In order to assess the width of crack propagation, the study employed 4 lin
variable differential transformers (LVDTs) (Manufacturer: Shenzhen Milont Technol
Co., Ltd., Shenzhen, China) arranged on both sides of the specimen along its central a
The displacement values d1, d2, d3, and d4 were measured using the four LVDTs, res
Materials 2024, 17, 1819 6 of 22

Wang et al. [34] found that concrete was a multiphase heterogeneous material, and
cracks typically first appear on one side of the concrete specimen and then extend to the
other side. In order to assess the width of crack propagation, the study employed 4 linear
variable differential transformers (LVDTs) (Manufacturer: Shenzhen Milont Technology Co.,
Ltd., Shenzhen, China) arranged on both sides of the specimen along its central axis. The
displacement values d1 , d2 , d3 , and d4 were measured using the four LVDTs, respectively.
The radial deformation (diameter variation, DV ) of the specimens under a splitting tensile
loading was calculated by Equation (1).

d1 + d2 + d3 + d4
DV = (1)
2
DV includes both the crack width and the elastic deformation of the concrete matrix
under load. Rastiello et al. [35] found that the two parts of concrete after cracking under
a splitting tensile loading can still be considered as elastic bodies. Therefore, the DCOD is
calculated using Equation (2).
DCOD = DV − ∆d,e (2)
∆d,e represents the elastic deformation of the concrete in the radial direction, which
can be calculated based on the load–DV curve. Equation (2) establishes the relationship
between the DV and the DCOD of the concrete specimen.
Akhavan et al. [8] found that, in order to accurately measure the crack permeability, it
is necessary to obtain the effective crack width (b) [26]. First, the crack area on the specimen
surface is required. However, as the crack on the specimen surface was covered with the
water vessels, rubber rings and waterproof tape during the permeability test, it is difficult
to capture the crack patterns on the specimen surface. Nonetheless, the DV of the concrete
specimen under splitting tensile load could be precisely measured. Therefore, the effective
crack area (Aeff ) was indirectly determined by the DCOD during the permeability test, based
on the statistical relationship between the DV and DCOD .
Figure 4 shows the data of DCOD and corresponding to Aeff of a representative concrete
specimen and demonstrates the statistical relationship between the DCOD and Aeff . From
Figure 4, the DCOD and Aeff have a good linear relationship, which can be expressed by
Equation (3).
Y = c·X (3)
where parameter c is the fitting parameter of the statistical relationship between the DCOD
and Aeff .
With the increasing of the DCOD of the specimen under loading, the Aeff on the surface
of the concrete specimen also increases linearly. Using the relationship between the DCOD
and Aeff of the concrete specimen, the Aeff on the concrete surface covered with water vessels
in the permeability test can be indirectly evaluated by the DCOD . The fitting parameter c
and the correlation coefficient R2 of the relationship between the DCOD and Aeff for each
group of specimens are listed in Table 3. The freeze–thaw cycles of the specimens were 0, 50,
100 and 150, respectively. From Table 3, the fitting parameter c of the statistical relationship
between the DCOD and Aeff of different specimens ranges between 0.02773 and 0.0564, with
a correlation coefficient of R2 ≥ 0.91. This implies that the Aeff can be determined from the
DCOD using the statistical relationship between the DCOD and Aeff .
Materials 2024,17,
Materials2024, 17,1819
x FOR PEER REVIEW 7 of
of 22
23

eff of representative concrete specimen.

Figure4.4.Statistical
Figure Statisticalrelationship betweenDDCOD
relationshipbetween COD and Aeff of representative concrete specimen.

c between eff of different specimens.

Table 3. Fitting
With theparameter
increasing of the DD COD
COD the Aspecimen
ofand under loading, the Aeff on the surface
of the concrete specimen also increases linearly. Using the relationship between the DCOD
Freeze–Thaw and Aeff of the concrete specimen, the Aeff on thec concrete surface covered with water ves-
Fitting Parameter
Cycles NC sels in SF20PP2.3
the permeability SF40 PP4.6
test can be indirectly SF40PP2.3
evaluated by the DCOD SF60 PP6.9
. The fitting parameter
0.02906 c and the correlation0.03538
0.0369 coefficient R 0.0441
2 of the relationship between the
0.0348 DCOD and A0.0424
0.04039 eff for each
0
(R2 = 0.99)group(Rof
2 =specimens
0.91) 2 = listed
(Rare (R2 = 0.95)
0.96) in Table (R2 = 0.98) cycles
3. The freeze–thaw 2 =the
(Rof (R2 = were
specimens
0.98) 0.97) 0,
0.02773 50, 100 and
0.0471150, respectively.
0.05042 From Table
0.0492 3, the fitting
0.0564parameter c of
0.03383 the statistical
0.0475 rela-
50 tionship between the D and A eff2 of different specimens ranges between 0.02773
2
(R = 0.97) 2
(R = 0.99) 2
(R = 0.99)
COD (R = 0.97) 2
(R = 0.96) 2
(R = 0.97) R = 0.96)and
2

0.04719 0.0564, 0.0512

with a correlation
0.03343coefficient of R ≥ 0.91. 0.0429
This implies that the A can0.0511 be deter-
2 eff
0.0358 0.05608
100 2 mined from
2 the D COD using
2 the statistical
2 relationship2 between the2 D COD and A eff2.
(R = 0.95) (R = 0.98) (R = 0.94) (R = 0.97) (R = 0.99) (R = 0.99) (R = 0.99)
0.03472 0.0461 0.04161 0.0377 eff 0.0476 0.04079 0.0294
150 Table 3.2Fitting parameter c between DCOD and A of different specimens.2
(R2 = 0.96) (R = 0.97) (R2 = 0.98) (R2 = 0.96) (R2 = 0.99) (R = 0.98) (R2 = 0.99)
Fitting Parameter c
Freeze–Thaw Cycles
NC SF20PP2.3
2.4. Permeability Test SF40 PP4.6 SF40PP2.3 SF60 PP6.9
0.02906As shown in0.0369
Figure 5, the 0.03538
device includes 0.0441 0.0348 a water 0.04039
a vacuum pump, 0.0424
pump, an electronic
0
(R2scale,
= 0.99)a container
(R2 =and
0.91) (R 2 = 0.96) (R2 = 0.95) (R 2 = 0.98) (R 2 = 0.98) (R2 = 0.97)
two water vessels. The principle of the device is to fill the water
0.02773 0.0471 with water
vessel in the upstream 0.05042using the0.0492
water pump, 0.0564
and to decrease0.03383 0.0475
the air pressure
50
(R2in the downstream
= 0.97) (R2 = using
0.99) the(Rvacuum
2 = 0.99)pump; a pressure(Rdifference
(R2 = 0.97) 2 = 0.96) of (R ∆p2 = 0.97)
90 kPaRfrom the
2 = 0.96)

