Fatigue of SFRC in compression: Size effect & autogenous self-healing

1. INTRODUCTION

Concrete is a quasi-brittle material, meaning its fracture behavior is highly dependent on the size and shape of the structural element (1Bažant ZP, Planas J. 1998. Fracture and size effect in concrete and other quasibrittle materials. CRC Press, Boca Raton, Florida, USA.). This size effect is a critical factor in concrete’s mechanical response, particularly because the energy released in the fracture process zone for a given crack increment varies with the size of the element, influencing both crack propagation and the stress at which failure occurs. Extensive research has established that concrete’s strength generally decreases as the size of the element increases, even when the geometry remains consistent across different scales (1–5SimJI, YangKH, JeonJK. 2013. Influence of aggregate size on the compressive size effect according to different concrete types. Constr. Build. Mater. 44: 716-725. 10.1016/j.conbuildmat.2013.03.066). This size effect complicates the direct application of strength data obtained from laboratory specimens to larger structural elements, as the experimentally measured strength is typically higher than the effective strength in real structural applications.

Compressive strength is the most widely used parameter for assessing concrete capacity, as concrete performs best under compressive loads. However, this property is also affected by the size effect. For large elements, compressive strength tends to a finite value, which is considered the intrinsic strength of the material (2Del VisoJR, CarmonaJR, RuizG. 2008. Shape and size effects on the compressive strength of high-strength concrete. Cem. Concr. Res. 38(3): 386-395. 10.1016/j.cemconres.2007.09.020).

While the size effect in concrete has been studied extensively under quasi-static loads, its influence on fatigue behavior, particularly under compressive fatigue, remains less explored. Given that concrete is primarily used in compression members in structures, understanding this relationship is crucial for predicting the long-term performance of concrete under cyclic loading.

Most research on the fatigue of concrete has focused on flexural behavior, examining crack propagation under cyclic loading conditions. This work has largely been based on the application of the Paris-Erdogan law (6ParisP, ErdoganF.1963. A critical analysis of crack propagation laws. J. Basic Eng. 85(4): 528-533. 10.1115/1.3656900), which relates the rate of crack growth to the range of stress intensity factors. Some studies have modeled fatigue crack growth by considering the cohesive stresses in the fracture process zone (7ZhangJ, LiVC, StangH. 2001. Size effect on fatigue in bending of concrete. J. Mater. Civ. Eng. 13(6): 446-453. 10.1061/(ASCE)0899-1561(2001)13:6(446)). However, there is a relative lack of studies focused on compressive fatigue, especially regarding the size effect on fatigue life in concrete cubes. While studies on prismatic specimens under axial load (8TaherSEDF, FawzyTM. 2000. Performance of very-high-strength concrete subjected to short-term repeated loading. Mag. Concr. Res. 52(3): 219-223. 10.1680/macr.2000.52.3.219) and cyclic tests on cylinders of varying diameters and slenderness ratios (9SinaieS, HeidarpourA, ZhaoXL, SanjayanJG. 2015. Effect of size on the response of cylindrical concrete samples under cyclic loading. Constr. Build. Mater. 84: 399-408. 10.1016/j.conbuildmat.2015.03.076) have provided valuable insights, much remains to be understood about how concrete behaves under compressive fatigue loading.

