Fatigue Resistance of Concrete Revisited: Wide Range Data from Low-cycle to High-cycle and Influence of Aggregate on Fatigue Lifetimes

Post-print/AAM November 15, 2025 Fatigue Resistance of Concrete Revisited: Wide Range Data from Lowcycle to High-cycle and Influence of Aggregate on Fatigue Lifetimes Authors’ Names and Affiliations Petr Miarkaa,b*, José D. Ríosc , Lucie Malíkováa,b, Vlastimil Bílekd aInstitute of Physics of Materials, Czech Academy of Sciences, Žižkova 22, 616 00 Brno, Czech Republic bBrno University of Technology, Faculty of Civil Engineering, Veveří 331/95, 602 00 Brno, Czech Republic cDepartment of Continuum Mechanics and Structural Analysis, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Seville, Spain dVŠB — Technical University of Ostrava, Faculty of Civil Engineering, Ludvíka Podéště 1875/17, 708 00 Ostrava-Poruba, Czech Republic *Corresponding Author: Petr Miarka (email: [email protected]z). Tel: +420 532 290 430 Publishing information: Miarka P., Ríos J.,D., Malíková L., Bílek V., Fatigue Resistance of Concrete Revisited: Wide Range Data from Low-cycle to High-cycle and Influence of Aggregate on Fatigue Lifetimes, Cement and Concrete Composites, 2026, 156, 106394, ISSN 0958-9465. DOI: https://doi.org/10.1016/j.cemconcomp.2025.106394. Abstract This paper presents the outcome of an experimental study focused on the investigation of the fatigue fracture behaviour of high-performance concrete (HPC) with granite aggregates under three-point bending. Notched beam specimens were tested under static, low-cycle, and high-cycle regimes to obtain load-CMOD responses, S-N curves, crack propagation and damage accumulation. Key parameters were derived from CMOD-controlled tests, Paris’ law fitting, and a compliance-based crack growth approach. Post-test microscopy was used to characterise the fracture process zone (FPZ). The results reveal a cohesive-like law governing the cyclic damage evolution, independent of the initial notch depth. Aggregate bridging and confinement effects from large granite aggregates (Dmax = 22 mm) significantly influenced crack trajectories, delaying failure and enhancing fatigue life. Fatigue crack growth analysis identified distinct propagation phases and multiple apparent threshold stress intensity factors (KI,th), closely linked to the material’s meso-structure. The findings suggest that despite accumulated damage, final failure occurs when local conditions reach the fracture toughness of the unfatigued material. This work provides new insights into the role of aggregate type in fatigue performance and proposes a robust methodology for correlating static and cyclic fracture parameters in conventional HPC. Keywords: Fatigue; High-Performance Concrete; S-N; Paris’ law; Fracture; Damage Copyright: ©2025. This version is made available under the CC-BY 4.0 license and all right remains to authors according to CZ/EU legislation. https://creativecommons.org/licenses/by/4.0/ Highlights • Fracture and fatigue properties of HPC concrete. • Low-cycle and high-cycle fatigue tests. • Fatigue crack growth rate based on cyclic creep curves. • Multiple apparent threshold SIF values.. 2 • Aggregate bridging effect enhances fatigue life. Abbreviations CM cementitious materials OPC ordinary Portland cement FCGR fatigue crack growth rate FPZ fracture process zone CMOD crack mouth opening displacement CTOD crack tip opening displacement CMODIR irreversible crack mouth opening displacement CTODc critical crack tip opening displacement SIF stress intensity factor ULS ultimate limit state SLS service limit state 3PBT thee-point bending test LVDT linear variable differential transformer LC load case SCM supplementary cementitious materials Nomenclature W specimen’s height [mm] T specimen’s thickness [mm] L specimen’s length [mm] S span between supports [mm] Alig ligament area [mm2] a0 initial notch length [mm] a crack length [mm] E0 elastic modulus [GPa] P force [kN]  relative crack length [-]   stress intensity factor [MPam1/2] KIC fracture toughness [MPam1/2] wc crack width [mm] wu ultimate crack width [mm] Fmax maximum force [kN] fc compressive strength [MPa] ft flexural strength [MPa] r volume density [kg/m3] Gf fracture energy [Nm] C, m Paris’ law coefficients A, B Wöhler’s curve coefficients da crack increment [mm] dN load cycle increment [cycle] YCMOD CMOD shape function [-] 1. Introduction The fracture behaviour of conventional concrete, particularly under cyclic or repetitive loading, is of fundamental importance for ensuring durability, sustainability and structural safety across multiple sectors of civil engineering. From critical infrastructure such as bridges, airports and dams to everyday structures like buildings and pavements, concrete is frequently exposed to variable loading, fatigue and environmental degradation resulting from vehicle traffic, industrial machinery vibration, seismic forces, wind loading and other influences [1,2]. Modern society demands structures that are more resilient, costeffective and long-lasting; thus, achieving a comprehensive understanding of crack initiation and fracture development under cyclical loading, and how to mitigate these phenomena, remains a key and unresolved challenge for the scientific and technological community [3,4]. Fatigue failure in concrete caused by cyclic loading is often unexpected, as it manifests as cumulative damage that is frequently overlooked. This is largely due to the fact that the primary parameters considered in structural design are the compressive and flexural strengths of the material. In such cases, fatigue is treated as a reduction in stress capacity at critical locations induced by external loading, and the service life of the structure is linked to the total number of load cycles, N [5,6]. This approach is directly related to the Ultimate Limit State (ULS), as defined by design codes such as the Eurocode [7] and ACI [8], and to the reduction in stress levels expressed as a percentage of the static strength in 3 compression or flexure, respectively. The stress–cycle relationship is commonly referred to as the S–N curve or Wöhler curve [9]. Structural design codes for concrete typically provide engineers with S–N curves to ensure that the structure can withstand the required service life. An example of an S–N curve adopted in Model Code 2010 [10,11] is shown in Figure 1. Figure 1: Model code MC2010 fatigue design recommendations. The S–N curves presented in Figure 1 suggest overly optimistic expectations for structural service life. Even under reduced stress levels, the anticipated fatigue life ranges between 10 × 10⁸ and 10 × 10¹² load cycles, whereas Lee and Barr [12] propose a more realistic fatigue life of approximately 2 × 10⁸ cycles for load-bearing structures. Another critical limitation is that the recommendation requires a loading frequency of 0.1 cycles/min, equivalent to 0.01667 Hz. This loading condition makes it virtually impossible to replicate the expected fatigue lives in laboratory testing environments. The exclusive