Multi-Scale Investigation of Fracture Behavior of Polypropylene Fiber-Reinforced Concrete Segment During Bending Test

Abstract

Polypropylene fibers provide an innovative solution for enhancing the crack resistance of tunnel lining segments. However, existing macro-models obscure the distinct effects of fibers on the mortar and ITZ, while explicit meso-modeling remains computationally prohibitive. This study develops a multi-scale modeling framework to investigate PFRC segment fracture under bending. The framework integrates a 3D meso-scale module for calibrating fracture-related material properties, a 3D macro-scale module for predicting global displacements, and a 2D meso-scale module for resolving local fracture processes. A full-scale bending test was performed to validate the framework and to examine the effects of fiber content at both scales. Both the full-scale test and numerical simulations show that the segment response exhibits three stages: elastic, damage development, and cracking at the design load. Numerical simulations further reveal that an optimal fiber content of 0.4% reduces the vertical displacement at the load point by 9.8% and the horizontal displacement at the edge point by 2.9% relative to the fiber-free case. Meso-scale simulations show that 0.4% fibers decrease the bottom crack width from 0.0868 to 0.0770 mm (−11.29%) and limit internal crack connectivity. Although fibers may locally promote ITZ cracking due to reduced mortar–aggregate bonding, a strengthened mortar matrix suppresses crack penetration and connected crack networks. A pronounced high-damage peak in the ITZ near the failure threshold confirms the ITZ as the governing weak link; therefore, further improvements may require ITZ-strengthening strategies.

1. Introduction

With rapid urbanization, the demand for urban metro systems continues to grow. Shield tunneling with precast segmental linings has become a prevalent solution for urban rail tunnels, benefiting from efficient prefabrication and quick segment erection [1,2]. Compared with conventional concrete, fiber-reinforced concrete (FRC) exhibits improved crack resistance when subjected to flexural tensile load [3,4,5,6,7]. Due to its low cost, low density, ease of mixing, and effectiveness in improving key mechanical properties, polypropylene fiber has been successfully introduced into a wide range of concrete composites, including lightweight, heavyweight, and cellular concretes [8,9,10,11]. Because of these versatile material advantages, incorporating fibers into reinforced concrete segments to form a hybrid reinforcement system can enhance the load-bearing capacity of tunnel segments [12]. A clear understanding of the mechanisms by which polypropylene fibers enhance the crack resistance of tunnel segments is essential for optimizing the design of polypropylene fiber-reinforced concrete (PFRC) segmental linings.

To clarify the crack-resistance enhancement mechanisms of polypropylene fibers, material-level laboratory testing is often the primary approach for quantifying how fiber dosage and characteristics modify the mechanical and fracture-related properties of PFRC [13]. Since polypropylene fibers mainly improve concrete crack resistance by restraining crack initiation and subsequent propagation, existing laboratory studies have primarily focused on how fiber dosage influences mechanical strength and cracking-related performance. Existing studies indicate that the improvement in crack resistance and mechanical performance of concrete provided by polypropylene fibers is strongly dosage-dependent. At an appropriate fiber content, both the fracture energy and fracture strength of concrete can be enhanced [14]. This toughening effect becomes more pronounced in material systems with higher brittleness, where strength and ductility may be improved simultaneously [15]. However, excessive fiber addition may deteriorate workability and dispersion and increase interfacial porosity, thereby weakening or even offsetting the crack-resistance benefits [14,16]. Under complex stress states such as triaxial compression, a similar dosage sensitivity has also been observed, further highlighting the importance of selecting a rational fiber dosage [17]. Besides fiber dosage, fiber length and length combinations also impact the mechanical performance and cracking behavior of concrete. Overall, the length effect is often less pronounced than the dosage effect, and variations in a single fiber length do not necessarily lead to a clear monotonic improvement in peak strength or fracture energy [18]. Nevertheless, an appropriate hybridization of long and short polypropylene fibers may produce more consistent benefits: long fibers tend to bridge macrocracks, whereas short fibers are more effective in restraining microcrack growth, resulting in a synergistic enhancement of both strength and fracture toughness [19].

A key advantage of laboratory testing is that it enables quantitative evaluation of how fiber dosage and characteristics affect the overall performance of fiber-reinforced concrete, thereby supporting mix-proportion optimization. However, laboratory conditions often differ from those in the field. Therefore, results from laboratory tests alone are often insufficient to reliably predict the mechanical behavior of concrete tunnel segments under real service conditions. In view of these limitations, previous studies have carried out full-scale tests under various loading and boundary conditions to investigate the mechanical response and cracking behavior of fiber-reinforced concrete segments. Bending loads and thrust jack loads are commonly included in such full-scale programs, as crack width under the design load is often adopted as a quality-control indicator for concrete segments, while localized cracking near thrust-jack contact areas is a typical form of construction-induced damage [20,21,22]. With systematically designed test programs, these full-scale experiments can not only quantify the fracture behavior of individual segments under various conditions, but also facilitate investigations of staggered segmental assemblies involving multiple rings [23,24]. Despite providing the most direct experimental evidence of segment fracture behavior, full-scale testing still has inherent drawbacks that may limit its application. It is often expensive and time-consuming to plan and execute, and instrumentation is typically confined to a limited number of critical locations, which makes it difficult to obtain comprehensive stress-strain and cracking data across the entire segment or lining system.