The incorporation of fibers into concrete as a distributed reinforcement leads to a more ductile material once the concrete matrix breaks, which is manifested in bending (10Di PriscoM, PlizzariG, VandewalleL. 2009. Fibre reinforced concrete: new design perspectives. Mater. Struct. 42: 1261-1281. 10.1617/s11527-009-9529-4), but also in tension (11MasihVW, RuizG, YuRC, De La RosaA. 2024. Crack initiation and growth in indirect tensile tests of steel fiber-reinforced concrete studies by means of DIC. In: MechtcherineV, SignoriniC, JungerD, editors. Transforming Construction: Advances in Fiber Reinforced Concrete, BEFIB 2024. RILEM Bookseries54. p. 43-50.) and compression (12RuizG, De La RosaA, PovedaE, ZanonR, SchäferM, WolfS. 2023. Compressive behaviour of steel-fibre reinforced concrete in Annex L of new Eurocode 2. Hormigón y Acero. 299-300:187-198. 10.33586/hya.2022.3092), and can also lead to modifying the effect of the size (13CarmonaJR, Cortés-BuitragoR, Rey-ReyJ, RuizG. 2022. Planar crack approach to evaluate the flexural strength of fiber-reinforced concrete sections. Materials. 15(17): 5821. 10.3390/ma15175821). Fiber reinforced concrete already plays a significant role in modern construction applications due to its enhanced ductility and post-cracking tensile strength, which makes it more enduring to fatigue compared to plain concrete (14PovedaE, RuizG, CifuentesH, YuRC, ZhangXX. 2017. Influence of the fiber content on the compressive low-cycle fatigue behavior of self-compacting SFRC. Int. J. Fatigue. 101(1): 9-17. 10.1016/j.ijfatigue.2017.04.005). Research shows that fiber reinforcement generally improves fatigue resistance, though the fibers’ influence under compressive stress is more limited, as they can sometimes act as crack initiation points (14–16LeeMK, BarrBIG. 2004. An overview of the fatigue behaviour of plain and fibre reinforced concrete. Cem. Concr. Compos. 26(4): 299-305. 10.1016/S0958-9465(02)00139-7). Fiber reinforced concrete demonstrates more uniform fatigue behavior across different loading frequencies, with steel fibers, in particular, helping to mitigate the detrimental effects of low-frequency loading (17MedeirosA, ZhangXX, RuizG, YuRC, VelascoMSL. 2015. Effect of the loading frequency on the compressive fatigue behavior of plain and fiber reinforced concrete. Int. J. Fatigue. 70: 342-350. 10.1016/j.ijfatigue.2014.08.005.). However, it has been observed that there is an optimal fiber content, beyond which the fatigue life of the material decreases (14PovedaE, RuizG, CifuentesH, YuRC, ZhangXX. 2017. Influence of the fiber content on the compressive low-cycle fatigue behavior of self-compacting SFRC. Int. J. Fatigue. 101(1): 9-17. 10.1016/j.ijfatigue.2017.04.005). Additionally, few studies have examined the size effect specifically in fiber reinforced concrete. Some research, such as (18LiVC, LinZ, MatsumotoT. 1998. Influence of fiber bridging on structural size-effect. Int. J. Solids Struct. 35(31-32): 4223-4238. 10.1016/S0020-7683(97)00311-9), suggests that fiber addition shifts the brittle-to-ductile transition in bending tests, while other studies have found that increasing both matrix strength and fiber content reduces the size effect in compressive fatigue tests (19FládrJ, BílyP. 2018. Specimen size effect on compressive and flexural strength of high-strength fibre-reinforced concrete containing coarse aggregate. Compos. Part B-Eng. 138: 77-86. 10.1016/j.compositesb.2017.11.032).

Parallel to the study of size effects on fatigue, there is growing interest in the phenomenon of autogenous self-healing in concrete under fatigue conditions. Contrary to the traditional view that cyclic loading degrades material properties over time, by the late 1900s and early 2000s some research showed that concrete can exhibit strength increases after undergoing fatigue loading. For example, Nelson et al. (20NelsonEL, CarrasquilloRL, FowlerDW. 1988. Behavior and failure of high-strength concrete subjected to biaxial-cyclic compression loading. ACI Mater. J. 85(4): 248-253.) observed a remarkable 39% strength increase in runout specimens, and Taliercio and Gobbi (21TaliercioALF, GobbiE. 1997. Effect of elevated triaxial cyclic and constant loads on the mechanical properties of plain concrete. Mag. Concr. Res. 49(181): 353-365. 10.1680/macr.1997.49.181.353) reported strength gains between 6% and 33% due to lateral confinement in axially fatigued concrete. Taher and Fawzy (8TaherSEDF, FawzyTM. 2000. Performance of very-high-strength concrete subjected to short-term repeated loading. Mag. Concr. Res. 52(3): 219-223. 10.1680/macr.2000.52.3.219) extended these findings, demonstrating that normal-strength concrete can exhibit strength gains of 15% to 57% after a limited number of fatigue cycles. Cachim et al. (22CachimPB, FigueirasJA, PereiraPAA. 2002. Fatigue behavior of fiber-reinforced concrete in compression. Cem. Concr. Compos. 24(2): 211-217. 10.1016/S0958-9465(01)00019-1) confirmed similar results for both plain and fiber reinforced concrete, noting improvements up to 16%. More recent studies, including those by Malek et al. (23MalekA, ScottA, PampaninS, MacRaeG, MarxS. 2018. Residual capacity and permeability-based damage assessment of concrete under low-cycle fatigue. J. Mater. Civ. Eng. 30(6): 04018081. 10.1061/(ASCE)MT.1943-5533.0002248), Qiu et al. (24QiuJ, Aw-YongWL, YangEH. 2018. Effect of self-healing on fatigue of engineered cementitious composites (ECCs). Cem. Concr. Compos. 94: 145-152. 10.1016/j.cemconcomp.2018.09.007), and Garijo et al. (25GarijoL, OrtegaJJ, RuizG, De La RosaA, ZhangXX. 2022. Effect of loading frequency on the fatigue life in compression of natural hydraulic lime mortars. Theor. Appl. Fract. Mech. 118: 103201. 10.1016/j.tafmec.2021.103201), have further corroborated these observations, with strength increases as high as 42%.

