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Article

Improving Mixed-Mode Fracture Properties of Concrete Reinforced with Macrosynthetic Plastic Fibers: An Experimental and Numerical Investigation

1
Department of Civil Engineering, Faculty of Engineering, Razi University, Kermanshah 6718958894, Iran
2
Department of Civil Engineering, Ramsar Branch, Islamic Azad University, Ramsar 4691966434, Iran
3
Department of Civil Engineering, University of Science and Culture, Tehran 1461968151, Iran
4
Department of Civil Engineering, Kermanshah Branch, Islamic Azad University, Kermanshah 6714656141, Iran
5
Collage of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China
*
Authors to whom correspondence should be addressed.
Buildings 2024, 14(8), 2543; https://doi.org/10.3390/buildings14082543
Submission received: 6 June 2024 / Revised: 9 July 2024 / Accepted: 16 August 2024 / Published: 18 August 2024
(This article belongs to the Collection Advanced Concrete Materials in Construction)

Abstract

:
This article offers a comprehensive analysis of the impact of MSPF on concrete’s mechanical properties and fracture behavior. Combining findings from numerical simulations and laboratory experiments, our study validates numerical models against diverse fiber percentages and aggregate distributions, affirming their reliability. Key findings reveal that mixed-mode fracture scenarios in fiber-reinforced concrete are significantly influenced by the mode mixity parameter (Me), quantifying the balance between mode I and mode II fracture components, ranging from 1 (pure mode I) to 0 (pure mode II). The introduction of the effective stress intensity factor (Keff) provides a profound understanding of the material’s response to mixed-mode fracture. Our research demonstrates that as Me approaches zero, indicating shear deformation dominance, the concrete’s resistance to mixed-mode fracture decreases. Crucially, the addition of MSPF considerably enhances mixed-mode fracture toughness, especially when Me ranges between 0.5 and 0.9, resulting in an approximately 400% increase in fracture toughness. However, beyond a specific threshold (approximately 4% FVF), diminishing returns occur due to reduced fiber–cement mortar bonding forces. We recommend an optimal fiber content of around 4% by weight of the total concrete mixture to avoid material distribution disruption and strength reduction. The practical implications of these findings suggest improved design strategies for more resilient infrastructure, particularly in earthquake-resistant constructions and sustainable urban development. These insights provide a valuable framework for future research and development in concrete technology.