To overcome these limitations of full-scale testing, numerical simulation has increasingly been adopted as a complementary approach for investigating the damage evolution and fracture behavior of shield tunnel segments [25,26,27,28,29]. Compared with full-scale experiments, numerical models can not only capture the mechanical response of concrete segments at virtually any location under applied loads, but also allow the effects of factors that are difficult to incorporate in full-scale tests (e.g., seismic loading) to be examined [30]. In most existing simulations, fibers are not explicitly modeled as discrete constituents. Instead, fiber-reinforced concrete is typically idealized as a homogeneous material whose effective constitutive parameters implicitly account for the presence of fibers [21,31,32]. The initiation and propagation of cracks in concrete segments can be numerically simulated using the extended finite element method (XFEM) or cohesive zone models (CZM), once the fracture-related material parameters have been determined. Therefore, under this homogenized-material assumption, the predicted fracture response of fiber-reinforced concrete segments becomes highly dependent on the prescribed (effective) material parameters adopted in the numerical model. However, it has been reported that the incorporation of polypropylene fibers, while enhancing the crack resistance of the cementitious matrix (e.g., cement mortar), may simultaneously weaken the mortar–aggregate interface (i.e., the interfacial transition zone) by reducing its bond strength [16]. If fiber-reinforced concrete is treated as a single homogeneous continuum, such differentiated effects on the matrix and the interface cannot be explicitly represented in the fracture analysis, which may in turn limit the reliability of numerical predictions for cracking and damage evolution. In some studies focusing on the fracture behavior of fiber-reinforced concrete at the material level, FRC is modeled as a heterogeneous composite consisting of cement mortar, coarse aggregates, the interfacial transition zone (ITZ), and fibers [16,33]. Such meso-scale representations make it possible to distinguish the respective roles of the matrix and interfaces and to gain more detailed insight into crack initiation and propagation. However, the geometric scale of shield tunnel segments is far larger than that of laboratory material specimens. Directly extending a similar heterogeneous (meso-scale) modeling strategy to full-size segments would lead to a dramatic increase in degrees of freedom and computational costs, often making the simulations prohibitively expensive or even intractable for practical analysis. In this context, a multi-scale modeling framework offers a promising compromise, enabling key meso-scale fracture mechanisms to be incorporated into segment-level analyses without incurring prohibitive computational costs. A recent study adopted a localized multi-scale approach in which the presence of steel fibers was explicitly represented within a selected subdomain located in the central region of a concrete segment, while the remaining part of the segment was still modeled at the structural (homogenized) level [34]. However, this model did not explicitly distinguish aggregates from the cementitious mortar. In addition, polypropylene fibers are typically much smaller than steel fibers, which poses additional challenges for multi-scale modeling, particularly in terms of geometric resolution and computational efficiency.

Considering that traditional homogenized macro-models fail to capture the distinct, contrasting effects of fibers on the mortar matrix and the interfacial transition zone (ITZ), alternative, computationally feasible multi-scale strategies for analyzing the fracture behavior of PFRC segments become imperative. To address this critical need, the primary objective of this study is to develop an innovative multi-scale modeling framework for investigating the fracture behavior of PFRC segments under bending. By treating PFRC as a heterogeneous meso-structure comprising aggregates and a homogenized fiber-enhanced mortar, this methodology circumvents the prohibitive computational costs of explicit fiber modeling while successfully capturing the distinct impacts of fibers on both the mortar matrix and the ITZ. The proposed framework integrates three components: a 3D meso-scale module to calibrate concrete fracture properties, a 3D macro-scale module to predict the global displacement response of the segment in the bending test, and a 2D meso-scale module to resolve local fracture processes. A full-scale bending test on PFRC segments was conducted to validate the developed framework. Finally, the effects of polypropylene fiber content on the macro- and meso-scale fracture behavior of PFRC segments were investigated.

2. Materials and Experiment

2.1. Materials

This study used P.II 52.5 Portland cement produced in Guangdong, China. The initial setting time of the cement was 480 min, and the final setting time was 600 min. The mix ratio of the polypropylene fiber-reinforced concrete is shown in

Table 1. Mix ratio of PFRC (kg/m³).

Cement	Water	Sand	Coarse Aggregate	Fly Ash	Water Reducer
336	134	650	1207	84	8.4

. The polypropylene fibers were produced by Tai’an Hengda Engineering Materials Co., Ltd., Tai’an, China. The appearance of the polypropylene fibers used in this study is shown in Figure 1. The properties of polypropylene fibers are shown in

Table 2. Properties of polypropylene fibers.

Length (mm)	Diameter (μm)	Tensile Strength (MPa)	Density (g/cm3)	Elastic Modulus (MPa)	Elongation (%)
19	43.6	346	0.91	3800	16

. The fiber content was 0.2% by volume for PFRC. Grade I fly ash was selected, with a fineness of 7.4%, water demand ratio of 94%, moisture content of 0.1%, and SO₃ content of 0.72%. The sand (0–4.75 mm) used in this study was river sand produced in Guangdong, China. The apparent density of the sand was 2610 kg/m³. The coarse aggregate (5~25 mm) used in this study was deep-seated igneous rock, with an apparent density of 2700 kg/m³, and the gradation met the requirements of continuous gradation. A polycarboxylic acid-based high-performance water reducer (early strength type) was used as an admixture.

2.2. Bending Test

A full-scale bending test for the PFRC segment was conducted. The schematic diagram of the experimental setup and a photo of the experimental setup are shown in Figure 2a and Figure 2b respectively. The PFRC segment used in the bending test was prepared based on the mix ratio presented in the Section 2.1. The outer and inner arc radii of the segment were 4400 mm and 4000 mm respectively. The horizontal span of the inner arc of the segment was 3800 mm. The length of the segment in the tunneling direction was 1800 mm. A total of 36 principal reinforcing steel bars with a diameter of 28 mm were embedded in the segment. These principal bars are arranged in three distinct layers, with 12 bars distributed in each layer. From the inner to the outer surface of the segment, the arc radii of these reinforcing layers are 4074 mm, 4231 mm, and 4341 mm, respectively. The circumferential spacing of the principal bars, measured sequentially from the edge of the segment along the longitudinal direction of the tunnel, exhibits a symmetrical distribution: 61, 140, 196, 145, 145, 133, 80, 133, 145, 145, 196, 140, and 61 mm. Furthermore, the yield strength and ultimate tensile strength of these reinforcing steel bars are 450 MPa and 610 MPa, respectively. During the molding process, the PFRC mixture was cast into steel molds and compacted using mechanical vibration to ensure uniform fiber distribution and minimize entrapped air. Following demolding, the segments were subjected to standard curing for 28 days prior to testing. During the bending test, the PFRC segment was firmly placed on two steel supports that could move laterally. A loading system was used to apply vertical load to the segment through two loading bars positioned 1340 mm apart. The load was applied in seven steps, gradually increasing until reaching the design load of the segment. The seven levels of load were 59.86 kN, 123.71 kN, 187.57 kN, 251.42 kN, 283.35 kN, 299.32 kN, and 315.28 kN respectively. The loading process was conducted in seven continuous increments without resetting to zero. Upon reaching each intermediate target load level, the load was held constant for 10 min to allow for stable structural deformation and data acquisition before proceeding to the next increment. Finally, once the ultimate design load was reached, it was maintained for 30 min to observe the sustained behavior of the segment. The vertical displacements at measurement point P1 and horizontal displacements at measurement point P2, shown in Figure 2a, were automatically recorded during the bending test. The initiation and propagation of cracks were observed and measured for subsequent analysis. The flowchart of the experiment is shown in Figure 3.