The underlying mechanism for this strength increase is thought to be autogenous self-healing, where microcracks generated by cyclic loading allow occluded water to penetrate the material, rehydrating dormant cement particles and sealing the cracks (25–28De La RosaÁ, OrtegaJJ, RuizG, García CalvoJL, Rubiano SánchezFJ, CastilloÁ. 2023. Autogenous self-healing induced by compressive fatigue in self-compacting steel-fiber reinforced concrete. Cem. Concr. Res. 173: 107278. 10.1016/j.cemconres.2023.107278). Various mechanisms have been proposed for this self-healing process, including calcium carbonate precipitation, the rehydration of unhydrated cement particles, the obstruction of cracks by impurities, and the swelling of the hydrated cement matrix (29–34De RooijM, Van TittelboomK, De BelieN, SchlangenE, editors. 2013. Self-healing phenomena in cement-based materials. RILEM Tech. Comm. 221-SHC. Vol. 11. Springer, Dordrecht.). Some researchers emphasize the role of calcium carbonate precipitation (33WuM, JohannessonB, GeikerM. 2012. A review: Self-healing in cementitious materials and engineered cementitious composite as a self-healing material. Constr. Build. Mater. 28(1): 571-583. 10.1016/j.conbuildmat.2011.08.086, 35EdvardsenC. 1999. Water permeability and autogenous healing of cracks in concrete. ACI Mater. J. 96(4): 448–454. 10.14359/645), while others highlight the importance of ongoing hydration reactions (36Roig-FloresM, FormaginiS, SernaP. 2021. Self-healing concrete: What is it good for?Mater. Construcc. 71(341): e237. 10.3989/mc.2021.07320). The ability of concrete to self-heal is dependent on several factors, including the availability of moisture, the size of the cracks, and the use of fibers to control crack width and propagate stress in a more distributed manner (29FerraraL, KrelaniV, MorettiF, Roig FloresM, Serna RosP. 2017. Effects of autogenous healing on the recovery of mechanical performance of high performance fibre reinforced cementitious composites (HPFRCCs): Part 1. Cem. Concr. Compos. 83: 76-100. 10.1016/j.cemconcomp.2017.07.010, 36–38SnoeckD, De BelieN. 2016. Repeated autogenous healing in strain-hardening cementitious composites by using superabsorbent polymers. J. Mater. Civ. Eng. 28(1): 1-11. 10.1061/(ASCE)MT.1943-5533.0001360).

In addition, advances in nanotechnology have provided opportunities to enhance the autogenous self-healing capabilities of concrete by incorporating nanomaterials that act as nucleation sites for the formation of new hydration products (39ChacónC, CifuentesH, Luna-GalianoY, RíosJD, Ariza P, LeivaC. 2023. Exploring the impact of graphene oxide on mechanical and durability properties of mortars incorporating demolition waste: Micro and nano-pore structure effects. Mater. Construcc. 73(352): e327. 10.3989/mc.2023.351623). This phenomenon has also been observed in ultra-high-performance concretes, where large amounts of cementitious material remain unhydrated. When exposed to moderately elevated temperatures, such as those exceeding 200°C, these materials exhibit strength increases due to the activation of dormant hydration processes (40Suescum-MoralesD, RíosJD, Martínez De La ConchaA, CifuentesH, JiménezJR, FernándezJM. 2021. Effect of moderate temperatures on compressive strength of ultra-high-performance concrete: A microstructural analysis. Cem. Concr. Res. 140: 106303. 10.1016/j.cemconres.2020.106303).