use of a stress-based design approach for fatigue assessment also poses significant challenges when evaluating the Serviceability Limit State (SLS), which primarily focuses on structural deflection or the calculation of crack width, wc. Structural deflection is directly related to stiffness, whereas the crack width wc reflects the allowable damage within the structure. The SLS is particularly critical in prestressed concrete structures, which must demonstrate enhanced durability in aggressive environments. To date, advances in characterising concrete fracture have focused mainly on the development and standardisation of monotonic loading experiments, among which the three-point bending test (3PBT) stands out as the most widely employed method [13–16]. Under both monotonic (static) and cyclic (fatigue) loading regimes, concrete exhibits progressive fracture behaviour marked by the nucleation, growth and coalescence of micro-cracks preceding the eventual formation of a dominant macro-crack. However, considerable distinctions exist between these two types of loading [17]. In monotonic loading, damage evolves gradually via cohesive fracture mechanisms, micro-cracking and aggregate bridging within the fracture process zone (FPZ), yet final failure tends to be abrupt once peak resistance is reached, resulting in a sudden collapse in load-bearing capacity [18]. By contrast, under cyclic loading the damage progression becomes even more gradual and complex, as existing cracks undergo repeated partial or complete opening and closure, producing cumulative degradation over an extended number of cycles before eventual fracture [19–21]. Concrete fatigue has traditionally been a complex phenomenon to study due to its inherently heterogeneous and quasi-brittle nature. Nevertheless, significant progress has been made through the adaptation of theoretical frameworks from fracture mechanics originally developed for metallic materials. In particular, a considerable body of research has employed Linear Elastic Fracture Mechanics (LEFM) to characterise crack propagation in concrete under cyclic loading [22], often utilising models 4 based on the Paris–Erdogan law [5,23–25]. As early as the 1980s, it was demonstrated that the Paris law, which relates the crack growth rate, da/dN, to the range of stress intensity, ΔK, can be applied to concrete provided that certain material-specific modifications are introduced [26]. Among the most notable contributions is the work of Bažant and colleagues [27,28], who incorporated size-dependent effects and an effective fracture toughness contingent on crack length to model stable fatigue crack growth in concretes of varying strengths. More recently, experimental investigations have focused on quantifying the residual load-bearing capacity of concrete under fatigue. A particularly effective means of assessing this progressive loss is through analysis of the remaining fracture energy following various levels of cyclic loading. In this context, Jia et al. [29] conducted an extensive experimental programme using 150 notched beams subjected to three-point bending fatigue. After reaching predetermined cycle counts, the beams were tested under static loading until failure to determine the residual fracture energy, GF,res. Their results showed that GF,res declines sharply during the initial few thousand cycles, falling to a significantly lower fraction of the original value, before entering a slower, quasi-linear degradation phase. Remarkably, despite cumulative deterioration, unstable fatigue fracture was found to occur when the crack reached a critical toughness level very close to the static KIC. This observation suggests that the fatigue fracture threshold does not imply a drastic reduction in fracture parameters, but rather an accumulative weakening of the remaining ligament until critical conditions matching those of the unfatigued material are reached. On another note, the type of coarse aggregate used in the mix plays a crucial role in the fracture performance of concrete [30,31]. Given that aggregate constitutes between 60 % and 80 % of the total volume [32,33], its physical properties—such as hardness, angularity, surface texture and mechanical strength, significantly influence the quality of the interfacial transition zone (ITZ) and the bridging mechanisms within the FPZ [33–35]. Recent studies have shown that concretes produced with crushed hard rock aggregates, such as granite or basalt [36], exhibit markedly higher fracture toughness. Both the critical stress intensity factor, KIc, and the critical crack-tip opening displacement, CTODc, increase significantly compared to concretes made with rounded gravel. In three-point bending tests, enhancements in toughness of around 30 % have been observed with granite and up to 70 % with basalt, when compared to reference concrete [37]. This improvement is attributed to the higher intrinsic strength of these rock types and their irregular geometry, which promotes stronger aggregate, matrix bonding and greater energy dissipation during fracture. It should be noted, however, that most of these studies have been conducted under monotonic loading, and there remains a limited number of investigations exploring such aggregate effects in the context of fatigue. The literature review also reveals that a substantial portion of studies on concrete fatigue have focused on compression conditions [38] or bending [39,40], encompassing a wide range of materials: fibrereinforced concretes [41–43], alkali-activated composites [44], and even concrete made with recycled aggregates [45–47]. Regarding the specific investigation of crack growth, most experimental campaigns focus on loading regimes around 10⁴ cycles [48–50], with notable contributions such as that of Baktheer [50], who provides an extensive database on crack evolution and toughness under cyclic loading using fracture mechanics approaches including numerical modelling of cyclic failure in [51]. The practical applications of these investigations range from large-scale bending tests [52] to real structural components such as railway sleepers [53] or wind turbine foundations [54,55]. Although fatigue does not often cause immediate structural collapse, it can initiate microcracking that, as it develops over time, compromises durability and significantly reduces the service life of the structure. The available literature on how fatigue modifies the residual fracture energy and progressively alters the FPZ remains limited, and a comprehensive understanding is still lacking regarding the influence of key parameters such as load amplitude, number of cycles, and, in particular, the type of coarse aggregate under prolonged cyclic loading [56]. Moreover, another significant gap lies in the absence of a clear and reliable correlation between static parameters, which considerably restricts the accuracy of predictive 5 models used in structural engineering. In addition, research specifically focused on aggregates such as granite under cyclic