3. Model Development

3.1. General Framework of Multi-Scale Modeling

The incorporation of polypropylene fibers can affect the fracture properties of cement mortar and the interfacial transition zone (ITZ), which may further impact the fracture behavior of concrete segments during the bending test. Meso-structures of concrete material have been widely used in related studies to investigate the overall fracture strength and localized crack propagation process. In these studies, concrete is treated as a composite material composed of cement mortar and aggregates. Since the nominal particle size of aggregates used for preparing concrete is relatively small (e.g., <30 mm), the maximum size of the concrete meso-structure must be controlled to avoid extensive computational costs. However, the sizes of typical concrete segments used in tunnel lining are usually much larger. For example, the thickness, lateral span, and radial length of the concrete segment used in the bending test of this study are 0.4 m, 3.8 m, and 1.8 m respectively. Therefore, it is technically infeasible to develop a full-scale meso-structure of a concrete segment. To solve this problem, a multi-scale modeling framework was developed in this section. A brief schematic diagram of the multi-scale modeling framework is proposed in Figure 4.

As shown in Figure 4, the entire multi-scale modeling framework comprised three modules that are sequentially connected, labeled Module 1, Module 2, and Module 3. The functions and interrelationships of these three modules are as follows:

(1)

Module 1 contained a series of 3D meso-scale finite element models of cubic concrete specimens. These models considered the heterogeneous structures of concrete and the fracture properties of cement mortar and ITZ with different contents of polypropylene fibers. The overall fracture properties of PFRC were determined through tensile fracture simulations and were then exported as necessary input parameters for Module 2.

(2)

In Module 2, a 3D finite element model of concrete segment was developed to simulate the bending test. To reduce computational costs, a quarter segment with principal reinforcement was modeled instead of the whole segment, and symmetric boundary conditions were set where necessary. In this concrete segment model, the concrete material was treated as a homogeneous material, without considering the distribution of aggregates in the cement mortar. Therefore, this 3D concrete segment model was classified as a macro-scale model. The fracture properties of the PFRC were obtained from the simulations of Module 1. During the simulation of the bending test, the concrete segment model deformed as the load increased. Then cracks initiated and propagated upwards from the segment bottom. The nodal displacements near the major crack were extracted and then imported to Module 3 for further analysis.

(3)

The role of Module 3 was to investigate the fracture propagation behavior inside the internal structure of the PFRC segment during the bending test. Unlike Modules 1 and 2, finite element models in Module 3 were developed based on a 2D meso-structure of concrete. The reason for using 2D models stemmed from two major considerations. The first was to reduce computational costs. The model size in Module 3 should cover the length of the major crack, which would result in a relatively larger size and more aggregates compared to Module 1. Using 3D models could introduce an excessive number of meshes, thereby imposing a significant burden on numerical calculations. The second consideration was that 2D models could present crack propagation paths much more clearly than the 3D model; thus, extensive studies simulated the fracture behavior of concrete material based on 2D finite element models [16]. In this study, the local cross-section near the major crack determined by Module 2 was extracted and modeled as heterogeneous structures containing aggregates and mortar. The nodal displacements at the boundaries of these cross-sections were also extracted from Module 2 and were then used as boundary conditions for the 2D models in Module 3. Finally, the local fracture behavior of the PFRC segment was analyzed by Module 3 from a meso-scale perspective.

3.2. Meso-Scale Modeling of Concrete

The finite element models in Module 1 and Module 3 are both meso-scale models considering the heterogeneous internal structure of concrete. Although there are some differences in modeling 3D and 2D heterogeneous structures, the general modeling strategy is the same. In this study, both the 3D and 2D heterogeneous structures of concrete were generated by integrating image processing technique and the random aggregate method. A brief modeling process is presented in Figure 5, and the detailed modeling steps are as follows.

(1)

Aggregate geometry collection

The Aggregate Image Measurement System (AIMS) was used to capture the 2D geometries of aggregates. As shown in Figure 5, the major components of the AIMS include a camera, a set of lights, and a rotatable tray. During measurement, several aggregates were placed in the groove of the tray, and the camera captured photos of these aggregates one by one. These photos were automatically processed by the system to get 2D aggregate projections in the X-Z plane shown in Figure 5.

(2)

2D digital aggregate generation

Each 2D aggregate projection obtained in the previous step was simplified into a polygon. By setting the Y-coordinate value of each vertex to zero, the 2D polygon was then positioned in a 3D space characterized by the X-Y-Z coordinate system. The geometry of the polygon could be determined by the spatial coordinates of all its vertices. These polygons would be used as 2D digital aggregates in the subsequent steps.

(3)

3D digital aggregate generation

Based on the 2D digital aggregate generated previously, additional vertices with random spatial coordinates were generated, as shown in Figure 5. These additional vertices, together with the original vertices of the 2D digital aggregate, enclosed a space representing a 3D digital aggregate. In this process, to ensure that the projection of the 3D digital aggregate matched the image captured by the AIMS, the projections of these vertices on the X-Z plane had to fall within the interior of the 2D polygon.