upstream
0.04719 to the downstream
0.0512 of the
0.03343 permeability
0.0358 setup was
0.0429 created. An electronic
0.05608 scale
0.0511
100
(R2with an accuracy
= 0.95) (R2 of 0.0001 g (R
= 0.98) was 2 =situated
0.94) (Rat2 =the upstream
0.97) (R2 =of0.99)
the permeability
(R2 = 0.99) setup,
(R2 =and
0.99)a
beaker with water
0.03472 was placed0.04161
0.0461 on the electronic
0.0377 scale. The pressure difference
0.0476 0.04079 caused 0.0294the
150 water from the beaker to flow through the concrete specimen and finally into the container.
(R2 = 0.96) (R2 = 0.97) (R2 = 0.98) (R2 = 0.96) (R2 = 0.99) (R2 = 0.98) (R2 = 0.99)
When the test started, the decrease in mass on the electronic scale was the mass of water
that flowed through the specimen. This enabled a real-time measurement of the mass and
2.4. Permeability Test
evaluation of the permeability of the concrete under loading.
As shown
During the in Figure 5, the
permeability device
test, whenincludes a vacuum
cracks appear pump,
in the a water
concrete pump, an
specimens, elec-
water
tronic scale,
permeates a container
through both theand two water
concrete cracksvessels.
and theThe principle
concrete matrixof [36].
the device is to fillthe
Consequently, the
water vessel in the upstream with water using the water pump, and
water flow rate (Q) included the water flow through the concrete crack (Qc ) and throughto decrease the air
pressure in the downstream using the vacuum pump; a pressure difference
the concrete matrix (Qm ). Seong et al. [37] found that the splitting tensile load does not of Δp = 90 kPa
from the upstream
affect Qm of thetoconcrete.
the downstream
Therefore, of the
thepermeability setup was
Qm can be obtained created.
from An electronic
the permeability
scale with an accuracy of 0.0001 g was situated at the upstream of the permeability setup,
Materials 2024, 17, 1819 8 of 22

test without a splitting tensile load at the beginning of the test. The Qc was calculated by
Equation (4).
Qc = Q − Qm (4)
During permeability tests, the hydraulic pressure gradient was kept below 1.8 MPa/m
to ensure laminar water flow into the concrete specimens. Darcy’s law could be used to
evaluate the crack permeability [35], represented by Equation (5).
Materials 2024, 17, x FOR PEER REVIEW 8 of 23
Qc µ ∆p −1 Qc µ ∆p −1
κc = e f f ( ) = ( ) (5)
A ∆x b · L ∆x

and a beaker
where with water
κ c represents the was
crackplaced
permeability, m2 ; ∆p/∆x
on the electronic scale.specifies
The pressure differencepressure
the hydraulic caused
the waterPa/m;
gradient, from the beaker
ρ is to flowof
the density through
water the concrete
at room specimen and
temperature, kg/m 3
finally eff
; A into the crack
is the con-
tainer. When
effective m2 ;test
area, the µ isstarted, the decrease
the dynamic in mass
viscosity on the0.001
of water, Pa·s; Lscale
electronic was thethe
represents mass of
crack
water oriented
length that flowed through the
perpendicular to specimen. Thisdirection;
the water flow enabled aand real-time
b is the measurement
effective crackof the
width,
mass
m, and evaluation
as shown in Figureof6.the permeability of the concrete under loading.

Materials 2024, 17, x FOR PEER REVIEW 9 of 23

Figure5.5.Schematic
Figure Schematicview
view of
of permeability
permeability setup.
setup.

During the permeability test, when cracks appear in the concrete specimens, water
permeates through both the concrete cracks and the concrete matrix [36]. Consequently,
the water flow rate (Q) included the water flow through the concrete crack (Qc) and
through the concrete matrix (Qm). Seong et al. [37] found that the splitting tensile load does
not affect the Qm of the concrete. Therefore, the Qm can be obtained from the permeability
test without a splitting tensile load at the beginning of the test. The Qc was calculated by
Equation (4).

Qc  Q  Qm (4)

During permeability tests, the hydraulic pressure gradient was kept below 1.8
MPa/m to ensure laminar water flow into the concrete specimens. Darcy’s law could be
used to evaluate the crack permeability [35], represented by Equation (5).

Qc p Qc p
Figure6. 6. Effective
Effective crack
crack width
width of c  effwhere
of concrete,
1
( )AAu/dis isthethecrack 1
( crack
)area ononboth concrete sides, m(5)
2; 2
Figure
Au/deff is
eff the effective crack area on the concretesurface,
A where
concrete,
x u/dm b2;L2and bxu/d isarea
the
both
effective
concrete
crack
sides,
width on
m ;
Au/d is the effective crack area on the concrete surface, m ; and bu/d is the effective crack width on
both concrete sides, m.
where
both κc represents
concrete sides, m. the crack permeability, m2; Δp/Δx specifies the hydraulic pressure
gradient, Pa/m; ρ is the density of water at room temperature, kg/m3; Aeff is the crack ef-
2.5. Collection of2Crack Surface Topography
fective area, m ; µ is the dynamic viscosity of water, 0.001 Pa·s; L represents the crack
lengthFollowing
orientedthe permeability to
perpendicular test,
thethewater
specimen
flow was bisected
direction; along
and b isthe
thecrack. The fibers
effective crack
on the crack surface of the
width, m, as shown in Figure 6.specimen were cut, and a laser scanner was used to perform
morphological scanning and information collection of the crack surface. The scanning area
Materials 2024, 17, 1819 9 of 22

2.5. Collection of Crack Surface Topography

Following the permeability test, the specimen was bisected along the crack. The fibers
Figure 6. Effective crack width of concrete, where Au/d is the crack area on both concrete sides, m2;
on the crack surface of the specimen were cut, and a laser scanner was used to perform
Au/deff is the effective crack area on the concrete surface, m2; and bu/d is the effective crack width on
morphological scanning and information collection of the crack surface. The scanning
both concrete sides, m.
area was 75 mm × 35 mm on the crack surface, as shown in Figure 7. The crack surface
roughness Rn of
2.5. Collection of Crack
the concrete
Surface specimen
Topography was calculated by Equation (6).

Following the permeability test, the specimen

St was bisected along the crack. The fibers
on the crack surface of the specimen wereRn =cut,Soand a laser scanner was used to perform (6)
morphological scanning and information collection of the crack surface. The scanning area
was 75Smm
where o is ×
the
35 projected
mm on thearea of surface,
crack the scanning region,
as shown 75 mm7.×
in Figure 35 crack
The St is therough-
mm; surface crack
surface area of the scanning region, mm 2.
ness Rn of the concrete specimen was calculated by Equation (6).

(a) (b)
Figure 7.
Figure 7. (a)
(a) Scanning
Scanning of
of crack
crack surface
surface of
of specimen,
specimen, (b)
(b) scanning
scanning path
path of
of crack
crack surface.
surface.