conditions is scarce, despite the widespread use of these materials in certain regions due to their local availability and favourable mechanical properties. Building upon these observations, the present study aims to fill the identified knowledge gaps and provide new experimental data regarding the behaviour of high-performance concrete with granitic aggregate under cyclic loading, using standardised 3PB tests. In particular, the measurement of static fracture energy, the degradation of residual toughness with the number of cycles, and the development and transformation of the FPZ under different loading conditions are investigated. To this end, notched 3PBT specimens were tested to conduct experimental tests under three regimes: static fracture, lowcycle fatigue, and high-cycle fatigue. Special attention was paid to crack trajectories to identify the crack bridging mechanisms leading to heterogeneous behaviour under cyclic loading. This experimental campaign establishes a clear link between the increasing heterogeneity observed in cyclic tests and the role of aggregate bridging is established. 2. Material and Experimental Procedures This section presents the composition of the high-performance concrete (HPC) mixture and provides a concise overview of the methods used for evaluating both fracture and fatigue behaviour. First, the theoretical background underlying the evaluation of fracture tests and fatigue damage analysis is introduced. The procedures for calculating fracture energy and performing inverse analysis under static conditions are described. Then, the methodology for calculating cyclic damage, including the fatigue crack growth rate (FCGR) approach and the compliance method used to assess the material constants in Paris’s law, is presented in detail. Subsequently, a detailed description of the experimental setup is provided, along with the applied load cases and the geometry of the HPC specimens. Finally, the methodology for post-test damage assessment is briefly outlined. This combination of theoretical and experimental approaches provides a solid foundation for a comprehensive analysis of the complex problem of fatigue failure mechanisms in HPC. 2.1 Mixture Composition Ordinary Portland cement (OPC) CEM I 42.5 R was used as the binder, with a constant dosage of polycarboxylate superplasticizer Glenium 300 (BASF, Germany) to ensure good workability. The water-to-cement ratio (w/c) was 0.3. Three different aggregate fractions were employed: fine sand (0/4 mm), and high-quality granite as coarse aggregates in two fractions, 4/8 mm and 8/22 mm, respectively. The mixture was prepared in a small laboratory mixer with a capacity of 0.5 m3 and directly poured into moulds. After one day, the concrete samples were demoulded, covered with polyethylene (PE) foil to prevent excessive moisture exchange with the environment, and stored in a room maintained at a controlled temperature of 20°C. The composition of the mixture per m3 is presented in Table 1. Table 1: Mix proportions of raw materials for HPC concrete per m3. CEM I 42.5R [kg] Water [kg] Sand 0/4 [kg] Aggregate 4/8 [kg] Aggregate 8/22 [kg] Superplasticizer (Glenium 300) [kg] 450 135 866 290 740 9 Table 1 shows the precise dosage of the mixture components. The two selected fractions of coarse aggregates were used in the mix. Mechanical properties were tested at different ages (1, 28, 91 and 365 days) according to the European standards. A summary of the measured mechanical properties is provided in Table 2. 6 Table 2: Measured mechanical parameters at various age. Age [days] Volume density  [kgm-3] Compressive strength fc [MPa] Flexural strength fc,t [MPa] 1 2418 ± 30 49.9 ± 0.5 - 28 2432 ± 3 91.5 ± 4.3 8.1 ± 0.4 91 2421 ± 16 90.4 ± 4 9.3 ± 1.1 365 2413 ± 18 109.6 ± 1.0 10.2 ± 0.7 Table 2 presents the measured compressive strengths obtained from 100 mm cubes [57] and flexural strengths obtained from beams with a square cross-section of 80 × 80 mm2 and a length of 480 mm [58], tested at different ages. The results show relatively consistent behaviour at later ages. Additional strength measurements at 91 and 365 days provide insight into long-term strength development, which is relevant given the time-consuming nature of fatigue testing and the potential variation in concrete strength over time. The compressive strength at 365 days exhibits a further increase of approximately 20 MPa, reaching 109 MPa, confirming that the mixture meets the criteria for high-performance concrete (HPC). 2.2 Fracture Characterisation Experimental tests were conducted to determine size-independent fracture energy by means of static three-point bending tests on prismatic specimens (specific dimensions are provided in subsection 2.2). Each specimen included an initial notch with a notch-depth-to-effective-height ratio (a0/W) of 0.3. Three repetitions were conducted for each test configuration. During the tests, mid-span deflection was measured using a linear variable differential transformer (LVDT), while the CMOD was recorded using a clip gauge (the experimental procedure is described in further detail in section 2.5). The tests were displacement-controlled via the CMOD signal, enabling accurate capture of the post-peak softening behaviour of the material. 𝐺𝑓=𝑊𝑓 𝐴𝑙𝑖𝑔, (1) a Additionally, an inverse analysis based on the non-linear hinge model was performed to derive the bilinear tension softening diagrams (Figure 2) [59]. The procedure was developed following the methodology proposed by Lennart, which builds upon the well-established framework of the non-linear hinge model originally introduced by Jenq and Shah [60] and subsequently adopted by numerous authors, such as Karihaloo [61], Planas et al. [62], and Roesler [63], for fracture characterization of concrete. The proposed bilinear cohesive laws presented in this study are fully compatible with discrete crack models, allowing for their direct implementation in numerical simulations aimed at accurately reproducing the fracture processes governing the structural behaviour of HPC. Figure 2: Schematic overview of experimental data transformation to dimensionless bilinear cohesive law. 7 The Young’s modulus was determined from three independent three-point bending tests by analysing the corresponding load–CMOD (L-CMOD) curves obtained experimentally. Following the procedure proposed by Jenq and Shah [64], the Young’s modulus was calculated as the slope of the initial linear segment of each load–CMOD response, which represents the elastic behaviour of the specimen prior to crack initiation according to following expression: 𝜎𝑡(𝑤)={𝐸𝜀 𝜎𝑡(𝑤)=𝑓𝑡(1− 𝑤 𝑤𝐶) (2) a The calculation accounted for the geometrical characteristics of the tested beams, including the span, cross-sectional dimensions, and notch depth. By applying this methodology individually to the three experimental load-CMOD curves, three values of Young’s modulus were obtained, reflecting the variability inherent to the material and testing conditions. This experimentally derived modulus was subsequently used as a key input parameter in the fracture analysis. 