(4)

Concrete model generation

By distributing the 2D or 3D digital aggregates within a prescribed boundary without any overlapping, meso-structures of the concrete specimen were generated. As shown in Figure 5, 2D aggregates were distributed in a rectangular area (about 85 mm × 200 mm) near the major crack in the concrete segment, while the 3D aggregates were placed within a cube with a side length of 50 mm.

3.3. Finite Element Model Development

3.3.1. Meso-Scale Model for Simulating Direct Tension

In Module 1, shown in Figure 4, a series of 3D meso-scale finite element models of cubic concrete specimens were developed to determine the overall fracture properties of PFRC based on the simulation of direct tension. The modeling details are as follows.

(1)

Model size and mesh properties

The size of the finite element model in this section is 50 mm × 50 mm × 50 mm, which is sufficient to represent the effective properties of PFRC reported by a previous study [16]. The cohesive zone model (CZM) was used to simulate the crack propagation behavior in the model. Considering that the fracture strength of the aggregate is relatively larger than that of the ITZ and the cement mortar, cracks were assumed to propagate within the ITZ and cement mortar, which could reduce computational costs. Based on this assumption, cohesive elements with zero thickness were inserted within the ITZ and cement mortar. Finally, about 6.5 × 10⁵ elements were used to mesh this model, including 2.6 × 10⁵ linear tetrahedral elements (C3D4) for aggregates and cement mortar, and 3.9 × 10⁵ linear wedge elements (COH3D6) for cohesive elements.

(2)

Loading and boundary conditions

An implicit dynamic analysis was conducted to simulate the fracture behavior of the model during direct tension. During the simulation, one face of the concrete cube model was fixed, and a 0.2 mm tensile displacement was applied to the opposite face in 1 s. With the gradual increase of the applied displacement, the reaction force increased to the peak value and then began to decrease. The ratio of the maximum reaction force to the cross-sectional area of the model was treated as the overall tensile strength of PFRC. Since the meso-structure of the concrete model was heterogeneous, when the tensile displacement was applied to different faces, the calculated tensile strength could vary due to anisotropy. In order to eliminate this influence, three parallel simulations were conducted for each model, as shown in Figure 4. The mean value of tensile strength along three axes was taken as the tensile strength of PFRC.

(3)

Material properties

For the meso-scale simulation of direct tension in this section, material properties of aggregate, cement mortar, and ITZ are needed. Existing literature [16] has investigated the changes in fracture parameters of cement mortar and ITZ with polypropylene fiber content of 0, 0.2%, 0.4%, and 0.6%, as shown in

Table 3. Fracture properties of cement mortar and ITZ [16].

Fiber Content (%)	Tensile Strength (MPa)		Fracture Energy (N/mm)
	Mortar	ITZ	Mortar	ITZ
0	4.02	2.32	0.169	0.049
0.2	4.4	2.15	0.199	0.038
0.4	4.9	2.02	0.28	0.029
0.6	4.45	1.78	0.319	0.019

. The polypropylene fiber content used in the bending test of this study was 0.2%, as covered by the data in Table 3. Therefore, the tensile strength and fracture energy for 0.2% polypropylene fiber content shown in Table 3 were used in the meso-scale model for analysis and model validation. In addition, fracture properties with other polypropylene fiber content in Table 3 were also used for further simulations, which were beneficial for a better understanding of the fracture mechanism of PFRC segments. Based on the material properties shown in Table 3, a bilinear traction-separation law was adopted for the cohesive elements to describe the relationship between stress and displacement. The material coefficient of the B-K (Benzeggagh–Kenane) criterion was taken as 1.2. The density of aggregate and mortar were taken as 2700 kg/m³ and 2200 kg/m³, respectively, based on the raw material testing report. The elastic modulus and Poisson’s ratio were taken as 72 GPa and 0.16 for aggregate, and 26 GPa and 0.2 for mortar, as recommended by a previous study [16].

3.3.2. Macro-Scale Model for Simulating Bending Test

In Module 2 shown in Figure 4, a 3D finite element model of a concrete segment was developed to simulate the bending test. The modeling details are as follows.

(1)

Model size and mesh properties

To reduce computational costs, a quarter concrete segment model was developed instead of a full-scale model. In this case, the length in the tunneling direction (900 mm) and the horizontal span (1900 mm) of the model were both half of those of the full-scale concrete segment, while the thickness of the model (400 mm) was the same as that of the full-scale concrete segment. A total of 18 principal reinforcing steel bars with a diameter of 28 mm were embedded in the model, and their distribution was consistent with that of the concrete segment used in the bending test. Cohesive elements with zero thickness were inserted within the concrete of the model to simulate crack propagation inside the concrete segment during loading. About 1.2 × 10⁵ elements were used to mesh this model, including 3.5 × 10⁴ linear hexahedral elements (C3D8) for concrete and steel reinforcement, and 8.5 × 10⁴ linear hexahedral elements (COH3D8) for cohesive elements.

(2)

Loading and boundary conditions

An implicit dynamic analysis was conducted to simulate the bending test of the PFRC segment. Seven levels of load, the same as those used in the full-scale bending test, were applied to the loading area of the model. Symmetry conditions were applied to the boundaries of the model corresponding to the center planes of the full-scale concrete segment. For the model boundary in contact with the steel support shown in Figure 2a, only horizontal displacement in the span direction was allowed.

(3)

Material properties

For the macro-scale simulation of the bending test in this section, the material properties of PFRC and steel are required. The fracture properties of PFRC were obtained from the numerical simulation in Module 1. The density and elastic modulus of PFRC were taken as 2400 kg/m³ and 38.1 GPa based on the raw material testing report. A typical value of 0.4 was used as Poisson’s ratio for concrete in this model. The density, elastic modulus and Poisson’s ratio of steel were taken as 7850 kg/m³, 207 GPa, and 0.25, respectively.

3.3.3. Meso-Scale Model for Analyzing Fracture Behavior

In Module 3, shown in Figure 4, a series of 2D meso-scale finite element models representing local concrete cross-sections were developed to investigate the fracture behavior of a PFRC segment during the bending test. The modeling details are as follows.