3. Results and Discussion

3.1. Crack Permeability of Concrete St
nR  (6)
Figures 8 and 9 present the effective crack Swidth–crack
o
permeability curves of the
HFRC specimens with a fiber volume content of 0.5 vol.% and 0.75 vol.%, respectively. The
mono FRC specimens with the same fiber volume content and the Poiseuille flow model
where So is the projected area of the scanning region, 75 mm × 35 mm; St is the crack sur-
serve as references. As the effective crack width values are 100 µm and 200 µm for the
face area ofwith
specimens the scanning region,content
a fiber volume mm2. of 0.5 vol.% and 0.75 vol.%, respectively, the crack
permeability values κ c-100 and κ c-200 are shown in Tables 4 and 5.
3. Results and Discussion
3.1. Crack
Table Permeability
4. Comparison of Concrete
of crack permeability of SF20PP2.3, SF40 and PP4.6 exposed to various freeze–
thaw Figures
cycles. 8 and 9 present the effective crack width–crack permeability curves of the
HFRC specimens with a fiber volume content of 0.5 vol.% and 0.75 vol.%, respectively.
Freeze–Thaw κc-100
The mono FRC(m2specimens
) κc-200 (m
with the same fiber volume content
2)
and the Poiseuille flow
Cycles SF20PP2.3model serveSF40as references. As the effectiveSF20PP2.3
PP4.6 crack width valuesSF40
are 100 µm and PP4.6
200 µm for
0 9.97 × 10 the specimens
− 12 1.38 × 10with a fiber
− 11 7.94volume
× 10 − 11 content
1.03of
× 0.5
10 vol.% and
− 10 1.680.75
× 10vol.%, respectively,
− 10 1.14 × 10−9 the
50 −
8.57 × 10 crack permeability
12 6.13 × 10 − 12 values κc-100
4.36 −
× 10and κc-200 are
11 6.66shown −
× 10 in Tables
11 8.924×and −
10 5.
11 6.88 × 10−10
100 4.31 × 10−12 3.55 × 10−12 2.56 × 10−11 4.25 × 10−11 4.16 × 10−11 2.10 × 10−10
150 8.92 × 10−13 6.84 × 10−13 8.27 × 10−12 5.49 × 10−12 5.27 × 10−12 1.01 × 10−10

Table 5. Comparison of crack permeability of SF40PP2.3, SF60 and PP6.9 exposed to various freeze–
thaw cycles.

Freeze–Thaw κc-100 (m2 ) κc-200 (m2 )

Cycles SF40PP2.3 SF60 PP6.9 SF40PP2.3 SF60 PP6.9
0 3.65 ×10−12 3.62 × 10−12 6.89 × 10−11 3.30 × 10−11 3.30 ×10−11 7.82 × 10−10
50 2.60 × 10−12 2.36 × 10−12 5.04 × 10−11 2.07 × 10−11 1.86 × 10−11 4.44 × 10−10
100 1.16 × 10−12 1.33 × 10−12 9.10 × 10−12 1.24 × 10−11 1.01 × 10−11 1.23 × 10−10
150 4.65 × 10−13 3.30 × 10−13 2.37 × 10−12 4.39 × 10−12 2.41 × 10−12 2.27 × 10−11
Materials 2024,
Materials 2024, 17,
17, 1819
x FOR PEER REVIEW 10
10 of 23
of 22

(a) (b)

(c) (d)
Figure 8.
Figure 8. Relationship
Relationship of
of the
the effective
effective crack
crack width
width and
and the
the crack
crack permeability
permeability of
of SF20PP2.3,
SF20PP2.3, SF40
SF40
and PP4.6 exposed to various freeze–thaw cycles: (a) 0 freeze–thaw cycle; (b) 50 freeze–thaw cycles;
and PP4.6 exposed to various freeze–thaw cycles: (a) 0 freeze–thaw cycle; (b) 50 freeze–thaw cycles;
(c) 100 freeze–thaw cycles; (d) 150 freeze–thaw cycles.
(c) 100 freeze–thaw cycles; (d) 150 freeze–thaw cycles.
Table 4. Comparison of crack permeability of SF20PP2.3, SF40 and PP4.6 exposed to various freeze–
From Figure 8 and Table 4:
thaw cycles.
Compared to specimens with a fiber volume content of 0.5 vol% (SF20PP2.3, SF40
Freeze–Thaw and PP4.6),
κc‐100it(m
can
2) be observed that when the freeze–thaw cycles range
κc‐200 (m 2) from 0 to 150, the
Cycles SF20PP2.3 SF20PP2.3 and
SF40 SF40 specimens
PP4.6 exhibit similar crack
SF20PP2.3 permeability–effective
SF40 crack
PP4.6width
0 curves
9.97 × 10−12 and a lower
1.38 × 10−11 crack permeability
7.94 × 10−11 than that of PP4.6.
1.03 × 10−10 1.68 × 10−10 1.14 × 10−9
When the freeze–thaw cycles−11are the same, SF20PP2.3 exhibits−11a higher crack perme-
50 8.57 × 10 −12 6.13 × 10 −12 4.36 × 10 6.66 × 10 −11 8.92 × 10 6.88 × 10−10
ability than that of SF40 for an effective crack width less than 50 µm. However, as the crack
100 4.31 × 10−12 3.55 × 10−12 2.56 × 10−11 4.25 × 10−11 4.16 × 10−11 2.10 × 10−10
of the concrete widens, the increased rate of crack permeability of SF20PP2.3 is less than
150 8.92 × 10−13 6.84 × 10−13 8.27 × 10−12 5.49 × 10−12 5.27 × 10−12 1.01 × 10−10
that of SF40. For example, when the freeze–thaw cycles are 50, as the effective crack width
increases from 100 µm to 200 µm, the crack permeability values of SF40 and SF20PP2.3
increase by 13.5 and 6.7 times, respectively. Hence, the crack permeability value of SF40 is
equal to that of SF20PP2.3 for certain effective crack widths. As shown in Figure 8, when the
freeze–thaw cycles are 0, 50, 100 and 150, the same crack permeability values for SF20PP2.3
and SF40 are found at an effective crack width of 76 µm, 148 µm, 186 µm and 212 µm,
respectively. It is evident that, as the freeze–thaw cycles increase, the effective crack width
of equal crack permeability values of SF20PP2.3 and SF40 increases. Moreover, as the crack
continues to expand, the crack permeability of SF20PP2.3 remains lower than that of SF40.
From the analysis above, it is evident that, when the effective crack width is small, the
crack permeability of SF20PP2.3 is higher than that of SF40. As the crack width increases,
the crack permeability of SF20PP2.3 gradually becomes lower than that of SF40 and shows
the superior impermeability performance.
Materials
Materials 2024,
2024, 17,
17, x FOR PEER REVIEW
1819 1111of
of 23
22

(a) (b)

(c) (d)
Figure
Figure 9.9. Relationship
Relationship of
of the
the effective
effective crack
crack width
width and
and the
the crack permeability of
crack permeability SF40PP2.3, SF60
of SF40PP2.3, SF60
and
and PP6.9 exposed to various freeze–thaw cycles: (a) 0 freeze–thaw cycle; (b) 50 freeze–thaw cycles;
PP6.9 exposed to various freeze–thaw cycles: (a) 0 freeze–thaw cycle; (b) 50 freeze–thaw cycles;
(c) 100 freeze–thaw cycles; (d) 150 freeze–thaw cycles.
(c) 100 freeze–thaw cycles; (d) 150 freeze–thaw cycles.