Given the importance of both the size effect and autogenous self-healing in understanding the fatigue behavior of steel fiber reinforced concrete, we planned to investigate these two phenomena in detail. We focused on the compressive fatigue life of cubic and cylindrical specimens of varying sizes and explore how cyclic loading-induced microcracks contribute to the autogenous self-healing process. The research has already led to some publications that are summarized in this paper. The first two of them dealt with the size effect of cubes and cylinders of a single steel fiber reinforced concrete in fatigue (26OrtegaJJ, RuizG, PovedaE, GonzálezDC, TarifaM, ZhangX, YuRC, VicenteMÁ, De La RosaÁ. 2022. Size effect on the compressive fatigue of fibre-reinforced concrete. Constr. Build. Mater.322: 126238. 10.1016/j.conbuildmat.2021.126238, 27GonzálezDC, MenaA, RuizG, OrtegaJJ, PovedaE, MínguezJ, YuR, De La RosaÁ, VicenteMÁ. 2023. Size effect of steel fiber–reinforced concrete cylinders under compressive fatigue loading: Influence of the mesostructure. Int. J. Fatigue. 167(B): 107353. 10.1016/j.ijfatigue.2022.107353). Their results manifested that there could be autogenous self-healing in the specimens that endured most, which made us prepare another research that eventually demonstrated the production of new hydration products due to fatigue (28De La RosaÁ, OrtegaJJ, RuizG, García CalvoJL, Rubiano SánchezFJ, CastilloÁ. 2023. Autogenous self-healing induced by compressive fatigue in self-compacting steel-fiber reinforced concrete. Cem. Concr. Res. 173: 107278. 10.1016/j.cemconres.2023.107278). In this paper, we want to give a comprehensive view of these coupled phenomena. By combining probabilistic analysis of fatigue life with detailed microstructural and fracture assessments, we seek to provide a comprehensive understanding of how size and self-healing influence the mechanical behavior of concrete under fatigue conditions.

We also aim to familiarize the reader with the concept of steel fiber reinforced concrete (SFRC) fatigue in compression. This understanding is crucial for recognizing the significance of size effect and the initiation of autogenous self-healing caused by compressive fatigue loads. To achieve this, we delve into the study of concrete fatigue in compression, drawing from our own scientific research. This research has been driven by collaboration with industrial partners interested in investigating the fatigue of structural components, such as pylons for wind turbines (see Figure 1a), slabs for high-speed train tracks (Figure 1b) (41TarifaM, ZhangXX, RuizG, PovedaE. 2015. Full-scale fatigue tests of precast reinforced concrete slabs for railway tracks. Eng. Struct. 100: 610-621. 10.1016/j.engstruct.2015.06.016, 42PovedaE, YuRC, LanchaJC, RuizG. 2015. A numerical study on the fatigue life design of concrete slabs for railway tracks. Eng. Struct. 100: 455-467. 10.1016/j.engstruct.2015.06.037), and encased composite beams with steel fiber reinforced concrete (Figure 1c) (43ZanonR, SchäferM, RuizG, YuRC, De La RosaA, MasihVW. 2024. Experimental bending tests on encased steel-concrete composite beams with SFRC. In: MechtcherineV, SignoriniC, JungerD, editors. Transforming Construction: Advances in Fiber Reinforced Concrete, BEFIB 2024. RILEM Bookseries54. p. 197-204.).

  

FIGURE 1. (a) SFRC-tower supporting a wind turbine. Fatigue tests of: (b) a reinforced concrete slab for high-speed ballastless tracks (41TarifaM, ZhangXX, RuizG, PovedaE. 2015. Full-scale fatigue tests of precast reinforced concrete slabs for railway tracks. Eng. Struct. 100: 610-621. 10.1016/j.engstruct.2015.06.016, 42PovedaE, YuRC, LanchaJC, RuizG. 2015. A numerical study on the fatigue life design of concrete slabs for railway tracks. Eng. Struct. 100: 455-467. 10.1016/j.engstruct.2015.06.037), and (c) an encased composite beam with steel fiber reinforced concrete (43ZanonR, SchäferM, RuizG, YuRC, De La RosaA, MasihVW. 2024. Experimental bending tests on encased steel-concrete composite beams with SFRC. In: MechtcherineV, SignoriniC, JungerD, editors. Transforming Construction: Advances in Fiber Reinforced Concrete, BEFIB 2024. RILEM Bookseries54. p. 197-204.), both made in the Materials and Structures Laboratory at UCLM. 