1. Introduction

Concrete is one of the most widely used building materials in the world today due to its high compressive strength, suitable seismic performance, and economic efficiency [1,2]. However, low tensile strength and susceptibility to cracking are significant challenges that threaten the application of these materials [3]. Conventional concrete lacks the ability to avoid cracking, which is inevitable during its service life and observed in the primary stages of use. The presence of these cracks can have various consequences for the safety, performance, and durability of structures, ranging from minor defects to significant quality concerns, with potentially catastrophic failures that incur high costs to human life and the economy [4,5]. Cracks can weaken structures by causing issues such as corrosion of internal reinforcement and reducing load capacity [6]. To overcome these challenges, various methods have been employed to enhance concrete strength [7,8,9,10]. Incorporating suitable fibers into concrete is an efficient method to improve its fracture properties and prevent the progression of microcracks into macrocracks, thus reducing its load-bearing capacity [11,12,13]. Some of the most commonly used fibers for enhancing the mechanical properties of concrete include steel fibers, polypropylene fibers, carbon fibers, basalt fibers, and polyvinyl alcohol fibers [14,15,16,17,18,19]. The effectiveness of these fibers stems from distinct underlying mechanisms that impede the initiation and propagation of cracks in the concrete structure. For example, polyvinyl alcohol fibers are widely favored due to their excellent dispersion and high hydrophilicity, while the performance of steel fibers is influenced by factors such as shape, aspect ratio, and surface texture [20,21]. Macrofibers have a length at least twice that of coarse aggregate particles, and their cross-sectional diameter is significantly larger than cement particles [22]. Synthetic fibers (SF) have been studied by many researchers to improve resistance to cracking, ductility, and strength, and various experiments have suggested appropriate amounts to enhance crack resistance and reduce crack width in concrete [23,24,25]. Researchers have also concluded in previous studies that plastic fibers improve the tensile strength, ductility, fracture toughness, absorption capacity, and blast resistance of concrete [26,27,28,29]. The widespread use of plastic fibers aims to increase the tensile strength and crack resistance of concrete. These fibers can carry internal forces by resisting crack propagation and movement on cracks, thereby enhancing the tensile and flexural strength of concrete [30]. However, some researchers have observed a reduction in compressive strength when using plastic fibers in concrete [31,32,33]. Both macro- and microfibers inhibit crack initiation and propagation at the micro level, leading to improved strength, ductility, and workability [34,35]. In general, the presence of macrofibers affects the desired location of inclined tensile cracks and their propagation path [36]. Additionally, the presence of microfibers increases flexural and shear behavior, particularly in reinforced concrete beams under static and cyclic loading [37,38,39]. In recent years, research on the application of steel fibers in ultra-high-performance concrete (UHPC) has primarily focused on fiber orientation and shape, significantly impacting material properties. These fibers are extensively studied and utilized to enhance compressive, tensile, and flexural strength of high-performance concrete (HPC) and UHPC [40]. However, an excessively high length-to-diameter ratio may reduce workability, pose construction challenges, and have a negative impact on mechanical properties [41]. Despite these challenges, fiber-reinforced concrete with steel fibers has garnered significant attention in civil engineering due to its improved tensile strength and resistance to shrinkage cracks. Various MSPF such as polypropylene and polyvinyl alcohol have also made considerable advancements in workability, crack resistance, and impact resistance [42]. Recent studies have shown that combining these fibers in hybrid fiber-reinforced concrete (HFRC) can lead to synergistic effects and further increase mechanical properties. Moreover, research on mineral and natural fibers such as basalt and cellulose, due to their high thermal stability and environmental compatibility, shows potential for sustainable construction methods [43]. Despite these advancements, challenges remain in optimizing fiber types and dosages to maximize mechanical properties and durability of concrete. In this regard, theoretical models can predict the behavior of fiber-reinforced concrete structures under different conditions and aid in practical applications such as bridge construction, where improved ductility and toughness significantly enhance structural flexibility [44]. However, further research is required to develop standardized testing methods, optimize hybrid fiber combinations, and reduce production costs. These efforts will ensure the continued evolution and application of fiber-reinforced concrete (FRC) in building sustainable and resilient infrastructure. Concrete exhibits complex behavior and is often seen as a two-phase material consisting of aggregates and mortar, but imaging techniques such as electron microscopy and X-ray computed tomography reveal its true nature as a multiphase material with heterogeneous microstructure [45]. Various models at the macroscopic level have been proposed to understand concrete cracking behavior. The representative volume element (RVE) concept assumes material homogeneity, leading to the development of fracture mechanics concepts such as linear elastic fracture mechanics (LEFM), the cohesive crack model, and the cohesive zone model [46,47,48]. Although these macroscale models effectively simulate behaviors of different materials, parameters such as fracture energy are heavily influenced by the size of the fracture process zone (FPZ), primarily determined by the mesostructure of materials. Mesoscale modeling provides a powerful tool to investigate details of the fracture process zone, including a three-dimensional representation of the microstructure that allows exploration of how heterogeneities affect mechanical responses [49]. Most studies at the mesoscale consider concrete as a two-phase or three-phase material, with aggregates, matrix, and interfacial transition zone (ITZ) considered as an intermediate element [50,51,52]. While studies involving additional heterogeneity surfaces are rare, they are not fully represented in the literature [53]. Mesoscale models can be classified into continuum models, discrete models, and discontinuous models based on their behavior with respect to the field. Continuum models directly integrate mesostructures into finite element frameworks using techniques such as random aggregates, random fields, or X-ray tomography images [51,54,55,56,57,58]. Discontinuous models introduce discontinuities into the displacement field and enable explicit representation of cracks [59,60]. Strategies such as extended finite element method (XFEM) and energy finite element method (EFEM) can also be used to study the precise behavior of concrete [61,62,63,64].
Recent studies highlight the significance of considering the role of pozzolanic materials in concrete mixtures for enhancing mechanical properties and sustainability. Research on the impact of natural pozzolan as a partial substitute for microsilica demonstrates the environmental and performance benefits of such materials [65]. Additionally, numerical methods in structural identification, such as advanced observability techniques for 2D plane structures, are crucial for addressing nonlinear variables in concrete systems [66].
The investigation of sea sand coral concrete with FRP bar reinforcement, high-performance concrete with multiple fiber types, ultraductile waterborne epoxy–fiber composites, and fiber-reinforced concrete under impact loading highlights how diverse materials and reinforcements significantly enhance concrete’s performance and durability under extreme conditions [67,68,69].
Recent advances in machine learning algorithms have introduced innovative methods for predicting concrete’s mechanical properties. Studies using genetic programming and ensemble learners have successfully optimized plastic concrete design by accurately forecasting its compressive and flexural strengths [70]. Improved random forest algorithms, for instance, have effectively predicted concrete compressive strength, enhancing design efficiency and cost-effectiveness [71]. These techniques not only improve prediction accuracy but also promote sustainability by incorporating plastic waste into concrete. Studies demonstrate that MSPF significantly enhance both plastic and hardened concrete, offering practical applications across various construction contexts. In infrastructure projects, they enhance durability and reduce maintenance in bridges, highways, and tunnels. For commercial and residential buildings, MSPF improve toughness and impact resistance in floor slabs and foundations, ensuring stability. In precast concrete elements, they increase tensile strength and reduce cracking in beams, panels, and columns. In shotcrete applications, MSPF provide internal support and reduce rebound in tunnels, retaining walls, and slope stabilization. For nonmetallic reinforcement, they are ideal for environments requiring high corrosion resistance, such as wastewater treatment plants. In composite steel deck construction, MSPF control shrinkage cracks, enhancing the bond between concrete and steel in high-rise buildings. Furthermore, they support sustainable construction practices by increasing concrete longevity and reducing maintenance needs, promoting sustainability in infrastructure. These applications underscore the critical role of MSPF in improving concrete quality, performance, and sustainability [72,73,74,75]. In summary, by incorporating these contemporary perspectives and methodologies, this research not only validates the mechanical improvements observed with MSPF but also situates our findings within the broader context of recent advancements in concrete technology. This comprehensive approach ensures the robustness and applicability of our recommendations for concrete design and construction.
This article presents a holistic analysis of the impact of macrosynthetic plastic fibers (MSPF) on concrete’s mechanical properties and fracture behavior, merging findings from both numerical simulations and laboratory experiments to substantiate the reliability of our numerical models. Our research spotlights the study of mixed-mode fracture scenarios in fiber-reinforced concrete, particularly focusing on the mode mixity parameter Me, which quantifies the equilibrium between mode I and mode II fracture components. Me’s continuum from 1 (pure mode I) to 0 (pure mode II) plays a defining role in understanding concrete’s response to mixed-mode fracture. Our study unveils that as Me tends toward zero, signifying the ascendancy of shear deformation, concrete’s resistance to mixed-mode fracture diminishes. Crucially, the incorporation of macrosynthetic plastic fibers (MSPF) significantly bolsters mixed-mode fracture toughness, especially within the range of Me from 0.5 to 0.9, resulting in an approximately 400% increase in fracture toughness. However, beyond a certain threshold (approximately 4% fiber volume fraction (FVF)), diminishing returns occur due to decreased bonding forces between fibers and the cement mortar. Practically, we recommend an optimal fiber content of around 4% by weight of the total concrete mixture, as exceeding this threshold risks material distribution disruption and strength reduction. In addition, we introduce a standardized presentation for mixed-mode fracture outcomes and align them with numerical data through a power law criterion. In summary, this research provides a deep understanding of the mechanics behind FRC (fiber-reinforced concrete)’s performance, suggesting practical guidelines for concrete design and construction, enhancing the resilience and robustness of structures.