TableFrom
5. Comparison
Figure 9 of
andcrack permeability
Table 5: of SF40PP2.3, SF60 and PP6.9 exposed to various freeze–
thaw cycles.
Compared to specimens with a fiber volume fraction of 0.75 vol% (SF40PP2.3, SF60
Freeze–Thaw and PP6.9),κc‐100it(mis2seen
) that, when freeze–thaw cycles range κfrom c‐200 (m0 2to
) 150, SF40PP2.3 and
Cycles SF40PP2.3 SF60 showSF60 similar crack permeability–effective
PP6.9 crack width curves
SF40PP2.3 SF60 and a lower crack per-
PP6.9
meability than that of PP6.9. When the effective crack width is small,−11 the crack permeability
0 3.65 × 10 −12 3.62 × 10 −12 6.89 × 10 −11 3.30 × 10 −11 3.30 × 10 7.82 × 10−10
of SF40PP2.3 is less than that of SF60. As the effective crack width widens, the increase
50 2.60 × 10−12 2.36 × 10−12 5.04 × 10−11 2.07 × 10−11 1.86 × 10−11 4.44 × 10−10
rate of crack permeability of SF40PP2.3 exceeds that of SF60. Consequently, the crack
100 1.16 × 10−12 1.33 × 10−12 9.10 × 10−12 1.24 × 10−11 1.01 × 10−11 1.23 × 10−10
permeability of SF60 becomes equal to that of SF40PP2.3 for certain effective crack widths.
150 4.65 × 10−13
For the 3.30 × 10−13 without
specimens 2.37freeze–thaw
× 10−12 4.39 × 10−12
damage, 2.41 × 10−12
the permeability value of2.27 × 10−11 is
SF40PP2.3
found to be lower than that of SF60. When the freeze–thaw cycles reach 50, 100 and 150, the
sameFrom
crackFigure 8 and Table
permeability values4: for SF40PP2.3 and SF60 are observed at an effective crack
width of 36 µm, 131 µm and 62with
Compared to specimens µm, arespectively.
fiber volume Ascontent of 0.5crack
the effective vol%width
(SF20PP2.3,
continuesSF40
to
and PP4.6), it can be observed that when the freeze–thaw cycles
expand, the permeability of SF40PP2.3 becomes greater than that of SF60. This indicates range from 0 to 150, the
SF20PP2.3
that SF40PP2.3 and withSF40hybrid
specimens
fibersexhibit
exhibitssimilar crack
a higher crackpermeability–effective
impermeability than that crackof width
mono
curves and a lower crack permeability than that of PP4.6.
FRC, when the concrete structure works in an environment without freeze–thaw cycles. For
When and
SF40PP2.3 the freeze–thaw cycles are the
SF60 with freeze–thaw same, SF20PP2.3
damage, exhibits aof
with the increasing higher crack perme-
the effective crack
ability
width, than that of SF40
the advantage for an effective
of hybrid fibers in crack widthconcrete
enhancing less than 50 µm.
crack However, asgradually
impermeability the crack
of the concrete
diminishes andwidens, the increased
disappears, rate of better
and SF60 shows crack crack
permeability of SF20PP2.3
impermeability is less than
than SF40PP2.3.
that of SF40. For example, when the freeze–thaw cycles are 50, as the effective crack width
increases from 100 µm to 200 µm, the crack permeability values of SF40 and SF20PP2.3
Materials 2024, 17, 1819 12 of 22

3.2. Permeability Performance Evaluation of FRC

When the crack surfaces are parallel and absolutely smooth, the permeability can be
estimated by the Poiseuille flow model (κ PFM ), as shown in Equation (7).

bp2
κ PFM = (7)
12
where, bp is the distance between parallel plates of the Poiseuille flow model.
However, researchers have found that cracks in cementitious materials are rough and
tortuous, which does not align with the conditions of the Poiseuille flow model [8,26,35].
In order to assess the crack permeability of concrete using the Poiseuille flow model, some
researchers [8,38] have introduced a modified factor ξ to quantify the influence of crack
surfaces on the crack permeability of cracked concrete. The modified factor ξ is calculated
by Equations (8) and (9):
bp2
κc = ξ (8)
12
κc
ξ= (9)
κ PFM
Figures 10 and 11 show the relationships of the modified factor ξ and the effective crack
width of specimens with a fiber volume content of 0.5 vol.% and 0.75 vol.%, respectively.
From Figures 10 and 11, the following can be seen:
(1) When the effective crack width of the specimens is ≤25 µm, the modified factor ξ
exhibits noticeable fluctuations with the increase in the effective crack width. This
phenomenon gradually disappears as the effective crack width increases. This can be
attributed to the small crack width and incomplete separation of the crack surface (see
Table 2) at the initial stage of crack formation. As the splitting tensile load increases
and the crack widens, the aggregates on the crack surface may interlock. This causes
the phenomenon of the “widening-closing-rewidening” of the local crack width and
results in significant fluctuations in the effective crack width-modified factor ξ curves.
A similar phenomenon has been confirmed in the literature [35]. As the crack width
of the specimens continues to increase, the crack surfaces fully separate, and the
fluctuations in the effective crack width-modified factor ξ curves gradually diminish.
(2) With the increasing of the effective crack width of the specimens, the crack permeabil-
ity of each specimen is closer to the κ PFM . This is because the crack permeability of
concrete is primarily determined by two factors: crack width and crack surface rough-
ness. According to the Poiseuille flow model, permeability is directly proportional to
the square of the distance between the parallel surfaces. This suggests that when the
effective crack width of concrete increases, a significant increase in crack permeability
may occur. For the concrete crack, the rough crack surface leads to significant reduc-
tions in crack permeability [26]. However, when the crack surfaces are fully separated,
the roughness of the crack surface remains almost constant. There is no significant
change in the effect of nearly constant surface roughness on crack permeability. There-
fore, the crack permeability of concrete becomes high, and closer to the κ PFM , with the
increasing of the effective crack width. Akhavan et al. [8] and Rastiello et al. [35] have
shown similar results in studying the permeability of cracked concrete.
(3) For HFRC specimens, the modified factor ξ of crack permeability is observed to
gradually increase with the expansion of the concrete cracks. When the fiber volume
content and freeze–thaw cycles of the specimens are the same, the ξ-ω curves of the
HFRC specimens are found to be closer to those of the SFRC specimens. This is similar
to the trend observed in their crack permeability–effective crack width curves.
Materials 2024, 17, x FOR PEER REVIEW 13 of 23

Figures 10 and 11 show the relationships of the modified factor ξ and the effective
Materials 2024, 17, 1819 13 of 22
crack width of specimens with a fiber volume content of 0.5 vol.% and 0.75 vol.%, respec-
tively.