2.3 Cyclic and Fatigue Damage Evaluation In addition to using the load-CMOD curve for the evaluation of fracture energy Gf, the low-cycle fatigue behaviour of concrete can be assessed by analysing the hysteresis loops generated during cyclic loading and unloading. This approach provides deeper insight into the dissipative mechanisms associated with crack propagation and progressive material degradation. A detailed characterization of the concrete’s flexural fracture response under cyclic loading, in terms of stiffness degradation observed in each load step or hysteresis loop, was performed following the methodology reported in [65,66]. A schematic representation of the recorded hysteresis loops is shown in Figure 3. Figure 3: Low-cycle behaviour and measured characteristics. Figure 3 illustrates the results of a low-cycle fatigue test, represented by a typical hysteresis loop. Each loop enables fracture characterisation through the analysis of unloading stiffness degradation, which is quantified by damage parameter ω, defined as follows: 𝜔=1−𝐸𝑖 𝐸0, (3) a where E0 is the initial elastic stiffness of the material, and Ei is the stiffness corresponding to i-th hysteresis loop of the load-CMOD curve. Additionally, the damage progression within each load cycle can be characterized by the irreversible, crack mouth opening displacement, denoted as CMODIR (see Figure 3). This parameter represents the 8 residual crack opening that does not recover upon loading, indicating that the elastic limit of the material has been exceeded. In concrete, CMODIR value arises primarily from the presence of the FPZ, rough ligament surfaces, and aggregate interlock across the crack. In contrast, for metallic materials, CMODIR is attributed solely to plastic deformation. Unlike static and low-cycle testing, the evaluation of fatigue damage lacks a standardized procedure due to the complex and not fully understood mechanisms of fatigue crack initiation in concrete. For this reasons, the S-N curve remains a practical tool for material characterization, providing a direct link to the engineering fatigue strength. Fatigue damage is evaluated using the following exponential term: 𝜎=𝐴𝑁𝐵 (4) a where N denotes the number of load cycles to fatigue failure, and A, B are material-specific constants typically obtained through least-squares fitting of a power-law function to the experimental data. The parameter A is associated with the static strength of the material, while B reflects the rate of progressive damage accumulation. A more sophisticated approach to fatigue characterization is offered by the framework of linear elastic fracture mechanics (LEFM), which employs the power-law expression known as Paris’ law: 𝑑𝑎 𝑑𝑁=𝐶𝐾𝐼𝑚, (5) a where C and m are the material constants associated with Paris’ law, obtained from experimental data; da/dN is the fatigue crack growth rate (FCGR); and KI is the stress intensity factor (SIF), which can be calculated as follows: 𝐾𝐼=3𝑃𝑆 2𝑇𝑊2√𝜋𝑎0 𝑌𝐼(𝛼), (6) a 𝑌𝐼(𝛼)=1 √𝜋1.99−𝛼(1−𝛼)(2.15−3.93𝛼+2.7𝛼2) (1+2𝛼)(1−𝛼)3/2 , (7) a where P is the applied load; S, T and W are the span, thickness and width of the tested specimen, respectively; a0 is the initial notch length; and YI(α) is the geometry shape function expressed in polynomial form as given in Eq. (7), which depends on the relative crack length α = a0/W. In this study, the geometry shape function was adopted from the handbook by Tada and Paris [67]. A recent study has proposed a link between the S–N curve parameter B = 1/m see [40,68], which offers extended applicability for engineering fatigue assessments. It is worth noting that both S-N and Paris’ curves are influenced by several factors, including the load cycle asymmetry ratio R = Pmin/Pmax, loading frequency f, environmental conditions, boundary effects such as free-edge singularities [69], sequence loading[70–72], loading rate [73–75] and temperature effects [76,77]. 2.4 Compliance Method The presence of a FPZ in concrete specimens during fracture testing limits the applicability of optical crack monitoring under cyclic loading, as multiple cracks tend to develop ahead of the main crack tip. Therefore, the compliance method is employed to estimate the stress intensity factor (SIF) required for the application of Paris’ law. This method relies on the progressive degradation of stiffness during cyclic loading by correlating the measured CMOD to the corresponding crack length a. The compliance, denoted as CMODc can be calculated as follows: CMOD𝑐=4𝜎𝑎 𝐸𝑌𝐶𝑀𝑂𝐷(𝛼), (8) a 9 where  is the applied nominal stress, a is the crack length, and YCMOD is the geometry-dependent shape function, which varies with the relative crack length a/W. In this study, the function YCMOD was again adopted from the handbook by Tada and Paris [67], and is defined as follows: 𝑌𝐶𝑀𝑂𝐷(𝛼)=0.76−2.28(𝛼)+3.87(𝛼)2−2.04(𝛼)3+0.66 (1−𝛼)1 2. (9) a Equations (8) and (9) follow a methodology similar to that used for the calculation of stress intensity factors (SIFs), making them suitable for crack length estimation. The approximated crack length a at a given load cycle is subsequently used in Equation (6) to calculate the SIF required for the evaluation of Paris’ law parameters. The crack length estimation process is illustrated in Figure 4. Figure 4: Workflow of compliance method for assessment of Paris’ law from cyclic creep curves CMOD-N curves. A similar methodological approach, as depicted in Figure 4, was previously employed by Bažant [48] and Le [68], both of whom focused on the size effect. However, their analyses were limited to test series involving up to ten thousand load cycles. 2.5 Experimental details This section provides a detailed description of the specimen geometry and test setup used for the comprehensive fatigue assessment. The load cases employed to achieve the intended objectives and their influence on the experimental outcomes are discussed. Finally, the methodology used for damage assessment via optical microscopy is presented. 