2. Dataset Materials and Specimen Preparation

The experimental samples were composed of gravel and sand with specific dimensions as shown in Figure 1a. The maximum aggregate size (dmax) was 16 mm, and the median size (d50) was 2 mm. The samples also included MSPF, water, and Portland cement. We maintained a constant aggregate-to-cement ratio of 0.82 and a water-to-cement ratio of 0.45. The concrete specimens were prepared following standard procedures at a controlled temperature of 20 °C and humidity above 90%. Aggregates, including gravel and sand, constituted 75% of the total volume, as depicted in Figure 1b, where the samples are referred to as SCB. After a curing period of 90 days, we subjected the samples to quasi-static loading tests at a rate of 0.001 mm/min. During these tests, we measured applied force, displacement beneath the sample, and crack mouth opening displacement (CMOD). Three cylindrical samples were evaluated for mechanical properties such as elastic modulus, compressive strength, tensile strength, and Poisson’s ratio. The results were 30.6 Gpa for elastic modulus, 3.16 Mpa for tensile strength, 35.3 Mpa for compressive strength, and 0.22 for Poisson’s ratio. The mechanical properties and the number of laboratory samples are shown in Table 1 and Table 2. Furthermore, flexural tests on SCB1 and SCB2 samples, which lacked fibers, showed mode I fractures, as illustrated in Figure 2.
A thorough analysis of the SCB1 and SCB2 samples indicated that cracks began at the notch and progressed towards both the aggregate and the interfacial transition zone (ITZ). The cracks subsequently grew vertically in alignment with the direction of the applied load. The highest vertical load measured during the experiments was about 2.2 kN, with the tensile strength varying between 3.05 and 3.3 MPa. The force–displacement and force–CMOD curves for the samples without fibers are displayed in Figure 2. To summarize, the initiation of cracks occurred at the notch and advanced towards the aggregate and ITZ, with the maximum vertical loads in fiber-free samples ranging from 2.1 to 2.3 kN, corresponding to a tensile strength of approximately 3.16 MPa.
In the subsequent phase of our research, we evaluated two concrete samples embedded with 1.5% FVF of MSPF. These fibers markedly improved the concrete’s structural integrity, inhibiting crack formation and elevating the fracture energy (GF) from GF = 0.095 N/mm to GF = 0.243 N/mm. Furthermore, the peak tensile force measured was about 33% greater compared to samples without fibers. The corresponding load–CMOD curves, depicted in Figure 3, demonstrate a shift in the balance of compressive and tensile forces as the fractures moved toward the point of load application, influencing the internal forces acting within the fibers. The integration of plastic fibers effectively prevented the samples from being completely bifurcated by the conclusion of the loading. Moving forward, our research will aim to validate numerical models initially and further investigate the influence of varying fiber volume percentages on concrete behavior in both mode I and mode II fractures, as well as their collective effect, through comprehensive numerical simulations.

3. Fracture Modeling

During this stage of the investigation, two-dimensional (2D) numerical models were employed. The analysis centered on two critical aspects. The primary concern was the selection of suitable failure criteria that accurately reflect the nonlinear behavior of the fibers and concrete. The second key aspect was the identification of an appropriate yield criterion to effectively characterize the interactions between these materials. In the 2D modeling process, Plane183 elements were utilized, each configured with eight nodes and two translational degrees of freedom. Additionally, interface elements were used to facilitate the modeling of interactions among various materials. The elasticity modulus was set at 25 GPa for cement mortar and 42.5 GPa for aggregates. Using Equation (1) [76], we calculated the homogeneous elasticity modulus of the concrete, which was determined to be 30.5 GPa. These values were integral to the simulations conducted.
E h = E c m + V a ( E a E c m ) 1 ( 1 V a ) E a E c m E c m 4 G c m 3   ,   G c m = E c m 2 ( 1 + ν c m )
In Equation (1), Eh is Young’s modulus of the homogeneous concrete, Ecm is Young’s modulus of the cement matrix, Va is the volume of the aggregate, Ea is Young’s modulus of the aggregate grains, and Gcm is the shear modulus of the cement matrix. In the numerical simulations, we configured the elasticity modulus at 42.5 GPa for aggregates and 25 GPa for cement mortar, with each material having a Poisson’s ratio of 0.22. To address the nonlinear behavior of the aggregate and cement mortar under both compressive and tensile stresses, we employed the Menetrey–Willam yield surface [77], as depicted in Figure 4. Additional information on this yield surface can be found in references [78,79]. Furthermore, the nonlinear behavior of plastic fibers was modeled using the Von Mises yield surface, which incorporates multilinear isotropic hardening, as outlined in the corresponding references [76].
Cohesive zone modeling (CZM) was utilized to analyze the interactions between cement mortar, aggregates, and plastic fibers within the composite. This method leverages Equation (2), an extension of Hooke’s Law, to characterize the behavior of cohesive elements in the material. By applying this technique, the model provides a detailed understanding of how these different components bond and respond under stress, thereby offering insights into the structural integrity and mechanical properties of the composite material. This approach is crucial for predicting and enhancing the durability and performance of concrete structures incorporating diverse materials.
t = k δ   T n T t T s = k n 0 0 0 k t 0 0 0 k s u n u t u s  
In CZM, the stiffness matrix K serves as a critical link between displacement values (δ) and stress values (Ti). In this context, Ti signifies the levels of stress experienced, Ki denotes the stiffness of the cohesive elements, and δi describes the relative displacement occurring between two interacting surfaces. For a clearer understanding of this interaction, Figure 5 illustrates the concept of relative displacement between these surfaces, visually explaining how displacement and stress are interrelated in CZM. This depiction aids in comprehending how materials behave under stress and the mechanical interactions at play within the matrix of composite structures.
In cohesive zone modeling (CZM), the initial stiffness values, represented as Ki0, indicate the stiffness of materials before any fracture occurs. For more comprehensive details on these initial stiffness parameters, refer to references [51,80]. The modeling incorporates both tangential and normal stiffness levels. When applying penalty stiffness, the relative deformation remains zero within the elastic range. However, postfracture, the stiffness values vary depending on the parameter d.
k n = ( 1 d ) k n 0 k t = ( 1 d ) k t 0 k s = ( 1 d ) k s 0
The yield surface regulates the way the crack expands [81]:
( G n G c n ) 2 + ( G t G c t ) 2 + ( G s G s t ) 2 = 1 G n = t n d u n G t = t t d u t G s = t s d u s
In our analysis, the critical fracture energies for modes I, II, and III are represented by the values Gcn, Gct, and Gcs, respectively. The maximum tensile strength for the ITZ is considered within the ratio range of ft, ITZ/ft, cm = 0.3 to 1, where Tn denotes the tensile strength of the cement mortar. For this study, a ratio of 0.5 was specifically considered. Generally, CZM in 2D is conducted as illustrated in Figure 6.
The interaction between plastic fibers and cement mortar was explored using an analytical model focused on the bond stress–slip relationship. This model utilizes specific mechanical characteristics: the slip at first crack (S1) is defined as 1 mm, and the slip at ultimate strength (S2) is 3 mm, with the distance between ribs noted as S3. Additionally, the model incorporates an alpha coefficient (α) of 0.4, which significantly influences the bond behavior. The ultimate bond stress (τu) is calculated using the formula 2.5 times the concrete’s compressive strength (f′c) plus 0.5, and the final bond stress (τf) is determined to be 0.4 times τu. These critical parameters were used to assess the bond performance between the composite materials. The formulation of this model is depicted in Figure 7a. We employed ANSYS, a comprehensive finite element software, to incorporate this model into the USERFRIC section for a detailed analysis. To verify the model’s accuracy, a pull-out test was conducted on an individual fiber, with the results displayed in Figure 7b. The two-step loading shown in the plot confirms a robust correlation between our analytical predictions and the numerical simulations. As a result, this model was further applied in subsequent studies to more thoroughly investigate the bonding interactions between the fibers and cement mortar.