(a) (b)

(c) (d)
Figure 10.
Figure 10. Relationship
Relationshipofofthe
theeffective crack
effective width
crack andand
width the the
modified factor
modified ξ of ξSF20PP2.3,
factor SF40
of SF20PP2.3,
and PP4.6 exposed to freeze–thaw cycles: (a) 0 freeze–thaw cycle; (b) 50 freeze–thaw cycles; (c) 100
SF40 and PP4.6 exposed to freeze–thaw cycles: (a) 0 freeze–thaw cycle; (b) 50 freeze–thaw cycles;
freeze–thaw cycles; (d) 150 freeze–thaw cycles.
(c) 100 freeze–thaw cycles; (d) 150 freeze–thaw cycles.

In previous studies [8,38], the modified factor ξ was often regarded as a constant to
estimate the permeability of cracked concrete. However, the modified factor ξ is observed
to gradually increase with crack propagation. Therefore, it can only serve as an evaluation
parameter of crack permeability performance with a specific crack width and cannot
characterize the overall trend of crack permeability performance with crack propagation.
In order to accurately estimate the effect of hybrid fibers on concrete crack permeability,
this study incorporated the prediction model of the Poiseuille flow model and derived the
permeability parameter α to evaluate crack permeability development trends in cracked
concrete, as shown in Equations (10) and (11).

ξ = ξ (ω ) = αω β (10)

αω β+2
κc = ξ · κp = ξ (ω ) · κp = αω β · κp = (11)
(a) (b) 12
where α is the permeability parameter and β is the constant factor of the fitted equation.
When the effective crack width and β are constant, the permeability parameter α ex-
hibits a direct linear correlation with the crack permeability. Additionally, the permeability
parameter α can characterize the development trend of crack permeability within a certain
range of crack widths. Therefore, compared to the modified factor ξ, the permeability
Materials 2024, 17, 1819 14 of 22

parameter α is suitable to assess the effect of hybrid fibers on the crack permeability of
cracked concrete.
To estimate the hybrid fibers’ influence on the evolution trend of crack permeability,
a linear fit between the modified factor ξ and the effective crack width of the concrete
(c)
specimens is performed by Equation (11). The value of (d) parameter β is 1.17 [35]. The fitted
curve of each specimen is shown in Figures 10 and 11. The permeability parameter α
Figure 10. Relationship of the effective crack width and the modified factor ξ of SF20PP2.3, SF40
values areexposed
and PP4.6 obtained from the linear
to freeze–thaw fit(a)
cycles: and are presented
0 freeze–thaw in(b)
cycle; Table 6. The barcycles;
50 freeze–thaw chart of
(c) the
100
permeability parameter
freeze–thaw cycles; of each specimen
(d) 150αfreeze–thaw cycles. group is shown in Figure 12.

Materials 2024, 17, x FOR PEER REVIEW 14 of 23

(a) (b)

(c) (d)
Figure 11.
Figure 11. Relationship
Relationshipofofthe effective
the crack
effective width
crack andand
width the the
modified factor
modified ξ of SF40PP2.3,
factor SF60
ξ of SF40PP2.3,
and PP6.9 exposed to freeze–thaw cycles: (a) 0 freeze–thaw cycle; (b) 50 freeze–thaw cycles;
SF60 and PP6.9 exposed to freeze–thaw cycles: (a) 0 freeze–thaw cycle; (b) 50 freeze–thaw cycles; (c) 100
freeze–thaw cycles; (d) 150 freeze–thaw cycles.
(c) 100 freeze–thaw cycles; (d) 150 freeze–thaw cycles.

TableFrom Figures 10
6. Comparison and 11, the following
of permeability parameter αcan be seen:specimens.
of different
(1) When the effective crack width of the specimens is ≤25 µm, the modified factor ξ
Freeze–Thaw Permeability
exhibits noticeable fluctuations withParameter
the increase
α in the effective crack width. This
Cycles NC phenomenon
SF20PP2.3 gradually disappears
SF40 as the
PP4.6 effective crack width increases.
SF40PP2.3 SF60 This can be
PP6.9
0 9971 attributed
754 to the small crack width and
1233 8324 incomplete232 separation of the crack5213
282 surface
50 8281 (see Table
487 2) at the initial
645 stage of crack
5473 formation.152As the splitting
137 tensile 3354
load in-
100 6078 creases 321
and the crack widens,
404 the aggregates
1589 on the crack
99 surface may76 interlock.
962 This
150 3608 41 phenomenon
causes the 49 of the “widening-closing-rewidening”
784 35 19 178crack
of the local
width and results in significant fluctuations in the effective crack width-modified fac-
tor ξ curves. A similar phenomenon has been confirmed in the literature [35]. As the
crack width of the specimens continues to increase, the crack surfaces fully separate,
and the fluctuations in the effective crack width-modified factor ξ curves gradually
diminish.
(2) With the increasing of the effective crack width of the specimens, the crack permea-
bility of each specimen is closer to the κPFM. This is because the crack permeability of
 Permeability Parameter α
Freeze–Thaw Cycles
NC SF20PP2.3 SF40 PP4.6 SF40PP2.3 SF60 PP6.9
0 9971 754 1233 8324 232 282 5213
50 8281 487 645 5473 152 137 3354
Materials 2024, 17, 1819
100 6078 321 404 1589 99 76 96215 of 22
150 3608 41 49 784 35 19 178

(a) (b)

Figure 12. Bar charts of permeability parameter α: (a) specimen with a fiber volume content of
0.5 vol.%; (b) specimen with a fiber volume content of 0.75 vol.%.

From Table 6 and Figure 12a, compared to PP4.6, the permeability parameter α of
SF20PP2.3 subjected to 0, 50, 100 and 150 freeze–thaw cycles decreases by 90.9%, 91.1%,
79.8% and 94.8%, respectively. In contrast, compared with SF40, the permeability parameter
α of SF20PP2.3 subjected to 0, 50, 100 and 150 freeze–thaw cycles decreases by 38.8%, 24.5%,
20.5% and 16.3%, respectively. Therefore, based on an analysis using permeability param-
eter α, SF20PP2.3 with hybrid fibers demonstrates a significant advantage in improving
crack impermeability.
From Table 6 and Figure 12b, compared to PP6.9 exposed to 0, 50, 100 and 150 freeze–
thaw cycles, the permeability parameter α of SF40PP2.3 decreases by 95.5%, 95.4%, 89.7%
and 80.3%, respectively. Compared to SF60 without freeze–thaw damage, the permeability
parameter α of SF40PP2.3 decreases by 17.7%, and compared to SF60 exposed to 50, 100
and 150 freeze–thaw cycles, the permeability parameter α of SF40PP2.3 increases by 10.9%,
30.3% and 84.2%, respectively. An analysis based on permeability parameter α indicates that
SF40PP2.3 without freeze–thaw damage exhibits a higher crack impermeability than mono
FRC with the same fiber volume content. However, as the freeze–thaw cycles increase, the
permeability parameter α of SF40PP2.3 is lower than that of PP6.9 but higher than that of
SF60. Therefore, the crack impermeability of SF40PP2.3 lies between that of the PP6.9 and
SF60, subjected to the same freeze–thaw cycles.