(1)

Model size, mesh and material properties

As described in previous sections, the 2D meso-scale model in Module 3 should encompass the major crack simulated in Module 2. An excessively large size would result in significant computational costs, while a too small size could fail to encompass the major crack. Therefore, by analyzing the length of the major crack calculated in Module 2, the size of the 2D meso-scale model was taken as 200 mm × 85 mm. Similar to the meso-scale models in Module 1, cohesive elements with zero thickness were inserted within the ITZ and cement mortar. About 8.7 × 10⁴ elements were used to mesh this model, including 4.4 × 10⁴ linear plane strain triangular elements (CPE3) for aggregates and cement mortar, and 4.4 × 10⁴ linear quadrilateral elements (COH2D4) for cohesive elements. The material properties used in this model were the same as those used in Module 1, as described in Section 3.3.1.

(2)

Boundary conditions

An implicit dynamic analysis was conducted to simulate the fracture behavior of meso-scale models during the bending test. Since the meso-scale models in this section represented the local cross-section of the concrete segment in Module 2, any displacement at the model boundaries should be consistent with the nodal displacement at the same location in Module 2. Therefore, the boundary conditions in this section were determined by extracting nodal displacements at the same locations in Module 2.

4. Model Validation

The meso-scale models in Module 1 were used to determine the overall fracture properties of PFRC, which were then imported into the macro-scale model to analyze the mechanical behavior of the PFRC segment. To validate the accuracy of these meso-scale models, the tensile strength calculated by these models was compared with the experimental values reported in previous literature [16], as shown in Figure 6a. For each fiber content, the tensile strength of PFRC along the X-axis, Y-axis, and Z-axis directions is presented as vertical bars. It is observed that the tensile strength along different directions deviates from each other even when the fiber content is the same. This indicates that the tensile strength of PFRC exhibits certain anisotropy, which is closely related to the heterogeneity of the material structure. Considering this anisotropy, the average values of tensile strength in the three directions were calculated for comparison with the experimental values. As shown in Figure 6a, with the increase in fiber content, the calculated values exhibit trends similar to those of the experimental values, which reach the peak when the fiber content is 0.4% and then decrease. This is because the polypropylene fibers enhance the tensile strength of the cement mortar but decrease the strength of the ITZ between cement mortar and aggregate at the same time.

The relative errors between the calculated and measured tensile strength are shown in Figure 6b. It is observed that the relative errors vary for different fiber contents and stretching directions. The average relative errors for the models with fiber content of 0, 0.2%, 0.4%, and 0.6% are −6.86%, −2.44%, −6.80%, and −1.62%, respectively, indicating that the meso-scale model tends to slightly underestimate the tensile strength of PFRC. Considering that the average relative error is not greater than 7% and the maximum relative error is not greater than 10% for any given case, it can be concluded that the accuracy of the 3D meso-scale models in Module 1 is acceptable.

Figure 6a also reveals an interesting trend: at low fiber content (e.g., 0%), the tensile strength is highest along the X-axis and lowest along the Z-axis. However, at higher fiber contents (such as 0.4% and 0.6%), this pattern reverses, with the Z-direction exhibiting the highest tensile strength and the X-direction the lowest. The authors hypothesize that this phenomenon may be attributed to the material’s meso-structure, particularly the spatial orientation of the aggregates. A schematic representation of this aggregate orientation is illustrated in Figure 7a. As depicted, the spatial orientation of an aggregate is defined by the direction of its longest axis, which can be quantified by the angles formed between this principal axis and the three coordinate axes. For instance, Figure 7a demonstrates the angle between the aggregate’s longest axis and the Z-axis, referred to as the orientation angle along the Z-axis. Furthermore, the spatial orientation of an aggregate directly influences its geometric projection onto different coordinate planes. For example, Figure 7a displays the aggregate’s projection onto the XOY plane, which corresponds to the view along the Z-axis, denoted as the projection along the Z-axis. Based on the definition in Figure 7a, the specific orientation angle and projection along each axis can be evaluated for every aggregate in the 3D meso-scale model. By incorporating the volumetric differences of various aggregate sizes, the volume-weighted averages of these angles and projection areas were calculated for the entire aggregate population, with the results illustrated in Figure 7b. As observed, both the volume-weighted average orientation angles and the projection areas exhibit a consistent trend: they are maximized along the X-axis and minimized along the Z-axis. Correlating this finding with the pattern presented in Figure 6a, it can be inferred that at relatively high fiber contents (such as 0.4% and 0.6%), when the spatial orientation of the aggregates within the concrete predominantly aligns with a specific direction, the apparent tensile strength in that direction tends to be higher than in others. The underlying reason for this anisotropy is that, as the fiber dosage increases, the strength disparity between the mortar matrix and the interfacial transition zone (ITZ) widens significantly, rendering the ITZ the governing weak link within the material [35,36,37]. Consequently, when aggregates preferentially align in a specific direction, the effective ITZ area in that direction is relatively small, thereby mitigating the probability of fracture propagation along these vulnerable interfaces. Notably, this directional bias in ITZ connectivity may also be reflected in transport-related properties, since the ITZ is widely recognized as a relatively porous and conductive pathway in cementitious composites [38,39].

The macro-scale model in Module 2 was used to analyze the overall deformation and determine the location of the major crack during the bending test. To validate the macro-scale model, the vertical displacements at point P1 and horizontal displacements at point P2, shown in Figure 2a, were measured during the full-scale bending test. These measured displacements were compared with the displacements calculated from the macro-scale finite element model in Module 2. The results are shown in Figure 8a. The blue profile with a square mark represents the vertical displacements at the load point P1 through simulation. The red profile with a round mark represents the calculated horizontal displacement at edge point P2. The vertical displacement at P1 and horizontal displacement at P2 during the full-scale bending test are represented by black marks and gray marks, respectively. In general, the simulated values match the experimental values for both the loading point P1 and edge point P2. The absolute errors between the measured and calculated values under different loading levels are shown in Figure 8b. It is observed that compared with the experimental measurements, the macro-scale model tends to overestimate the horizontal displacement at the edge of the concrete segment during the entire experiment. For the vertical displacement at the loading point, the macro-scale model tends to underestimate the values for the first six levels of load but overestimates the value for the final level of load. The maximum absolute error is 0.11 mm for the loading point P1 and 0.16 mm for the edge point P2. Based on the comparison in Figure 8, it is believed that the overall accuracy of the macro-scale model in Module 2 is acceptable for the analysis in this study.