4. Verification of Numerical Models

4.1. Shape Form Factor

To accurately compute the stress intensity factors for SCB samples under mixed fracture modes, it is crucial to determine the shape factors for these specimens. Figure 8 illustrates this configuration, highlighting point A as the crack tip. By using the nodal displacements [82], it is possible to calculate the fracture mode shape factors using the provided Equations (5)–(7).
u = u ¯ A + ( 3 u ¯ A + 4 u ¯ B u ¯ C ) r L + ( 2 u ¯ A + 2 u ¯ B 4 u ¯ ) r L  
v = v ¯ A + ( 3 v ¯ A + 4 v ¯ B v ¯ C ) r L + ( 2 v ¯ A + 2 v ¯ B 4 v ¯ ) r L
K I K I I = 1 2 2 G κ + 1 2 π L 0 1 1 0 3 u ¯ A + 4 ( u ¯ B u ¯ D ) ( u ¯ C u ¯ E ) 3 v ¯ A + 4 ( v ¯ B v ¯ D ) ( v ¯ C v ¯ E )
To calculate the stress intensity factors for SCB samples under various fracture modes, we employed both the finite element method and the displacement correlation method. These approaches helped us accurately determine the stress intensity factors for the specific fracture combinations in the samples. The analysis involved setting appropriate boundary conditions, creating a mesh for the geometry, and conducting numerical simulations to gather the necessary data for these calculations.
A numerical model was developed to replicate the geometrical specifications of the laboratory experiment and to analyze the samples under various fracture mode combinations. Figure 9 illustrates the stress distribution within this model under the combined fracture modes. The ratio a/R is set at 0.26, with additional parameters ranging from 0.31 to 0.95 for S1/R and from 0.04 to 0.95 for S2/R, aligning with laboratory specifications. The shape factor for a semicircular disc under the first mode is calculated using Equations (8) and (9). For combined fracture modes, the shape factor is determined based on the support positions, as shown in Figure 10.
K I = Y 1 P π a 2 R B
Y 1 = 1.297 + 9.516 ( S R ) ( 0.47 + 16.457 ( S R ) ) ( a R ) + ( 1.071 + 34.401 ( S R ) ) ( a R ) 2 = 4.49

4.2. Production of Mesh Fragmentation

In mesoscale numerical modeling, the mesh fragmentation technique was utilized [83,84,85]. This method enables the visualization of the initiation and growth of cracks between elements. Initially, the numerical model is constructed using continuous elements. Subsequently, using the coordinates of each node and the connections between elements and nodes, a discrete model is generated. In this discrete model, both contact and target elements are employed, which exhibit cohesive zone model (CZM) behavior. Figure 11 provides a general two-dimensional and three-dimensional overview of the mesh fragmentation approach using contact elements.
For instance, in the modeling of fiber-reinforced concrete, aggregates and fibers are first distributed within the cement–mortar matrix. Then, using the mesh fragmentation technique, contact elements are strategically placed between all elements based on the mechanical properties of each component: matrix–matrix interface (MMI), aggregate–matrix interface (ITZ), and fiber–matrix interface (MFI). Figure 12 illustrates a small segment of fiber-reinforced concrete, showing an aggregate, fibers, and their associated contact elements.