3.3. Morphological Analysis of Crack Surface

Crack surfaces of specimens from each group are scanned for morphological analysis
using the laser-scanning setup. Crack surface morphology data for each specimen are
processed with Origin software to reconstruct 3D graphics of the crack surface. A consistent
color scale is used for the 3D reconstructions of all specimen groups. The 3D reconstruction
images of the crack surfaces of HFRC are shown in Figure 13, and the mono FRC specimens
with the same fiber volume content and NC specimens are the reference. The crack surface
roughness (Rn ) of each group is listed in Table 7.
Materials 2024, 17, 1819 16 of 22

Materials 2024, 17, x FOR PEER REVIEW 16 of 23

Table 7. Comparison of crack surface Rn of different specimens.

Freeze–Thaw Figure 12. Bar charts of permeabilityCrack Surface

parameter Rnspecimen with a fiber volume content of 0.5
α: (a)
Cycles NCvol.%; (b) specimen
SF20PP2.3with a fiber volume content
SF40 of 0.75 vol.%.
PP4.6 SF40PP2.3 SF60 PP6.9
0 1.138 1.371 1.319 1.202 1.480 1.466 1.304
50 1.280 From Table
1.456 6 and Figure
1.39512a, compared
1.319 to PP4.6, the
1.510permeability
1.511parameter
1.460α of
100 SF20PP2.3
1.318 subjected
1.548 to 0, 50, 100
1.529 and 150 freeze–thaw
1.374 cycles
1.675 decreases by
1.680 90.9%, 91.1%,
1.589
150 79.8% and 1.758
1.366 94.8%, respectively.
1.692 In contrast,
1.539 compared with SF40,1.838
1.812 the permeability
1.732
parameter α of SF20PP2.3 subjected to 0, 50, 100 and 150 freeze–thaw cycles decreases by
38.8%, 24.5%, 20.5% and 16.3%, respectively. Therefore, based on an analysis using
From Figure 13 and Table 7, the following can be seen:
permeability parameter α, SF20PP2.3 with hybrid fibers demonstrates a significant
(1) Wheninthe
advantage specimenscrack
improving are exposed to the same freeze–thaw cycles, the crack surface of
impermeability.
FRC is rougher than that of NC.
From Table 6 and Figure 12b, compared to PP6.9 exposed to 0, 50, 100 and 150 freeze–
(2) The
thaw crack
cycles, thesurface Rn values
permeability of all groups
parameter increase with
α of SF40PP2.3 the increment
decreases by 95.5%,in95.4%,
freeze–thaw
89.7%
cycles.
and 80.3%, respectively. Compared to SF60 without freeze–thaw damage, the
(3) When theparameter
permeability addition ofαfiber volume content
of SF40PP2.3 is 0.5 vol.%,
decreases by 17.7%,the crack
and surface
compared Rn values of
to SF60
SF20PP2.3 consistently exceed those of SF40 and PP4.6 exposed to 0,
exposed to 50, 100 and 150 freeze–thaw cycles, the permeability parameter α of SF40PP2.3 50, 100 and 150
freeze–thaw
increases by 10.9%, cycles.
30.3% and 84.2%, respectively. An analysis based on permeability
parameter α indicatesvolume
(4) When the fiber content without
that SF40PP2.3 of the specimens
freeze–thaw is 0.75 vol.%,
damage the crack
exhibits surface
a higher Rn
crack
value of SF40PP2.3 without freeze–thaw damage is consistently higher than those of
impermeability than mono FRC with the same fiber volume content. However, as the
SF60 and PP6.9. However, with the increment in freeze–thaw cycles, the crack surface
freeze–thaw cycles increase, the permeability parameter α of SF40PP2.3 is lower than that
Rn value of SF40PP2.3 becomes higher than that of SF60 but lower than that of PP6.9.
of PP6.9 but higher than that of SF60. Therefore, the crack impermeability of SF40PP2.3
From thethat
lies between discussion above,
of the PP6.9 andit SF60,
is concluded thattoan
subjected theincrease in crack surface
same freeze–thaw roughness
cycles.
correlates with a gradual decrease in crack permeability. A similar phenomenon has
beenMorphological
3.3. observed inAnalysis
previous studies
of Crack [3,39]. To estimate the relationship between crack
Surface
permeability and the crack surface morphology, the relationship between the crack surface
Rn and permeability parameter α of different specimens is illustrated in Figure 14.

NC(0) NC(50) NC(100) NC(150)

SF20PP2.3(0) SF20PP2.3(50) SF20PP2.3(100) SF20PP2.3(150)

SF40(0) SF40(50) SF40(100) SF40(150)

Figure 13. Cont.

Materials 2024, 17, x FOR PEER REVIEW 17 of 23
Materials 2024, 17, 1819 17 of 22

PP4.6(0) PP4.6(50) PP4.6(100) PP4.6(150)

SF40PP2.3(0) SF40PP2.3(50) SF40PP2.3(100) SF40PP2.3(150)

SF60(0) SF60(50) SF60(100) SF60(150)

PP6.9(0) PP6.9(50) PP6.9(100) PP6.9(150)

Color scale (mm):

Figure 13. Reconstruction images of crack surface.

Figure 13. Reconstruction images of crack surface.
From Figure 14, it is apparent that an exponential functional relationship exists be-
Table
tween7.the
Comparison of crack
permeability surface Rαn of
parameter anddifferent specimens.
the crack surface Rn , which can be expressed by
Equation (12). The fitted parameters are presented in Table 8.
Crack Surface Rn
Freeze–Thaw Cycles
NC SF20PP2.3 SF40 Y = PP4.6
γ · e(τ ·X) SF40PP2.3 SF60 PP6.9(12)
0 1.138 1.371 1.319 1.202 1.480 1.466 1.304
50 1.280 γ and τ are
where 1.456 1.395
the parameters 1.319
obtained through fitting 1.510 1.511
experimental data. 1.460
100 1.318 1.548 1.529 1.374 1.675 1.680 1.589
150 1.3668. Fitted parameters
Table 1.758 of an exponential
1.692 functional
1.539 relationship.
1.812 1.838 1.732
Curve γ τ R2
From Figure 13 and Table 7, the following
7
can be seen:
α–Rn 1.0415 × 10 −6.025 0.76
(1) When the specimens are exposed to the same freeze–thaw cycles, the crack surface of
FRC is rougher than that of NC.
2
The crack
(2) The correlation
surface Rn valuesRof of
coefficient all the curve iswith
α–Rnincrease
groups 0.76;the
a similar phenomenon
increment was
in freeze–thaw
confirmed in the literature [26]. This indicates a significant correlation between the crack
cycles.
permeability and the roughness of the crack surface. Specifically, the rougher the surface
of the crack, the lower its crack permeability is. Therefore, the crack impermeability of
FRC can be characterized by its crack surface roughness. This also demonstrates that
 SF20PP2.3 consistently exceed those of SF40 and PP4.6 exposed to 0, 50, 100 and 150
freeze–thaw cycles.
(4) When the fiber volume content of the specimens is 0.75 vol.%, the crack surface Rn
value of SF40PP2.3 without freeze–thaw damage is consistently higher than those of
SF60 and PP6.9. However, with the increment in freeze–thaw cycles, the crack surface
Materials 2024, 17, 1819 Rn value of SF40PP2.3 becomes higher than that of SF60 but lower than that of PP6.9. 18 of 22
From the discussion above, it is concluded that an increase in crack surface roughness
correlates with a gradual decrease in crack permeability. A similar phenomenon has been
observed in previous studies [3,39]. To estimate the relationship between crack permea-
HFRC specimens
bility and can effectively
the crack surface morphology,increase the crackbetween
the relationship surfacetheroughness and
crack surface Rn enhance
and the
crack impermeability of concrete.
permeability parameter α of different specimens is illustrated in Figure 14.