We will first discuss the phenomenology of fatigue of steel fiber reinforced concrete in compression. This will include subsections covering basic aspects of representing fatigue results (Subsection 2.1), as well as the effects of experimental scatter (2.2), frequency (2.3), fiber content (2.4), and load eccentricity (2.5). Section 3 will focus on the impact of size on the compressive fatigue of the material, while Section 4 will deal with autogenous self-healing due to fatigue (Section 4). It follows a discussion on the principal effects influencing autogenous self-healing, namely mineral additions (Subsection 5.1), carbonation (5.2), steel fiber reinforcement (5.3), size effect (5.4), and time (5.5). Lastly, we will draw conclusions emphasizing the importance of taking these phenomena into account.

2. FATIGUE PHENOMENOLOGY IN CEMENTITIOUS MATERIALS

Cyclic loading in concrete, whether compressive or tensile, causes the material to experience strain increments with each cycle. Figure 2a illustrates a typical setup of a compressive fatigue test on a concrete cube, whereas Figure 2b plots some compressive load-strain loops in a specimen that has been subjected to cycles running from 5% to 70% of its compressive strength. They correspond to a 76 × 76 × 302 mm³ prism of a 36 MPa concrete tested by Jinawath (44JinawathP. 1974. Cumulative fatigue damage of plain concrete in compression [Ph.D. thesis]. University of Leeds, Leeds, United Kingdom.). The compression loops initially exhibit a convex shape during loading and a concave shape during unloading, but soon they feature a concave loading branch. The secant modulus of elasticity decreases as the number of cycles increases, as illustrated in Figure 2b. The strain increase is rapid during the first few cycles, then levels off and becomes linear with the number of cycles during a second phase, and eventually grows rapidly again before failure, as depicted by the strain-number of cycles curve in Figure 3. This curve is often referred to as the “cyclic creep curve” because it resembles the deformation caused by a sustained load on the material. The average of the cyclic loading affects the specimen, and the resulting deformation somehow responds to this mean load.

  (adapted from Figure 6.4, P. Jinawath PhD Thesis Report, University of Leeds, 1974 (44), & Figure 8.23 in (45)).

FIGURE 2. (a) Concrete cube tested in compressive fatigue and (b) stress-strain cycles in compression plotted with the stress-strain monotonic curve of the same material (adapted from Figure 6.4, P. Jinawath PhD Thesis Report, University of Leeds, 1974 (44), & Figure 8.23 in (45)).

  

FIGURE 3. Cyclic creep curve in cementitious materials. 