4.3. Result of Verification

Building on data from laboratory samples SCB1 and SCB2, numerical models were analyzed at the mesoscale to validate their accuracy. The finite element models with different discretization are shown in Figure 13. The numerical models incorporated the aggregate distribution from the SCB1 sample and dispersed plastic fibers in a normal distribution within the mortar matrix, as depicted in Figure 14. This ensures that the model dimensions and aggregate sizes accurately reflect those of the experimental specimens. The material properties employed in these finite element simulations are as follows: the modulus of elasticity (E) for aggregates is set at 42 GPa, for the cement matrix at 25 GPa, and for plastic fibers at 5.3 GPa. The Poisson’s ratio (υ) is 0.2 for aggregates, 0.21 for the cement matrix, and 0.36 for fibers. The compressive strength (f’c) of aggregates is 93 MPa, and for the cement matrix, it is 25.6 MPa. The tensile strength (ft) of aggregates is measured at 11 MPa and for the cement matrix at 3.25 MPa, while the yield strength (fy) for fibers is set at 310 MPa. In terms of the cohesive elements within the model, the stiffness (Kn, Kt) for the matrix–matrix interface (MMI), the aggregate–mortar interface (ITZ), and the mortar–fiber interface (MFI) is uniformly 106. The tensile strength (ft) for the MMI is 3.16 MPa, and for the ITZ, it is 1.58 MPa. The fracture energy (GF) for the MMI is 0.04 N/mm, and for the ITZ, it is 0.02 N/mm. Additionally, the parameters for bond behavior in plastic fibers include an alpha coefficient (α) of 0.4, ultimate bond stress (τu) of 14.8 MPa, and residual bond stress (τf) of 5.91 MPa, with the distances between ribs (S1, S2, S3) set at 1 mm, 3 mm, and 5 mm, respectively. For modeling, the Willam–Warnke and Von Mises yield surfaces were used to represent the nonlinear behavior of cement mortar, aggregates, and plastic fibers, respectively. Furthermore, the interactions between cement mortar and aggregates, as well as fibers and cement mortar, were simulated using CZM as detailed in Section 3. The mechanical properties were considered in FEM according to Table 3. This comprehensive approach guarantees that the numerical models accurately capture the complex material behaviors and their interactions, providing a robust framework for understanding the mechanics of fiber-reinforced concrete.
In addition to validating the numerical models with experimental data, a comprehensive sensitivity analysis was conducted to assess the impact of mesh size on the accuracy of the simulation results (Figure 13). Various mesh sizes were tested to ensure that the numerical models accurately captured the intricate interactions between the cement mortar, aggregates, and MSPF. The sensitivity analysis revealed that finer mesh sizes provided more detailed stress and displacement distributions, particularly around the fiber–matrix interface and the interfacial transition zone (ITZ). However, an optimal balance between computational efficiency and accuracy was achieved with a mesh size that adequately resolved the critical regions without excessively increasing the computational load. This rigorous sensitivity analysis confirmed that the chosen mesh size was appropriate for accurately simulating the fracture behavior of fiber-reinforced concrete, thereby enhancing the reliability of the numerical findings. To investigate the sensitivity analysis of the numerical models, the SCB2 numerical model was examined with three different element sizes. The element sizes in the different models were considered to be 1 mm, 2 mm, and 4 mm. As we know, damage variables increase gradually based on the amount of energy dissipated for the various damage modes. To achieve an objective response, the dissipated energy for each damage mode is regularized as follows: gv = gc/Le, where gv is the energy dissipated per unit volume, gc is the energy dissipated per unit area (fracture toughness), and Le is the characteristic length of the element. The characteristic length Le is calculated from the element area A via the following expressions:
L e = 1 . 12 A ,   for   a   square   element 1 . 52 A ,   for   a   triangular   element Δ  
With this characteristic length, the constitutive relation is converted from a stress–strain to a stress–displacement relation.
As shown in Figure 13, the numerical model with an average mesh size of 1 mm demonstrates higher accuracy. Furthermore, the initiation of the crack at the notch in the numerical model with a 1 mm mesh size closely matches the experimental results. In contrast, in the numerical models with an average mesh size of 2 mm, the crack initiates from the opposite side compared to the experimental sample. In the numerical model with an average mesh size of 4 mm, the crack initiates and propagates from almost all sections of the notch.
The numerical simulations demonstrated strong concordance with the outcomes of laboratory tests, especially in terms of the force–CMOD curves and the patterns of crack propagation. For instance, in the SCB1 numerical model, the initiation of cracks closely replicated that of the laboratory sample, commencing from the left side of the notch, advancing towards the ITZ, and subsequently progressing towards the load application point. In a similar vein, the SCB2 model showed that crack propagation was significantly influenced by the aggregates’ presence, which directed the crack towards the ITZ before moving towards the load application point. Furthermore, the models were designed to include a normal distribution of the concrete fibers, as depicted in Figure 14. This figure illustrates the meticulous meshing of the model and the deliberate placement of fibers and aggregates within it. Particularly, in the MSF 1.5% FVF model where a single fiber is positioned to the left of the notch, the initiation of the crack occurs on the right side and moves vertically towards the load application point, spreading beneath it. The corresponding laboratory sample, also depicted in Figure 14, is shown to split into two parts due to the crack extending beneath the load application point. The integration of fibers plays a crucial role in modifying the material’s behavior; specific fibers display areas of stress concentration which significantly boost the concrete’s fracture energy by effectively decelerating the growth of cracks. This strategic incorporation of fibers into the concrete matrix not only strengthens the material but also enhances its durability by mitigating the rapid progression of fractures.

5. Parametric Study

5.1. Effect of MSPF on the Behavior of Concrete in Mode I Fracture

To investigate how fibers affect the mechanical properties of concrete in mode I fractures, a study involved creating fifty numerical models with fiber contents ranging between 0.5% and 4% FVF. Within each fiber content category, ten distinct models were generated to explore various fiber and aggregate configurations. The outcomes, displayed as force–CMOD curves, can be seen in Figure 15. These curves reveal a clear pattern: as the percentage of fibers in the concrete increases, both its tensile strength and fracture energy also rise. Moreover, at higher fiber contents, the influence of the particular arrangement of fibers and aggregates on the samples’ properties diminishes. This reduced dependency is attributed to the presence of multiple fibers in the crack growth path, which effectively hinders crack propagation and boosts fracture energy. However, it is important to note that higher fiber percentages can reduce the interfacial binding strength between the fibers and the cement mortar. This weakening of the bond may lead to premature failure of the specimens, especially in models with 3% and 4% FVF, as demonstrated in the load–CMOD outcomes shown in Figure 15.
Figure 16 illustrates that the integration of fibers into concrete significantly enhances its mechanical properties, notably tensile strength and ductility. For example, the addition of a minimal 0.5% FVF resulted in a notable 4.17% increase in tensile strength, demonstrating the effective role of fibers in reinforcing concrete. Remarkably, a higher concentration, such as 4% FVF, produced an even more impressive increase, boosting tensile strength by 204.5%. This indicates a clear, though nonlinear, relationship between fiber content and mechanical enhancement. It is essential to recognize that while the fracture energy improves with fiber content, it tends to plateau at around 4% FVF concentration. This plateau likely results from the fibers compromising the interfacial bond strength between themselves and the cement mortar as their concentration increases. This reduction in bond strength can potentially lead to earlier structural failure of the concrete samples under stress, emphasizing the need for careful optimization of fiber content in concrete formulations to avoid diminishing returns.
For instance, Figure 17 presents a numerical example with a 5% FVF. As shown in Figure 17e, the tensile strength increases by approximately 234% (point A). However, stress will be increased in fibers due to high fiber congestion, and the concrete sample will experience a sudden failure (point B). Increasing the fibers beyond 4% FVF leads to the behavior of the sample becoming brittle. Moreover, reputable standards limit the maximum FVF to 0.5–2% [86,87,88], as exceeding this limit not only induces brittle behavior in the concrete but also creates challenges during the concrete mixing process.