Materials 2024, 17, x FOR PEER REVIEW 19 of 23

Figure Relationship
14.Relationship
Figure 14. between
between crack
crack surface
surface Rnpermeability
Rn and and permeability parameter
parameter α of concrete.
α of cracked cracked concrete.

3.4. Analysis of Positive Synergistic Effect of Hybrid Fibers on Crack Impermeability

From 3.4.
Figure 14, it of
Analysis is Positive
apparent that an exponential
Synergistic Effect of Hybrid functional
Fibers onrelationship exists be-
Crack Impermeability
Tothe
tween investigate
permeabilitytheparameter
effects of αpolypropylene
and the crack surfacefibers and steel fibers
Rn, which can beon crack formation
expressed by in
concrete To investigate thetypes
effects offibers,
polypropylene mainfibers
crackand steel group
fibers on
ofcrack formation
Equation reinforced with
(12). The fitted both
parameters areofpresented thein Table 8. in each specimens is
in concrete reinforced with both types of fibers, the main crack in each group of specimens
propagated. Figure 15a,b illustrate the interface ( X) zone between polypropylene fibers and
the concrete matrix on the crack surface. Y   The
is propagated. Figure 15a,b illustrate ethe interface zone between polypropylene
blue line shows the micro-crack (12) fibers and
between
the concrete matrix on the crack surface. The blue line shows the micro-crack between the
the polypropylene
where γ and fiber and
τ are the parameters
polypropylene
the concrete
fiber andobtained
matrix.
through
the concrete
Figure
fitting
matrix.
15c,d illustrate
experimental
Figure
the interface
data. the interface zone
15c,d illustrate
zonebe-
between the steel fibers and the concrete matrix on the crack
tween the steel fibers and the concrete matrix on the crack surface. surface.
Table 8. Fitted parameters of an exponential functional relationship.

Curve γ τ R2
α–Rn 1.0415 × 107 −6.025 0.76

The correlation coefficient R2 of the α–Rn curve is 0.76; a similar phenomenon was
confirmed in the literature [26]. This indicates a significant correlation between the crack
permeability and the roughness of the crack surface. Specifically, the rougher the surface
of the crack, the lower its crack permeability is. Therefore, the crack impermeability of
FRC can be characterized by its crack surface roughness. This also demonstrates that
HFRC specimens can effectively increase the crack surface roughness and enhance the
crack impermeability of concrete.
(a) (b)

(c) (d)
Figure 15.zone
Figure 15. Interface Interface zone between
between fiber and fiber and concrete
concrete matrix:matrix: (a) and (b) polypropylene
(a,b) polypropylene fiber andfiber and
concrete
concrete matrix; (c) and (d) steel fiber and concrete matrix.
matrix; (c,d) steel fiber and concrete matrix.
From Figure 15, a micro-crack can be observed at the interface zone between the pol-
ypropylene fiber and the concrete matrix. However, the interface zone between steel fiber
and concrete matrix is sound. This implies that the anchorage of the steel fibers with the
concrete matrix is higher than that of the polypropylene fibers. De Alencar Monteiro et al.
[40] and Biao et al. [41] have shown similar results in their studies on the mechanical be-
 (c) (d)
Materials 2024, 17, 1819 19 of 22
Figure 15. Interface zone between fiber and concrete matrix: (a) and (b) polypropylene fiber and
concrete matrix; (c) and (d) steel fiber and concrete matrix.

From Figure 15,

From 15, aamicro-crack
micro-crackcan canbebeobserved
observed at the interface
at the zone
interface between
zone the pol-
between the
polypropylene
ypropylene fiber fiber
andandthe the concrete
concrete matrix.
matrix. However,
However, the interface
the interface zonezone between
between steel steel
fiber
fiber and concrete
and concrete matrix matrix is sound.
is sound. This This implies
implies that that the anchorage
the anchorage of steel
of the the steel fibers
fibers withwith
the
the concrete
concrete matrix
matrix is higher
is higher thanthan
thatthat of polypropylene
of the the polypropylenefibers.fibers. De Alencar
De Alencar Monteiro
Monteiro et al.
et al.and
[40] [40] Biao
and Biao
et al.et[41]
al. have
[41] have
shownshown similar
similar results
results in their
in their studies
studies on the
on the mechanical
mechanical be-
behavior
havior of SFRC and PFRC. The interface zone between the polypropylene fiberfiber
of SFRC and PFRC. The interface zone between the polypropylene and
and con-
concrete matrix
crete matrix is the
is the weakweakareaarea of concrete.
of concrete.
Figure
Figure 1616 shows
shows the the crack
crack surface
surface topographies
topographies of HFRC, SFRC
of HFRC, SFRC and
and PFRC
PFRC specimens,
specimens,
respectively.
respectively. TheThe blue
blue lines
lines represent
represent micro-cracks
micro-cracks on on the
the crack
crack surface,
surface, the
the red
red circles
circles
represent steel fibers, and the yellow circles represent polypropylene
represent steel fibers, and the yellow circles represent polypropylene fibers. fibers.

Materials 2024, 17, x FOR PEER REVIEW 20 of 23

(a) (b)

(c) (d)

(e) (f)
Figure 16.
Figure 16. Crack
Crack surface
surface topographies
topographies of
of (a)
(a) SF20PP2.3,
SF20PP2.3, (b)
(b) SF40PP2.3,
SF40PP2.3, (c)
(c) SF40,
SF40, (d)
(d) SF60,
SF60, (e)
(e) PP4.6
PP4.6
and (f) PP6.9.
and (f) PP6.9.