2.5.1 Specimen Geometry In this experimental study, a geometry widely recognised in the research community was selected: the three-point bending test (3PBT) configuration. This setup is suitable for both static and cyclic loading, 16 Figure 12 presents a non-linear relationship between the irreversible CMODIR and versus the total CMOD, with the size of the inelastic component increasing with notch depth a0. This difference in CMODIR is attributed to the varying extent of the FPZ, which is influenced by edge effects [78] that become boundary [78] is more pronounced in specimens with deeper notches. In shallow-notched (a0/W = 0.15), the FPZ can expand more freely due to the larger remaining ligament area Alig, allowing both elastic and inelastic regions to form distinctly. In contrast, for deeper notches (a0/W = 0.3), the reduced ligament restricts FPZ development, resulting in a less distinguishable separation between elastic and inelastic behaviour. Interestingly, when the coordinate system is changed from CMODIR to CMOD-CMODIR, a previously unrecognised trend emerges. This representation reveals a cohesive-like law that governs cyclic inelastic damage evolution. The observed relationship suggests that geometric confinement plays a key role in regulating damage accumulation during repeated loading. Figure 13 presents the experimental data and the fitted power function, which supports the existence of this phenomenon and provides a reliable model for characterising inelastic crack growth under cyclic loading. Figure 13: Relationship between CMODIR and CMOD for both notch depths, with fitted power-law functions in valid interval. Figure 13 shows the relationship between CMODIR and CMOD for both relative notch depths a0/W = 0.15 and a0/W = 0.3, along with power-law functions fitted using least-squares regression CMOD = k‧CMODIRn . While the exponent n remains the same for both geometries, the coefficient k varies depending on the notch depth. Specifically, lower a0/W ratios (i.e. longer remaining ligaments) result in smaller values of k, indicating that a given CMODIR corresponds to a lower total CMOD, or in other words, a smaller reversible component of the crack opening. For specimens with short ligaments (e.g. a0/W = 0.3), the FPZ interacts with the free edge, producing a geometric confinement effect that limits the energy dissipation capacity in the FPZ. Consequently, part of the damage that would normally contribute to the irreversible component CMODIR, cannot fully develop. This effect is quantitatively reflected in the exponent of the fitted power law, which takes the same value of n = 0.582 for both geometries. As previously discussed, CMODIR is associated with the non-closing portion of the crack, primarily governed by aggregate interlock and the detachment of coarse particles from the cement matrix. In this context, CMODIR is related to the FPZ, which enables the formulation of power-law relationships that are independent of the initial notch tip length. The general expression for this relationship is given by: 17 𝐶𝑀𝑂𝐷=(0.815−0.1267𝛼)∙𝐶𝑀𝑂𝐷𝐼𝑅 0.58, (10) a where α is the relative notch ratio a0/W. The relationship presented in Eq. (10) indicates that the lowcycle fatigue behaviour is governed by a cohesive law, similarly to what is observed in static fracture tests. This observation may open new avenues in the fracture assessment of concrete, potentially enabling the development of alternative constitutive models. Although Eq. (10) may be valid only for this specific type of concrete, the robustness of the results, supported by a high coefficient of determination R2, provides justification for the proposed relationship. Nonetheless, the influence of the free-edge boundary on the size and development of the FPZ during cyclic loading warrants further detailed investigation through complementary techniques. For example, digital image correlation (DIC) could provide additional validation of the observed phenomena and may help to explain some unexpected results that were not initially anticipated when planning the experimental campaign [79,80]. 3.3 High-Cycle Fatigue Behaviour The assessment of the material’s fatigue resistance begins with the construction of the S–N curve, which provides insight into the progressive reduction in mechanical performance under cyclic loading. The S– N representation also spans a broad range of fatigue regimes, from low-cycle to high-cycle domains. In this study, a total of 16 specimens were tested to construct the experimental S-N curve. Stresses were calculated based on the measured specimen dimensions and the applied loads using LC3. The resulting curve is shown in Figure 14. Figure 14: Experimentally measured S-N curve for HPC under force-controlled loading. Figure 14 presents the experimental results along with the evaluated S-N curve, derived by fitting the data and excluding the runout specimens (i.e. samples that did not fail before reaching 2 × 10⁶ cycles). The data show relatively consistent behaviour in the static regime, but exhibit increasing scatter as the fatigue life approaches the runout threshold. This scatter is reflected in the relatively low coefficient of determination, R2=0.62. Notably, two specimens subjected to a maximum applied stress of σmax = 5.5 MPa (labelled FAT_01 and FAT_02) displayed unexpectedly high fatigue lives of 985 and 1 092 779 cycles (or three orders of magnitude difference), respectively. These results highlight the inherent variability in high-cycle fatigue behaviour, which was not observed in static or low-cycle tests. We attribute this scatter to the influence of the coarse granite aggregate used (Dmax=22 mm). When large aggregate particles are located near the notch, they promote crack bridging and increase the effective crack path length, thereby delaying fatigue 18 failure and extending the specimen's service life. The fitted parameters A and B from Eq. (4), obtained through least-squares regression of the experimental S–N data, are summarised in Table 4. Table 4: Summary of fatigue parameters evaluated from the experimental S-N curve. ft [MPa] A [MPa] B [-] R2 [-] σfat [MPa] σfat /A[%] 5.745 5.759 -0.017 0.62 4.0 70 The evaluated fitting parameter A, derived from the S–N curve, closely matches the mean measured tensile strength ft, indicating consistent and homogeneous behaviour of the material under static loading. The slope coefficient B, with a value of -0.017, aligns with previously reported results for highperformance concrete (HPC); for instance, the same value was obtained in the study by [40]. By contrast, typical values of B for normal-strength concrete (fc < 60 MPa) fall in the range of -0.03 to -0.04, reflecting a more rapid mechanical degradation under cyclic loading. The relatively flat slope observed here suggests a good resistance of fatigue loading. This is further supported by the measured fatigue limit stress σfat at 2 × 106 cycles, which was found to be 4.0 MPa, representing a 30% reduction relative to the mean static tensile strength ft. Additionally, the large scatter in fatigue performance remains evident, with fatigue limits ranging from 3.8 MPa to 4.75 MPa across the tested specimens (see empty symbols in Figure 14). As the 3PBT specimens were instrumented with a clip gauge during the fatigue tests, CMOD-N curves were recorded for each specimen. This continuous monitoring of CMOD during cyclic loading enabled the application of the compliance method (described in Figure 4 and Eqs. (8)-(9)) to evaluate fatigue crack growth in accordance with Paris’ law. This approach allows for a more comprehensive analysis, as CMOD is directly linked to the specimen’s stiffness and its progressive degradation under fatigue loading. Subsequently, the experimental results provide the necessary information for the calibration of numerical models [81–84] or serve as a dataset for training neural networks [85]. Depending on the phase of the load cycle, either CMODmin or CMODmax can be used, corresponding to the minimum Pmin and maximum Pmax applied loads, respectively. Both values, CMODmin and CMODmax, result in the computation of KI,min and KI,max, minimum and maximum SIFs. Accurate calculation of the SIFs using the compliance method depends on the appropriate value of the Young’s modulus E, a value of 24.5 GPa, (as listed in Table 3) was therefore used in Eq. (8). The fatigue crack growth curves derived using KI,max are shown in Figure 15. Figure 15: Measured fatigue crack growth rates related using KI,max values from compliance-based calculations. 