5.2. Effect of Fibers on the Behavior of Concrete in Mode I and Mode II Fractures

To investigate the effect of fibers on concrete behavior across various fracture modes, 60 numerical models were analyzed for each fiber percentage, from 0.5% to 4% FVF, totaling 300 models. These models varied parameters such as S1/R and S2/R from 0.95 to 0.31. Figure 18 displays several examples that illustrate crack initiation and stress distribution in models with 1.5% FVF under mixed-mode fracture. These examples show that as cracks propagate, the fibers within the plastic zone prevent further crack growth and enhance energy absorption. This behavior underscores the significant role of fibers in bolstering concrete’s response to combined fracture modes.

5.3. Investigation of Fracture Parameters under a Combination of Fracture Modes

Table 4 details the Me parameter, a dimensionless value created to assess the effects of various fracture modes in scenarios involving mixed-mode fractures. The value of Me varies from 1, which denotes a pure mode I fracture scenario, down to 0, which indicates a pure mode II scenario. The specific mathematical formula used to calculate Me is provided in Equation (11). This parameter is instrumental in gauging the relative influence of mode I and mode II fractures, thereby offering essential insights into how materials behave under mixed-mode conditions. Understanding this balance helps in predicting and analyzing the structural integrity of materials when subjected to different types of stresses.
M e = 2 π tan 1 ( K I K I I )
To determine the stress intensity factors, refer to the fracture load and geometry factor values listed in Table 4 and incorporate them into Equation (12). This table delineates the critical stress intensity factors (KIF and KIIF) across various combinations of mode mixities and jute fiber concentrations. In this framework, the parameter Keff, known as the effective stress intensity factor, quantifies the overall fracture toughness for scenarios involving mixed-mode I/II. Keff is pivotal in assessing the collective effects of various fracture modes, thereby providing a comprehensive analysis of the material’s response in mixed-mode conditions. This parameter, summarized in Table 4, enhances the coherence of the results, enabling a holistic understanding of the material behaviors under diverse stress conditions.
K e f f = K I F 2 + K I I F 2  
Figure 19a depicts how the effective stress intensity factor (Keff) of fiber-reinforced concrete correlates with the loading mode (Me) at varying fiber concentrations. The graph demonstrates that Keff reaches its lowest value under pure mode II, where Me approaches zero. This pattern indicates that shear deformation significantly compromises the material’s ability to withstand mixed-mode fractures. Additionally, the integration of MSPF into the concrete markedly improves its mixed-mode fracture toughness, as evidenced in Figure 19. A more detailed analysis in Figure 19b shows the relationship between relative mixed-mode fracture toughness (ρeff) and Me, as detailed in Equation (13). This visualization emphasizes the critical role of fibers in enhancing the concrete’s resilience against mixed-mode fractures, thereby significantly boosting its durability and performance across diverse stress scenarios.
ρ e f f = K e f f ( w i t h   f i b e r ) K e f f ( w i t h o u t   f i b e r )  
Figure 19b highlights that the inclusion of MSPF in concrete can boost its mixed-mode fracture toughness by an impressive 400% (ρeff = 1 signifies no fiber content). This enhancement is particularly significant in conditions that involve mixed-mode I/II scenarios (0.5 < Me < 0.9) compared to pure mode I or II conditions. It has also been noted that adding more than 4% FVF does not yield further significant improvements in mixed-mode fracture toughness, indicating that the ideal fiber concentration is around 4% FVF. Higher concentrations of fibers are not recommended as they may lead to uneven material distribution, which could compromise the structural integrity of the concrete. Furthermore, Figure 19c elaborates on this concept through a fracture locus graph, which displays the mixed-mode fracture toughness across different fiber contents. Each curve on the graph delineates a safety zone defined by the vertical and horizontal axes. For example, the curve corresponding to Me = 0.7 emphasizes how the critical stress intensity factors (KIf and KIIf) increase with the applied load until they meet the fracture curve. This visual representation highlights the advantageous impacts of MSPF, demonstrating that an increase in fiber content leads to a broader safety zone, thereby enhancing the concrete’s fracture resistance.
The outcomes for mixed-mode situations are typically presented in a standardized manner, comparing (KII) to (KIIC) with (KI) to (KIC). The criteria for fracture in such cases can be expressed as follows, based on research (Equation (14)):
( K I I K I I C ) p + ( K I K I C ) q = 1
In the specified equation, KI and KII represent the stress intensity factors for mode I and mode II at the fracture point, respectively, while KIC and KIIC denote the critical fracture toughness values for these pure modes. The parameters p and q within the equation are crucial in determining the fracture behavior. This equation can be further interpreted in terms of fracture energies, where KI is calculated from (E*×GI)0.5 and KII from (E*×GII)0.5. Here, E* is equivalent to E under plane stress conditions and E/(1 − ν) under plane strain conditions. The extensive research provides numerical insights for mixed-mode scenarios, blending modes I and II, as depicted in Figure 20. The figure also illustrates that the power law criterion, to match the numerical fracture data effectively, specifies values of p = 1 and q = 3. These values align with the observed behavior in the numerical simulations for mixed-mode fractures, affirming the model’s consistency with empirical data.