From Figure
From Figure 16,
16, itit is
is evident
evident that
that HFRC
HFRC (SF20PP2.3
(SF20PP2.3 andand SF40PP2.3)
SF40PP2.3) specimens
specimens are are
prone to
more prone to micro-crack
micro-crack formation
formation thanthan SFRC
SFRC (SF40
(SF40 and
and SF60)
SF60) and
and PFRC
PFRC (PP4.6
(PP4.6 andand
PP6.9) specimens. Moreover,
Moreover,atatthe thelocations
locationsofofmicro-cracks,
micro-cracks, polypropylene
polypropylene fibers cancan
fibers be
observed
be observed to to
bebedistributed
distributed along
alongthethedirection
directionofofmicro-crack
micro-crack propagation
propagation (blue
(blue lines),
while many
while many steel
steel fibers are embedded in the concrete matrix and bridge the micro-cracks.
contrast, the
In contrast, the main
main crack
crack surfaces
surfaces ofof mono
mono FRCFRC specimens
specimens show
show no
no micro-cracks.
micro-cracks. ThisThis
phenomenon may be
phenomenon be attributed
attributedto tothe
thehigh
highelastic
elasticmodulus
modulusand andhooked
hooked ends
endsof of
steel fi-
steel
bers, which
fibers, whichprovide
provideaahigh high anchorage
anchorage between
between steel
steel fibers
fibers and the concrete
concrete matrix.
matrix. In In
comparison, polypropylene fibers, with a low elastic modulus and straight straight ends,
ends, show
show
weak
weak anchorage
anchoragebetween
betweenthe thepolypropylene
polypropylenefibers and
fibers concrete
and matrix.
concrete The
matrix. polypropylene
The polypropyl-
fibers represent the weak area of concrete in the concrete matrix. As the
ene fibers represent the weak area of concrete in the concrete matrix. As the concrete concrete is loaded, is
the interface
loaded, zone between
the interface the polypropylene
zone between fibers and
the polypropylene concrete
fibers matrix is matrix
and concrete more prone
is moreto
form
pronemicro-cracks than the concrete
to form micro-cracks than thematrix.
concrete matrix.
For HFRC specimens with polypropylene fibers and steel fibers, the steel fibers bear
tensile stress on both crack faces and effectively transmit the stress into the concrete ma-
trix, while polypropylene fibers induce the formation of micro-cracks. The synergistic ac-
tion of the two types of fibers promotes the formation of micro-cracks in the concrete ma-
Materials 2024, 17, 1819 20 of 22

For HFRC specimens with polypropylene fibers and steel fibers, the steel fibers bear
tensile stress on both crack faces and effectively transmit the stress into the concrete matrix,
while polypropylene fibers induce the formation of micro-cracks. The synergistic action
of the two types of fibers promotes the formation of micro-cracks in the concrete matrix
and leads to the propagation of micro-cracks into macro-cracks in the concrete. Therefore,
micro-cracks increase the total surface area of concrete cracks. A large crack surface area
effectively increases the actual path length for water flow through the concrete specimens.
This results in the increased head loss and improved crack impermeability of the cracked
concrete. Moreover, compared to mono FRC (SFRC and PFRC), HFRC is more prone to
both micro-cracks and macro-cracks. This is one of the key factors of the positive synergistic
effect on the crack impermeability of cracked concrete.

4. Conclusions
This study investigated hybrid fiber’s influence on the crack permeability of cracked
concrete exposed to freeze–thaw cycles. The experimental and analytical results led to the
following conclusions:
(1) The modified factor ξ of crack permeability gradually increases and is closer to 1 with
the expansion of the concrete crack. When the effective crack width of the specimens
is less than 25 µm, the two sides of the crack surface are not completely separated
and the “widening-closing-widening” of the local crack width significantly reduces
the crack permeability. When the effective crack width of the specimens is beyond
25 µm, the crack surfaces are completely separated and the roughness of the crack
surface remains almost unchanged. The effect of crack surface roughness on crack
permeability gradually decreases, and the crack permeability of each specimen is
closer to the κ PFM .
(2) Compared with the modified factor ξ of crack permeability, the permeability parame-
ter α can effectively evaluate and quantify the development trend of crack permeability
within a certain range of crack widths and be used as a quantitative parameter for the
durability design of concrete materials.
(3) For specimens with a fiber volume content of 0.5 vol.%, compared to PP4.6 and SF40,
the permeability parameter α of SF20PP2.3, subjected to the same freeze–thaw cycles,
decreases by 16.3–94.8%. SF20PP2.3 demonstrates a positive synergistic effect on the
crack impermeability of cracked concrete. For specimens with a fiber volume content
of 0.75 vol.%, compared to PP6.9 and SF60, the permeability parameter α of SF20PP2.3
without freeze–thaw damage decreases by 17.7–95.5%, and the crack impermeability
of SF40PP2.3 subjected to the same freeze–thaw cycles lies between that of PP6.9
and SF60.
(4) The roughness of the crack surface and the crack permeability is highly correlated and
follows an exponential curve (Y = 1.0415 × 107 ·e−6.025·X , Y is the crack permeability
parameter α, X is the crack surface Rn ) in concrete. This indicates that hybrid fibers
enhance crack impermeability by increasing the crack surface roughness. Based on
this relationship, the permeability of cracked concrete can be quickly evaluated by
crack surface roughness in real engineering applications.
(5) The combination of steel fibers and polypropylene fibers effectively promotes the
formation of micro-cracks in the concrete matrix. This increases the actual path length
of water flow through concrete specimens and enhances the hydraulic head loss. This
is a key reason for the evident positive synergistic effect on the crack impermeability
of cracked concrete.

Author Contributions: Conceptualization, W.Z. and Y.T.; Methodology, W.W.; Software, L.Z. and
M.L.; Validation, W.Z., M.L. and Q.W.; Formal analysis, W.W.; Investigation, W.W.; Resources, W.W.;
Data curation, W.Z. and L.Z.; Writing—original draft preparation, W.Z.; Writing—review and editing,
W.Z. and L.Z.; Visualization, M.L. and L.Z.; Supervision, Q.W. and Y.T.; Project administration, Q.W.;
Funding acquisition, Y.T. All authors have read and agreed to the published version of the manuscript.
Materials 2024, 17, 1819 21 of 22

Funding: This research was funded by the National Natural Science Foundation of China [grant
number: 52108256], the Zhejiang Provincial Natural Science Foundation of China [grant number:
LQ22E080014], and the Science Fund Project of Department of Education of Zhejiang Province [grant
number: Y202146479].
Institutional Review Board Statement: Not Applicable.
Informed Consent Statement: Not Applicable.
Data Availability Statement: Data available on request from the authors. The data that support the
findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest: The authors declare no conflict of interest.

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Concr. Compos. 2022, 133, 104716. [CrossRef]
2. Divanedari, H.; Eskandari-Naddaf, H. Insights into surface crack propagation of cement mortar with different cement fineness
subjected to freezing/thawing. Constr. Build. Mater. 2020, 233, 117207. [CrossRef]
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