19 Each data set in Figure 15 corresponds to a single specimen selected from the S-N data set in Figure 14. A total of 10 specimens were analysed, with applied maximum stresses ranging from 5.5 MPa to 3.8 MPa. Consequently, each dataset is associated with a different KI,max range due to the variation in load. A closer examination of the results in Figure 15 reveals two distinct phases of crack growth: deceleration followed by acceleration, which is characteristic of brittle matrix composites [86,87]. In this analysis, KI,max was chosen as the reference SIF since it corresponds to the portion of the load cycle most likely to initiate and propagate fatigue cracks. Moreover, the computed KI,max values span from 0.9 MPam1/2 to 3.2 MPam1/2. These relatively high value of SIF, especially when compared to the typical fracture toughness KIC of conventional concrete (ranging from 0.8 to 1.3 MPam1/2 [88,89]), occur near final failure, when CMOD values are elevated and the crack length exceeds a0/W > 0.7. The results in Figure 15 also indicate the presence of multiple threshold stress intensity values Kth, which may be attributed to variations in applied stress and the influence of coarse granite aggregates (Dmax = 22 mm). The phenomenon of crack bridging becomes particularly evident when aggregate particles are located at or near the notch. In such cases, the crack is forced to propagate around the aggregate, increasing the effective crack path length and enhancing fatigue life. This difference in fatigue performance is especially noticeable in the two specimens tested at σmax = 5.5 MPa, (see Figure 14), which are further examined in detail from a fracture mechanics perspective. Their corresponding fatigue crack growth rates are presented in Figure 16. Figure 16 shows a comprehensive analysis of the FCGRs for two selected concrete specimens, progressing from raw experimental data to the derivation of Paris’s law parameters. Figure 16(a-b)-(i) show the raw data, illustrating typical deceleration and acceleration phases with an identifiable apparent threshold value KI,th [90]. Figure 16 (a-b)-(ii) segment the crack propagation into four distinct stages (IIV), while Figure 16(a-b)-(iii) and Figure 16 (a-b)-(iv) focus on Paris’ law fits used to extract the material parameters C and m. From the analysis in Figure 16 (a-b)-(ii), the crack propagation process can be divided into four distinct stages: Stage I corresponds to an initial deceleration phase, associated with the consumption of fracture energy during early crack formation within the cement matrix and around aggregates, commonly referred to as crack bridging. This is followed by Stage II, the crack growth plateau, which represents a near-arrest condition where propagation slows significantly due to mechanisms such as aggregate interlocking or bridging. Once the apparent threshold KI,th is exceeded, the crack enters Stage III, characterised by stable growth that is well-suited for Paris’ law characterisation. Finally, the process culminates in Stage IV, defined by unstable crack propagation leading to the ultimate failure of the specimen. The deceleration of fatigue crack growth observed in Stage I is attributed to the dissipation of fracture energy during crack initiation within the cement matrix and around the aggregates, a mechanism commonly referred to as crack bridging. This process leads to the appearance of an apparent threshold stress intensity factor, KI,th. Since the crack does not completely arrest, the FCGR remains low, and the crack gradually transitions into the plateau phase identified as Stage II. This plateau is indicative of temporary crack arrest, primarily caused by aggregate bridging and, potentially, aggregate interlocking A wider plateau, as observed in specimen FAT_02(Figure 16(b)-(ii)), suggests the presence of large aggregates near the notch tip and multiple aggregates distributed in the surrounding region. In contrast, specimen FAT_01(Figure 16(a)-(ii)) exhibits a shorter plateau, indicating less effective bridging and resulting in faster crack propagation and earlier failure. A clear distinction between Stage III and Stage IV can be drawn from the Paris’s law parameters m and C, as presented in Figure 16(a)-(iii) and Figure 16(b)-(iii). Upon entering Stage IV, both specimens, FAT_01 and FAT_02, exhibited similar FFCRs of approximately 10-2 mm/cycle, despite experiencing significantly different SIF values. The scatter observed in Stage III for FAT_01 (Figure 16(a)-(iii)) for 20 suggests multiple crack bridging events occurring simultaneously, leading to propagation at a relatively constant rate. In contrast, specimen FAT_02 (Figure 16(b)-(iii)) displays isolated data points that indicate a sharp increase in FCGR, followed by a brief crack arrest, as also evidenced in Figure 16(b)- (iv). This short crack arrest precedes a phase of unstable crack growth, ultimately resulting in specimen failure. Notably, the FCGR observed in Stage IV for FAT_02 nearly doubles that of Stage III, highlighting the transition to unstable propagation. Furthermore, having accurate knowledge of the SIF allows the estimation of crack length a at each stage of the fatigue process. Using the threshold values KI,th corresponding to each stage, it gives us thorough understanding, when the sample failed. Both specimens have a threshold value KI,th ≈ 1.5 MPam1/2, which corresponds to a crack length a ≈ 22 mm. his length coincides with the maximum aggregate size Dmax = 22 mm used in the mixture, suggesting a direct link between mesostructural features and fracture behaviour. This correlation has previously been observed by Susmel [91] in both static and cyclic with constant amplitude loading conditions, supporting the hypothesis that aggregate size plays a critical role in defining the fracture process zone and fatigue thresholds. The measured FCGR curves revealed multiple regions where Paris’ law could be applied, allowing the evaluation of distinct sets of parameters. The resulting Paris’ law parameters C and m for each stage are summarised in Table 5. Table 5: Paris’s law parameters evaluated for Stages III and IV of the fatigue crack growth curves. Sample Nf [cycle] Stage III Stage IV C [mm/(cycle‧MPam1/2) m [-] C [mm/(cycle‧MPam1/2) m [-] FAT_01 985 -3.637 11.732 -3.804 12.57 FAT_02 1 092 779 -8.067 15.036 -14.289 29.66 The parameters are reported separately for Stage III (stable crack growth) and Stage IV (unstable growth) for both specimens. For specimen FAT_01, which failed after 985 cycles, the values of exponent m in Stages III and IV are relatively similar (11.7 and 12.6, respectively), confirming the observation from Figure 16(a) that the crack propagated at a comparable rate during both final stages. In contrast, specimen FAT_02, which endured 1 092 779 cycles, shows a significant difference in the m values between Stage III (15.0) and Stage IV (29.7), indicating a more abrupt transition to unstable crack propagation. This behaviour is consistent with the longer plateau observed in Stage II and suggests a more effective aggregate bridging mechanism, which contributes to extended fatigue life. 21 Figure 16: Comparison of measured FCGRs for specimens subjected to the same applied stress but exhibiting different fatigue lives – (a) 985 load cycles and (b) – 1,092,778 load cycles. 