Besides the quantitative absolute error shown in Figure 8b, the overall trends shown in Figure 8a also prove the reliability of the macro-scale model in simulating the bending test. For both the simulation and the experiment, the increase in displacement is relatively slow for the first four levels of load and then becomes faster in the last three levels of load. This occurs because, when the load level is low, the stress and strain in the concrete segment are relatively small, which cannot cause significant damage to the structure. In this case, the integrity of the concrete segment is still good, effectively resisting deformation. However, at the last three levels of load, the damage in the segment accumulates to a certain extent, which reduces the load-bearing capacity of the concrete segment, resulting in a rapid increase in displacement. Based on the crack measurements during the full-scale bending test, the width of the major crack was 0.08 mm at load level 6 and 0.10 mm at load level 7. Before that, no measurable crack was observed. It is interesting because at load level 5, the displacement rate noticeably increased, without an obvious crack. This finding indicates that damage inside the concrete segment occurs before the appearance of a crack and proves that the macro-scale model can effectively simulate the mechanical behavior during the bending test.

5. Analysis and Discussion

5.1. Effect of Polypropylene Fiber on Macro-Scale Fracture Behavior of Concrete Segment

With the validated multi-scale simulation framework, the fracture behavior of PFRC segments was analyzed at both the macro- and meso-scales. Figure 9 illustrates the displacement responses of selected points on the segment during the incremental loading process up to the design load for different fiber contents. Specifically, Figure 9a presents the vertical displacement at the load point P1, while Figure 9b shows the horizontal displacement at the edge point P2. It can be seen that, for both the vertical and horizontal displacements, the evolution throughout the stepwise loading process can be divided into three distinct stages: an elastic stage (shaded in gray in Figure 9), a damage-development stage (yellow shading), and a cracking stage (blue shading). From the first to the third loading level, the computed displacements for different fiber contents are almost indistinguishable, and the load–displacement response in this range remains essentially linear. This is because the applied loads are relatively low in this stage, leading to modest stress levels within the segment that are insufficient to trigger damage. Consequently, the displacement curves show negligible sensitivity to fiber content. As the load increases from the third to the fifth level, the load–displacement curves gradually deviate from linearity, with an increasing slope that indicates an accelerated displacement response. Meanwhile, the curves corresponding to different fiber contents begin to separate, although the differences remain subtle. This behavior can be attributed to the higher stress demand at this stage, which initiates localized damage and reduces the effective carrying capacity of the segment, leading to a faster growth of displacement. Nevertheless, no pronounced cracks are observed in this interval, and the influence of fiber content on the displacement response is still relatively limited. During the final two loading levels, the discrepancies among the load–displacement curves become increasingly pronounced, indicating that the influence of fiber content on the deformation response is amplified as the load approaches the design level. As shown in Figure 9a, at the design load, the segment without polypropylene fibers exhibits the largest vertical displacement, whereas the segment with a fiber content of 0.4% shows the smallest value, corresponding to a reduction of approximately 9.8% relative to the fiber-free case. However, when the fiber content is further increased to 0.6%, the vertical displacement rises rather than continues to decrease, suggesting that the fiber-enhancement effect is not monotonic with dosage. This is consistent with the trend predicted by the meso-scale model in Module 1 of this study, which shows how the strength of polypropylene-fiber-reinforced concrete varies with fiber content. For the horizontal displacement shown in Figure 9b, a fiber content of 0.4% reduces this displacement by approximately 2.9%. These results imply that a fiber content of around 0.4% may represent an optimal dosage under the present loading conditions.

5.2. Effect of Polypropylene Fiber on Meso-Scale Fracture Behavior of Concrete Segment

5.2.1. Effect of Polypropylene Fiber on Crack Propagation

In the bending test of tunnel segments, cracking typically initiates at the inner surface under tension and then propagates upward as the load increases. The crack width measured at the design load is widely adopted as a key criterion for assessing segment quality and serviceability. To capture the influence of polypropylene fibers on crack propagation, the meso-scale model (Module 3) described in the previous sections was adopted to track crack development in segments with different fiber contents over successive load levels. It should be noted that, as indicated in Figure 9, no evident damage is observed within the segment during the first three load steps. Therefore, the discussion in this section focuses on the subsequent four load cases, namely 251.42 kN, 283.35 kN, 299.32 kN, and 315.28 kN, which are denoted as load levels 4–7, respectively. Figure 10 summarizes the evolution of crack width at the bottom region of the meso-scale model as a function of load level and fiber content.

As shown in Figure 10, for fiber contents of 0.0%, 0.2%, and 0.4%, cracking at the bottom region is first observed at load level 6. In contrast, for the specimen with a fiber content of 0.6%, the onset of bottom cracking occurs earlier, becoming evident at load level 5. For a given fiber content, the crack width in the bottom increases progressively as the load level rises. At a fixed load level, however, the crack width varies with fiber content, indicating a clear dependence of crack opening on the amount of polypropylene fibers incorporated. When the fiber content does not exceed 0.4%, the crack width decreases with increasing fiber content, indicating that polypropylene fibers enhance the crack resistance of the segment. However, once the fiber content reaches 0.6%, the crack width increases rather than continues to decrease. This observation again suggests that the crack-resistance improvement provided by polypropylene fibers is not a monotonically increasing function of dosage. At the design load of 315.28 kN (load level 7), the segment with 0.4% polypropylene fiber exhibits a smaller bottom crack width than the fiber-free case. The crack width decreases from 0.0868 mm to 0.0770 mm, representing a reduction of 11.29%. A piecewise function is proposed to describe the correlation between the fiber dosage and the bottom crack width of the PFRC segment at the design load. As shown in Figure 10, the coefficient of determination (R²) for this relationship is 0.8306, where F and CW stand for fiber content (%) and crack width, respectively. Overall, these results indicate that incorporating polypropylene fibers at an appropriate dosage can effectively enhance the crack resistance of tunnel segments.