6. Conclusions

This study investigated the effects of macrosynthetic plastic fibers on the mechanical properties and behavior of concrete through numerical simulations and experimental observations. The research focused on mixed-mode fracture conditions and the role of the mode mixity (Me) parameter, with Me values ranging from 1 (pure mode I) to 0 (pure mode II). Key findings are as follows:
  • Effective stress intensity factor (Keff) was crucial for understanding the material’s response to mixed-mode fractures.
  • As Me approaches zero and shear deformation becomes dominant, the resistance to mixed-mode fractures decreases.
  • Adding macrosynthetic fibers significantly boosts mixed-mode fracture toughness, especially in mixed-mode I/II conditions (0.5 < Me < 0.9).
  • There was an approximately 400% increase in mixed-mode fracture toughness with fiber content (ρeff = 1 signifies no fiber content).
  • Optimal fiber content for enhancing concrete durability is around 4%; beyond this percentage, the increase in energy absorption plateaus due to diminished bond strength between fibers and cement mortar.
  • A standardized method for presenting mixed-mode outcomes was developed, comparing stress intensity factors (KII and KIIC) with (KI and KIC).
  • A power law criterion was established with specific values for p and q that align with numerical data on fractures.

7. Future Research

  • Further exploration of the optimal distribution and fiber content for specific construction scenarios.
  • Deeper understanding of the complex interactions between fibers and the cement matrix in concrete.

Author Contributions

Conceptualization, A.P., M.P., I.S. and S.P.; methodology, S.G., H.A., I.S. and S.P.; software, A.P., M.P., N.K. and S.G.; validation, M.P. and N.K.; formal analysis, A.P., N.K., S.G. and H.A.; investigation, N.K, S.G. H.A. and I.S.; resources, M.P., N.K., S.G., H.A. and S.P.; data curation, A.P., N.K., M.P. and S.G.; writing—original draft preparation. A.P. and N.K.; writing—review and editing, A.P. M.P. and S.P. visualization, N.K., S.G., H.A. and S.P.; supervision, A.P., M.P., I.S. and S.P.; project administration, A.P. and M.P. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

The authors declare no competing interests.

Abbreviations

The following abbreviations are used in this manuscript:
SCBSemicircular bend
CMODCrack mouth opening displacement
XFEMExtended finite element method
EFEMEnergy finite element method
ITZInterfacial transition zone
CZMCohesive zone model
MFIFiber–matrix interface
MMI
SF
Matrix–matrix interface
Synthetic fibers
FVFFiber volume fraction
MSPFMacrosynthetic plastic fibers
Notation
dmaxMaximum aggregate size
d50Median aggregate size
EModulus of elasticity
S1Slip at first crack
S2Slip at ultimate strength
S3Distance between ribs
αAlpha coefficient
τuUltimate bond stress
τfFinal bond stress
a/RRatio of crack length to radius
S1/RSlip at first crack to radius ratio
S2/RSlip at ultimate strength to radius ratio
EhYoung’s modulus of homogeneous concrete
EcmYoung’s modulus of cement matrix
VaVolume of aggregate
EaYoung’s modulus of aggregate grains
GcmShear modulus of cement matrix
υPoisson’s ratio
f’cCompressive strength
ftTensile strength
fyYield strength
KeffEffective stress intensity factor
ΡeffMixed-mode fracture toughness
KIStress intensity factor for mode I
KIIStress intensity factor for mode II
KICCritical fracture toughness for mode I
KIICCritical fracture toughness for mode II
p, qParameters in fracture criterion equation
GIFracture energy for mode I
GIIFracture energy for mode II
MeMode mixity parameter
gvEnergy dissipated per unit volume
gcEnergy dissipated per unit area (fracture toughness)
LeCharacteristic length of the element
AElement area