22 Regarding the coefficient C, the values are –3.6 and –8.1 for specimens FAT_01 and FAT_02, respectively, which is in compliance with results present by Kirane and Bažant [26]. This difference, of nearly five orders of magnitude, further supports the conclusions drawn from Figure 16 and reinforces the link between the width of the crack arrest plateau (Stage II) and the observed fatigue life. The observed effects of aggregate bridging and interlocking and their effects on fatigue performance were identified using a relatively simple yet effective experimental approach. However, these mechanisms warrant further investigation using advanced techniques such as DIC or acoustic emission [92]. These methods could offer a more detailed characterisation of internal damage evolution, potentially leading to improved fatigue models that incorporate the nonlinear effects of the FPZ under cyclic loading leading to numerical-based adjustments to Paris’s law [93,94] or explicitly incorporate aggregate-bridging phenomena [95–97]. Furthermore, the influence of size effect [98,99] warrants dedicated investigation, which was beyond the scope of the present study. 3.4 Cyclic Stiffness Degradation To further support the findings derived from the Paris’s law analysis, CMOD-N cyclic creep curves are presented for two studied specimens, FAT_01 and FAT_02 measured by LC3. These curves serve as both experimental output and input to Paris’ law formulation, as they reflect the evolution of CMOD with respect to the number of load cycles. More importantly, they offer insight into the degradation of cyclic stiffness resulting from progressive crack propagation under repeated loading [100]. The experimentally measured CMOD–N curves are shown in Figure 17. Figure 17: CMOD–N cyclic creep curves obtained during fatigue testing for specimens FAT_01 and FAT_02. The cyclic creep curves presented in Figure 17 compare the experimental responses of specimens FAT_01 and FAT_02 with that of the runout specimen. The results are expressed as mean CMOD values to clearly illustrate the progression of stiffness degradation during fatigue testing. The comparison reveals that, during the initial phase of testing (up to approximately 4 cycles), the stiffness degradation in FAT_01 closely follows that of the runout specimen. Beyond this point, however, the curves diverge significantly, with FAT_01 failing at 985 load cycles. In contrast, the cyclic creep curve for FAT_02 shows consistently higher CMOD values throughout the test, indicating a larger crack opening amplitude. This elevated CMOD in FAT_02 supports previous observations regarding aggregate bridging near the notch tip. The presence of a coarse aggregate in the initial crack path contributes to temporary crack arrest, which manifests as a plateau in crack growth. In addition to this meso-structural effect, a 23 geometric factor must also be considered: the width of specimen FAT_02 was W = 70.52 mm, FAT_01 and the runout specimen had widths close to W ≈ 79 mm. This dimensional difference may also have influenced the CMOD response. Interestingly, the CMOD value at which FAT_02 enters the unstable crack growth regime (≈ 0.017 mm) is comparable to that reached by the runout specimen at 2 × 10⁶ cycles. This finding suggests that the conventional definition of the fatigue limit as a fixed cycle count is, in this case, a practical rather than fundamental limit, selected to constrain testing time in view of the demanding nature of fatigue experiments. This behaviour highlights an important design consideration: disregarding the continued accumulation of damage in runout specimens may lead to unsafe predictions of long-term structural performance. Once again, the results underscore the need for fatigue analysis of concrete to receive greater attention, particularly through the development of reliable constitutive models capable of capturing the complex evolution of fatigue damage in quasi-brittle materials. 3.5 Damage Assessment To validate the findings regarding fatigue crack propagation, specimens FAT_01 and FAT_02 were examined using optical microscopy following the methodology described in Figure 7. Post-test analysis of the damaged specimens enabled the localisation of aggregates within the cementitious matrix and allowed the correlation between the experimentally calculated crack lengths and the meso-structural features of the concrete. As the specimen thickness T does not meet the RVE-type criterion, the posttest crack growth analysis was carried out along the width W direction, which conforms to the required size ratio. Each specimen was sectioned along Planes I and II, as depicted in Figure 7, providing spatial information about aggregate placement within the material. The separation between these planes was approximately. 10 mm, offering a representative depth for structural interpretation. The crack path identified in the damaged specimen was visualised under high magnification, with granite aggregates highlighted by dashed yellow lines to clearly show their location in relation to the fracture surface and the failure trajectory. It is important to note that this analysis is limited to the observation of the crack path and aggregate positioning; no attempt was made to characterise the full extent or morphology of the FPZ. The meso-structural analysis of specimen FAT_01 is presented in Figure 18. Figure 18: Optical analysis of the crack path in specimen FAT_01 (number of cycles to failure: Nf = 985). Figure 18 presents the damaged meso-structure of the specimen FAT_01, showing the identified granite coarse aggregates and the experimentally calculated crack lengths. The crack path observed in Plane IA reveals the presence of coarse aggregate located directly in the notch region. This coarse aggregate, with its edge terminating approximately 18 mm from the specimen boundary, confirms the crack length calculated at Stage II of propagation, as shown in Figure 16(a)-(iii). Beyond this point, no further aggregates were detected in the ligament that could contribute to crack bridging or arrest mechanisms. 