It should be noted that while the bottom crack width provides a useful indicator of how polypropylene fibers affect the segment’s crack resistance, it is not sufficient to describe the fracture behavior solely in terms of crack width. This measure mainly reflects the severity of cracking at the inner surface, but it does not reveal how cracks initiate and propagate within the segment’s interior. To this end, the meso-scale model (Module 3) was used to investigate the internal crack-growth process and its characteristics. For a clearer comparison, two representative cases are selected, namely the fiber-free model (0%) and the model with 0.4% fibers, as the latter corresponds to the best crack-resistance performance. Their crack propagation patterns within the damaged region under load levels 5–7 are presented in Figure 11, where the gray phase corresponds to the cement mortar matrix, the blue phase represents the aggregates, and the white traces delineate the cracks. As illustrated in Figure 11, at the relatively low load level (e.g., load level 5), microcracks first emerge near the lower part of the segment, close to the inner surface. This is expected, as this region is subjected to higher tensile stress in bending. At this stage, the overall crack paths in the 0% and 0.4% fiber models are largely similar, with damage localized primarily along the ITZ and no visible surface-opening crack on the inner surface. Notably, the 0.4% fiber model shows slightly wider localized cracks near the ITZ, which can be attributed to the reduced mortar–aggregate bond strength at the ITZ associated with polypropylene fiber incorporation. As the load is further increased to load level 6, the localized cracks continue to propagate upward. In the model with a fiber content of 0.4%, cracking remains largely confined to the ITZ surrounding the aggregates. By contrast, in the model with no fiber, cracks progressively extend through the mortar matrix, linking adjacent aggregates and forming more continuous, connected crack paths. This difference can be explained by the role of polypropylene fibers in enhancing the tensile strength of the mortar matrix, which suppresses the penetration of ITZ-initiated microcracks into the surrounding mortar. In the fiber-free model, the lower tensile capacity of the mortar facilitates crack growth within the matrix, thereby promoting the development of through-mortar, connected cracking. When the load is further increased to load level 7, a crack-growth pattern similar to that at load level 6 is observed. The cracked region continues to migrate upward, while cracks in the 0.4% polypropylene fiber model remain less continuous and less connected than those in the fiber-free segment. Overall, the above observations suggest that polypropylene fibers enhance the crack resistance of concrete segments primarily by improving the cracking resistance of the mortar matrix. Although fiber incorporation may reduce the tensile strength of the ITZ to some extent, an appropriate fiber dosage can strengthen the mortar sufficiently to hinder the penetration of ITZ-initiated microcracks into the matrix, thereby effectively suppressing the formation of connected crack networks.

5.2.2. Effect of Polypropylene Fiber on Stiffness Degradation

During stepwise loading, progressive damage is induced as the stress level increases, accompanied by progressive stiffness degradation. Accordingly, damage evolution can be effectively characterized from the perspective of stiffness loss, which provides an important basis for analyzing the fracture behavior of the segment. To quantify this process in the meso-scale model for the bending test simulation, the stiffness degradation variable (SDEG) was employed. SDEG ranges from 0 to 1 and measures the extent of stiffness loss in an element: SDEG = 0 represents an undamaged state, whereas values approaching 1 indicate that most of the stiffness has been lost, corresponding to severe damage. In this study, an element is considered failed when SDEG reaches 0.9, ensuring numerical stability and avoiding convergence issues near complete stiffness loss.

Using the meso-scale model (Module 3), the SDEG value of each element within the local meso-structure was obtained at the design load, and its distribution was statistically examined. It should be noted that only a small fraction of the elements experience damage at the design load, meaning that the vast majority of elements remain intact, with SDEG remaining zero. If all elements were included in the statistical analysis, the distribution would be dominated by zeros and the characteristics of the damaged elements would be obscured. According to the simulation results, among all meso-scale models with different fiber contents, the maximum number of damaged elements (SDEG > 0) is 1738. Therefore, in the following analysis, a total of 1800 elements with the highest SDEG values are selected for statistical evaluation. This selection not only covers all damaged elements but also retains a small number of undamaged ones, facilitating a clearer characterization of damage.

Figure 12 presents the distribution characteristics of SDEG for the selected elements at the design load. In particular, Figure 12a shows the probability density distribution of SDEG. Two pronounced peaks can be identified, located near SDEG = 0 and SDEG = 0.9, indicating that a considerable portion of the sampled elements is either essentially undamaged or close to the failure threshold. Notably, for the model with a fiber content of 0.4%, the probability density near SDEG = 0 is higher than that of the other cases, whereas the density near SDEG = 0.9 is lower. This suggests that, compared with the other fiber dosages, a 0.4% polypropylene fiber content generally reduces the damage severity at the element level and effectively suppresses crack formation.

Figure 12b summarizes the number of elements with SDEG exceeding the prescribed thresholds (0.2, 0.4, 0.6, and 0.8) at the design load. It can be seen that for any of these thresholds, when the fiber content does not exceed 0.4%, the number of damaged elements decreases with increasing fiber content, indicating that polypropylene fibers mitigate damage at the meso-scale. In contrast, once the fiber content reaches 0.6%, the counts of damaged elements become consistently higher. This trend is particularly evident at higher damage levels (e.g., SDEG > 0.6), where the 0.6% fiber content exhibits more severely damaged elements than all other fiber contents. These results imply that when the fiber dosage exceeds the optimal range, polypropylene fibers may impair crack resistance and aggravate damage development.

5.2.3. Effect of Polypropylene Fiber on Degradation of ITZ and Mortar

To further clarify whether polypropylene fibers exert a distinct influence on the crack resistance of the ITZ and the mortar matrix, the SDEG values of ITZ and mortar elements at the design load were statistically analyzed based on the meso-scale model (Module 3). The resulting probability density distributions are presented in Figure 13. Similar to the previous analysis, to avoid the statistics being dominated by a large number of undamaged elements with zero stiffness degradation, only elements with relatively high damage levels were considered. According to the simulation results, the maximum number of damaged elements is 961 for the ITZ and 927 for the mortar across all cases. Therefore, for both phases, the 1000 elements with the highest SDEG values were selected for the following statistical evaluation.