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Figure 1. (a) Concrete beams subjected to three-point bending: geometry and boundary conditions. (b) Grading curve aggregates.
Figure 1. (a) Concrete beams subjected to three-point bending: geometry and boundary conditions. (b) Grading curve aggregates.
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Figure 2. (a) Force–displacement, (b) force–CMOD, and (c) patterns of SCB1 and SCB2 cracks.
Figure 2. (a) Force–displacement, (b) force–CMOD, and (c) patterns of SCB1 and SCB2 cracks.
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Figure 3. (a) Crack pattern MSF 1.5% FVF. (b) Force–CMOD diagram of experimental samples.
Figure 3. (a) Crack pattern MSF 1.5% FVF. (b) Force–CMOD diagram of experimental samples.
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Figure 4. The Menetrey–Willam yield surface model with a cap model [77].
Figure 4. The Menetrey–Willam yield surface model with a cap model [77].
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Figure 5. Schematic of interface elements.
Figure 5. Schematic of interface elements.
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Figure 6. Sliding curve and contact gap versus normal and shear contact stress.
Figure 6. Sliding curve and contact gap versus normal and shear contact stress.
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Figure 7. (a) Stress of bond–slip as proposed by the CEB fib model code; (b) numerical model of pull-out test conducted in two distinct steps.
Figure 7. (a) Stress of bond–slip as proposed by the CEB fib model code; (b) numerical model of pull-out test conducted in two distinct steps.
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Figure 8. Nodes used for the approximate crack-tip displacements.
Figure 8. Nodes used for the approximate crack-tip displacements.
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Figure 9. The stress distribution in the SCB sample under a combination of fracture modes.
Figure 9. The stress distribution in the SCB sample under a combination of fracture modes.
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Figure 10. Variations of modes I and II geometry factor.
Figure 10. Variations of modes I and II geometry factor.
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Figure 11. The process of mesh discretization: (a) 3D and (b) 2D.
Figure 11. The process of mesh discretization: (a) 3D and (b) 2D.
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Figure 12. The trend of discretization and the inclusion of contact elements between concrete components.
Figure 12. The trend of discretization and the inclusion of contact elements between concrete components.
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Figure 13. (a) Numerical model with 1 mm average mesh size, (b) numerical model with 2 mm average mesh size, (c) numerical model 4 mm average mesh size, and (d) force–COMD graph of numerical models with different average mesh sizes.
Figure 13. (a) Numerical model with 1 mm average mesh size, (b) numerical model with 2 mm average mesh size, (c) numerical model 4 mm average mesh size, and (d) force–COMD graph of numerical models with different average mesh sizes.
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Figure 14. (a) SCB1 meshing, (b) SCB2 meshing, (c) SCB1 cracking pattern, (d) SCB1 cracking pattern, (e) MSF meshing, (f) MSF 1.5% FVF cracks pattern, and (g) comparative analysis of force–COMD among numerical and experimental outcomes.
Figure 14. (a) SCB1 meshing, (b) SCB2 meshing, (c) SCB1 cracking pattern, (d) SCB1 cracking pattern, (e) MSF meshing, (f) MSF 1.5% FVF cracks pattern, and (g) comparative analysis of force–COMD among numerical and experimental outcomes.
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Figure 15. Force–COMD diagram of each plastic fiber percentage.
Figure 15. Force–COMD diagram of each plastic fiber percentage.
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Figure 16. (a) Average force–COMD by percentage of steel fibers; (b) correlation between fracture energy and fiber percentage.
Figure 16. (a) Average force–COMD by percentage of steel fibers; (b) correlation between fracture energy and fiber percentage.
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Figure 17. (a) View of the numerical model and meshing with 5% FVF, (b) step of crack initiation from the notch, (c) step just before collapse, (d) step of collapse, and (e) force–CMOD diagram.
Figure 17. (a) View of the numerical model and meshing with 5% FVF, (b) step of crack initiation from the notch, (c) step just before collapse, (d) step of collapse, and (e) force–CMOD diagram.
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Figure 18. Analysis of mixed-mode fracture: plastic strain contours and Von Mises stresses in numerical models.
Figure 18. Analysis of mixed-mode fracture: plastic strain contours and Von Mises stresses in numerical models.
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Figure 19. (a) Variation of Keff with Me across various plastic fiber concentrations. (b) Comparative ρeff as a function of Me. (c) Fracture behavior chart for concrete reinforced with plastic fibers.
Figure 19. (a) Variation of Keff with Me across various plastic fiber concentrations. (b) Comparative ρeff as a function of Me. (c) Fracture behavior chart for concrete reinforced with plastic fibers.
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Figure 20. Mixed-mode fracture envelope.
Figure 20. Mixed-mode fracture envelope.
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Table 1. Mechanical properties of each component.
Table 1. Mechanical properties of each component.
ComponentModulus of ElasticityPoisson’s Ratio Compressive StrengthMean Tensile Strength
Concrete30.60 Gpa0.2335.10 Mpa3.16 Mpa
Mortar25.00 Gpa0.2125.60 Mpa3.25 Mpa
Aggregate42.00 Gpa0.2093.00 Mpa11.00 Mpa
Fiber5.30 Gpa0.36-310.00 Mpa
Table 2. Number of laboratory samples.
Table 2. Number of laboratory samples.
SampleNumber of Samples Constructed
Cylindrical 3
SCB (without fibers)2
SCB (with 1.5% FVF)2
Table 3. Material properties used in FEM (Ansys software, V24).
Table 3. Material properties used in FEM (Ansys software, V24).
Solid Elements
AggregateCement MatrixFibers
E (GPa)42.0025.005.30
υ0.200.210.36
f’c (MPa)93.0025.60-
ft (MPa)11.003.25-
fy (MPa)--310
Cohesive Element
MFIITZMMI
Kn, Kt106106106
ft (MPa)3.161.58-
GF (N/mm)0.040.02-
S1--1 mm
S2 3 mm
S3 5 mm
α 0.4
τu (MPa) 14.80
τf (MPa) 5.91
Table 4. Assessing the mixed-mode fracture resistance of concrete with varied plastic FVF.
Table 4. Assessing the mixed-mode fracture resistance of concrete with varied plastic FVF.
0%0.5%1.5%2.5%3%4%
MeKIFKIIFKeffKIFKIIFKeffKIFKIIFKeffKIFKIIFKeffKIFKIIFKeffKIFKIIFKeff
149049760761620171225022525602722820300
0.94674772117215525157218352212473925027143274
0.840134265216813945146191622012217223224881261
0.731163552265810352116145741631728819319398217
0.62417303827467454911047612912389152139101172
0.518182628283954547676761079090127102102144
0.4131823202835405467557694669111274102126
0.391821142831285562397686479110252102115
0.2618199283018555825778130919533102107
0.131919428289555612777815929316102104
0.0511919228284555567777791928102102
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Permanoon, A.; Pouraminian, M.; Khorami, N.; GanjiMorad, S.; Azarkhosh, H.; Sadrinejad, I.; Pourbakhshian, S. Improving Mixed-Mode Fracture Properties of Concrete Reinforced with Macrosynthetic Plastic Fibers: An Experimental and Numerical Investigation. Buildings 2024, 14, 2543. https://doi.org/10.3390/buildings14082543

AMA Style

Permanoon A, Pouraminian M, Khorami N, GanjiMorad S, Azarkhosh H, Sadrinejad I, Pourbakhshian S. Improving Mixed-Mode Fracture Properties of Concrete Reinforced with Macrosynthetic Plastic Fibers: An Experimental and Numerical Investigation. Buildings. 2024; 14(8):2543. https://doi.org/10.3390/buildings14082543

Chicago/Turabian Style

Permanoon, Ali, Majid Pouraminian, Nima Khorami, Sina GanjiMorad, Hojatallah Azarkhosh, Iman Sadrinejad, and Somayyeh Pourbakhshian. 2024. "Improving Mixed-Mode Fracture Properties of Concrete Reinforced with Macrosynthetic Plastic Fibers: An Experimental and Numerical Investigation" Buildings 14, no. 8: 2543. https://doi.org/10.3390/buildings14082543

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