24 In contrast, the meso-structure observed in plane II-B reveals a similar aggregate in the same notch location as in Plane I-A, but with a shorter length extending into the ligament. This is followed by another aggregate approximately 25 mm from the specimen edge. However, the orientation and position of this second particle suggest that it had no significant influence on the crack path, as the fracture surface passes along its edge without signs of retardation. similar analysis conducted on specimen FAT_02 revealed a greater number of aggregates located near the notch tip and a noticeably rougher fracture surface. These features suggest more extensive crack– aggregate interaction. The results of the meso-structural characterisation and the corresponding crack path for FAT_02 are presented in Figure 19. Figure 19: Optical analysis of the crack path in specimen FAT_02 (number of cycles to failure: Nf = 1 092 778). A clear distinction in the meso-structure of specimens FAT_01 and FAT_02 can be observed from Figure 19. The analysis of Plane I-A captured from sample FAT_02 reveals a coarse aggregate (Dmax = 8 mm) located directly within the notch region, followed by a second, larger aggregate (Dmax = 22 mm) and a deep intervening valley. This specific arrangement of aggregates is associated with crack arrest behaviour, as evidenced by the extended plateau observed in Stage II of the Paris’ law curve (see Figure 16(b)-(ii)). Moreover, the meso-structural observations corroborate the calculated crack lengths marking the transitions from Stage II to Stage III (crack initiation), and subsequently to unstable growth in Stage IV. The aggregate responsible for the observed crack arrest terminates approximately 38 mm from the specimen’s edge and is followed by a relatively deep valley in the fracture surface. This morphology again reflects a segment of the crack path where the crack bypassed the coarse aggregate without interaction, leading to local acceleration of crack growth. In Plane II-C, the crack trajectory exhibits a notably rough surface, with direct damage to a granite aggregate located approximately 22 mm from the specimen’s edge. This damage indicates a substantial accumulation of fracture energy, ultimately sufficient to cause the crack to propagate through the aggregate, rather than around it. Such behaviour implies a temporary increase in local resistance and is consistent with the crack retardation observed during Stages III and IV of fatigue propagation, as documented in Figure 16(b)-(ii)). When comparing static (LC1) and cyclic fatigue results (LC2 and LC3), a drastic shift in material behaviour is observed, from homogenous under static loading to highly heterogenous under fatigue conditions. The material’s homogeneity, evidenced by the low scatter in measured static strengths, is lost under cyclic loading with constant amplitude, where the variability increases markedly. This straightforward yet powerful post-mortem optical analysis confirms that the observed high scatter in fatigue lifetimes can be intrinsically linked to the loading conditions, which vary in applied load. Both static (LC1) and low-cyclic (LC2) tests were performed under CMOD-controlled conditions, which aligns with the initial hypothesis of aggregate bridging and offers valuable insight into the crack 25 initiation phase. In these cases, the increasing CMOD value corresponds to an extended crack initiation period during which fracture energy is steadily dissipated at a near-constant rate. In contrast, under cyclic loading with constant amplitude (LC3), specimens are subjected to a constant stress amplitude with a load cycle asymmetry ratio R = 0.1, resulting in cyclic crack opening and closing at a frequency of 10 Hz. Applying constant load amplitude produces constant crack opening under which the fracture energy is slowly dissipated, as document by low FCGR measured from Paris’ law curves. It also validates the calculated crack lengths a associated with the transitions between propagation stages. More broadly, the results establish a clear link between meso-structural heterogeneity and fatigue crack behaviour, offering a mechanistic explanation. The dissipation mechanism of fracture energy during monotonic cyclic loading still remains an under-explored area and an interesting topic for explaining the significant scatter observed in the fatigue performance of quasi-brittle materials such as HPC. Conclusions This experimental study re-evaluated the fracture and fatigue behaviour of high-performance concrete (HPC) under a wide range of loading conditions, spanning from static to low-cycle and high-cycle regimes. The material response was investigated through notched three-point bending tests, employing varying notch lengths and. For cyclic loading, a testing frequency of 10 Hz was applied. All specimens were cast using a mix incorporating high-quality granite aggregates with a maximum size of 22 mm. The tests were conducted in CMOD controlled mode, whereas the load controlled mode was used for fatigue tests with constant amplitude. Under static loading, the material exhibited relatively homogeneous behaviour. In contrast, under high-cycle loading, its response was markedly more heterogeneous, highlighting the influence of fatigue mechanisms and meso-structural variability. Static fracture properties were assessed using notched 3PBT specimens with a relative notch length of a0/W = 0.3, which helped to reduce the apparent brittleness of the material and allowed the post-peak softening phase to be captured. Furthermore, the resulting load-CMOD curves enabled the determination of fracture energy Gf and the cohesive law parameters through inverse analysis. The material parameters were obtained with low variability and good internal consistency, confirming the reliability of the fracture characterisation. Low-cyclic fatigue tests were performed on specimens with a0/W = 0.15 and a0/W = 0.3, providing insight into crack propagation under different initial ligament geometries. From these tests, the damage parameter  was derived from the stiffness degradation, and the irreversible crack opening displacement (CMODIR) was thoroughly analysed. A power-law relationship between CMOD and CMODIR was identified, which was shown to be independent of the initial notch length and attributed to the geometric confinement effect of the free-edge boundary on the fracture process zone. High-cycle fatigue tests were performed on specimens with a0/W =0.1, instrumented with clip gauges to monitor CMOD during sinusoidal loading. The resulting S–N curve revealed significant scatter in both the fatigue life and the runout stress σfat, emphasising the stochastic nature of fatigue failure in HPC. From a fracture mechanics perspective, the cyclic CMOD–N curves were used to evaluate fatigue crack growth rates using the compliance method. This enabled the determination of Paris’s law parameters and the correlation of meso-structural features, most notably the maximum aggregate size Dmax, with the critical crack length a at which unstable crack growth was triggered. 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