Figure 13a,b show the SDEG distributions for the ITZ and the mortar matrix, respectively, and a clear difference between the two phases can be observed. For the ITZ, in addition to the peak at SDEG = 0, a pronounced peak also appears near SDEG = 0.9. In contrast, the mortar does not exhibit a similarly prominent peak at SDEG = 0.9; instead, the SDEG values of damaged mortar elements are distributed more evenly. This indicates that, under the design load, once damage is initiated in the ITZ (SDEG > 0), it is more likely to rapidly evolve toward near-failure levels (SDEG > 0.9) than in the mortar. These observations further suggest that the ITZ constitutes a relatively weak link in the bending-induced fracture process of PFRC segments.

In addition to the probability density distributions of SDEG for the ITZ and mortar shown in Figure 13, the number of ITZ and mortar elements falling within different SDEG intervals was also quantified. These results are summarized in Figure 14 in the form of stacked radar charts. As can be seen, the numbers and proportions of ITZ and mortar elements vary markedly across different SDEG ranges. For lightly damaged elements, namely those in the intervals 0 < SDEG < 0.25 (Figure 14a) and 0.25 < SDEG < 0.5 (Figure 14b), the mortar phase accounts for a substantially larger share than the ITZ. In contrast, for more severely damaged elements, i.e., 0.5 < SDEG < 0.75 (Figure 14c) and 0.75 < SDEG < 1 (Figure 14d), ITZ elements become dominant in both number and proportion. This trend indicates that, within the fiber content range considered in this study, the ITZ generally experiences more severe damage than the mortar at the design load, regardless of fiber dosage. This observation is reasonable, as the ITZ typically exhibits lower strength than the mortar matrix for all fiber contents.

To better highlight how the ITZ and mortar phases contribute across different damage levels, Figure 15 compares their element proportions within each SDEG interval. A clear trend can be observed. For elements with lower damage (smaller SDEG), the mortar phase accounts for a larger proportion, whereas for elements with higher damage (larger SDEG), the ITZ becomes increasingly dominant. This observation is consistent with the statistical results discussed above. Moreover, within each SDEG interval, the mortar proportion at a fiber content of 0.4% is the lowest among all cases, further supporting the idea that 0.4% represents the optimal dosage for the loading condition considered in this study.

Finally, it should be noted that, given the substantial improvement in mortar crack resistance provided by polypropylene fibers, further enhancement of the overall crack resistance of PFRC segments may require strategies aimed at strengthening the ITZ. Otherwise, the ITZ is likely to remain the governing weak link in fracture under bending.

6. Conclusions

In this study, a multi-scale modeling framework for investigating the fracture behavior of PFRC segments under bending was proposed. A full-scale bending test on PFRC segments was conducted to validate the developed framework. The effects of polypropylene fiber content on the macro- and meso-scale fracture behavior of PFRC segments were investigated. The following conclusions were drawn from the analysis:

(1)

The meso-scale module of the framework reproduced the experimentally observed tensile strength of PFRC, with an average relative error below 7% and a maximum error below 10%, providing reliable fracture-related inputs for subsequent analyses. The macro-scale module captured the global deformation response of the PFRC segment during the full-scale bending test. The maximum absolute differences between the simulated and measured displacements were 0.11 mm at the load point and 0.16 mm at the edge point. Overall, the accuracy of the proposed multi-scale modeling framework is acceptable.

(2)

The PFRC segment response under incremental bending showed three stages: elastic, damage, and cracking. At the design load, 0.4% fibers yielded the smallest deformation, reducing vertical displacement at the load point by 9.8% and horizontal displacement at the edge point by 2.9% compared to the fiber-free case. Increasing fiber content to 0.6% increased displacement, confirming a non-monotonic benefit and an optimal dosage around 0.4%.

(3)

The meso-scale results show that polypropylene fibers markedly affect crack initiation and internal crack connectivity. At the design load, a fiber content of 0.4% reduced the bottom crack width from 0.0868 mm to 0.0770 mm (−11.29%). Although fiber incorporation may locally promote ITZ cracking due to reduced mortar–aggregate bonding, the strengthened mortar matrix at an appropriate dosage effectively suppresses crack penetration through the mortar and limits the development of continuous, connected crack networks.

(4)

The SDEG-based meso-scale statistics at the design load show that polypropylene fibers can effectively mitigate stiffness degradation up to an optimal dosage of 0.4%, as reflected by a higher fraction of nearly undamaged elements, a lower fraction of near-failed elements, and consistently reduced counts of damaged elements above multiple SDEG thresholds. When the fiber content increases to 0.6%, the number of severely degraded elements rises markedly (especially for SDEG > 0.6), indicating aggravated damage and confirming a non-monotonic fiber benefit.

(5)

The SDEG statistics evaluated separately for the ITZ and mortar at the design load reveal that the ITZ undergoes more severe and more abrupt stiffness degradation than the mortar matrix. The ITZ exhibits a pronounced high-damage peak near the failure threshold, whereas a more gradual degradation process is observed for mortar. Moreover, mortar dominates the low-damage ranges, while ITZ dominates the high-damage ranges, confirming the ITZ as the governing weak link in the bending-induced fracture of PFRC segments. Therefore, further enhancement of the overall crack resistance of PFRC segments may require strategies aimed at strengthening the ITZ.

While this study provides valuable insights, future research will address current methodological limitations. Specifically, as the current model homogenizes fibers within the mortar, future work will develop explicit micro-scale fiber models to capture localized interactions. Additionally, while the full-scale tests successfully simulated design load conditions, future experiments will apply ultimate loads to investigate the complete post-peak fracture morphology. These advancements will further bridge the gap between structural behavior and meso-mechanisms, promoting the reliable application of PFRC in engineering practice.