The stress-strain loops show that there are internal surfaces and microcracks present in the concrete, caused by the loads, which then grow. These internal surfaces come into contact again when the test piece is compressed anew in the following cycles, leading to overall hardening that is reflected in the concavity of the load branch in compression (interestingly, no hardening is observed in tension because the surfaces do not come into contact (46HordijkDA, ReinhardtHW. 1993. Numerical and experimental investigation into the fatigue behavior of plain concrete. Exp. Mech. 33(4): 278-285. 10.1007/BF02322142)). The area enclosed in the loops represents energy dissipation due to the generation and propagation of microcracks, and friction between internal surfaces. Failure occurs when the concrete reaches a critical deformation that it can no longer withstand. For example, in Figure 2b, the critical elongation of the last cycle approximately coincides with the elongation recorded in the static test for that level of compression, schematically represented by the solid red curve. This observation suggests that the static resistance curve can be considered as a failure curve of the fatigue loops, like an envelope of the cyclic loops. Indeed, it is usually considered that fatigue failure takes place once the loops reach the quasi-static envelope (47–49American Concrete Institute (ACI). 2022. Concrete structure design for fatigue loading--Report. ACI PRC-215-21. American Concrete Institute.). It is confirmed that the deformations at the end of the second phase are similar to those of the envelope, while the critical deformations exceed those obtained in static tests (48ZanuyC, de la FuenteP, AlbajarL. 2007. Effect of fatigue degradation of the compression zone of concrete in reinforced concrete sections. Eng. Struct. 29(11): 2908-2920. 10.1016/j.engstruct.2007.01.030). However, due to the scatter of the post-peak part of the monotonic quasi-static curve, especially in concrete with fibers, and its dependence on the geometry of the specimen, this observation cannot be profitably used for predicting accurately the fatigue life (12RuizG, De La RosaA, PovedaE, ZanonR, SchäferM, WolfS. 2023. Compressive behaviour of steel-fibre reinforced concrete in Annex L of new Eurocode 2. Hormigón y Acero. 299-300:187-198. 10.33586/hya.2022.3092, 26OrtegaJJ, RuizG, PovedaE, GonzálezDC, TarifaM, ZhangX, YuRC, VicenteMÁ, De La RosaÁ. 2022. Size effect on the compressive fatigue of fibre-reinforced concrete. Constr. Build. Mater.322: 126238. 10.1016/j.conbuildmat.2021.126238, 50RuizG, De La RosaÁ, PovedaE. 2019. Relationship between residual flexural strength and compression strength in steel-fiber reinforced concrete within the new Eurocode 2 regulatory framework. Theor. Appl. Fract. Mech. 103: 102310. 10.1016/j.tafmec.2019.102310, 51De La RosaÁ, RuizG, PovedaE. 2019. Study of the compression behavior of steel-fiber reinforced concrete by means of the response surface methodology. Appl. Sci. 9(24): 5330.).

2.1. Stress/strain vs. number of cycles curves in concrete

 ⌅

Figure 4 illustrates the relationship between the logarithm of the secondary strain rate per cycle (which corresponds to the constant strain rate during the second stage of the cyclic creep curve, as shown in Figure 3) and the logarithm of the number of cycles to failure for three different types of concrete with the same matrix (17MedeirosA, ZhangXX, RuizG, YuRC, VelascoMSL. 2015. Effect of the loading frequency on the compressive fatigue behavior of plain and fiber reinforced concrete. Int. J. Fatigue. 70: 342-350. 10.1016/j.ijfatigue.2014.08.005.). The types are as follows: C1 is plain concrete, with a compressive strength of 56 MPa at 28 days; C2 is polypropylene fiber-reinforced concrete, with a compressive strength of 66 MPa and a fiber volume of 0.56%; and C3 is steel fiber-reinforced concrete, with a compressive strength of 67 MPa and a fiber volume of 0.64%.

  

FIGURE 4. Logarithm of the number of cycles to failure versus the logarithm of the secondary strain rate for three concretes with a shared matrix (17MedeirosA, ZhangXX, RuizG, YuRC, VelascoMSL. 2015. Effect of the loading frequency on the compressive fatigue behavior of plain and fiber reinforced concrete. Int. J. Fatigue. 70: 342-350. 10.1016/j.ijfatigue.2014.08.005.). 

Tests were conducted on 100 mm cubes at four different frequencies: 4 Hz, 1 Hz, 1/4 Hz, and 1/16 Hz. All data points generally align along a straight line, a trend first noted by Sparks and Menzies (52SparksPR, MenziesJB. 1973. Effect of rate of loading upon static and fatigue strengths of plain concrete in compression. Mag. Concr. Res. 25(83): 73-80. 10.1680/macr.1973.25.83.73) for plain concrete. Medeiros et al. (17MedeirosA, ZhangXX, RuizG, YuRC, VelascoMSL. 2015. Effect of the loading frequency on the compressive fatigue behavior of plain and fiber reinforced concrete. Int. J. Fatigue. 70: 342-350. 10.1016/j.ijfatigue.2014.08.005.) further demonstrated that this relationship holds true regardless of the type of fiber or the testing frequency. The linear relationship can be expressed as:

$log\frac{ϵ}{f}=m+slogN$ [1]  

where is the secondary strain rate per cycle, f is the frequency, and N is the number of cycles endured before failure (note that =, where n represents the cycle number, not the cycle at failure).

This feature of cyclic creep curves enables the derivation of a strain-based failure criterion. Poveda et al. (14PovedaE, RuizG, CifuentesH, YuRC, ZhangXX. 2017. Influence of the fiber content on the compressive low-cycle fatigue behavior of self-compacting SFRC. Int. J. Fatigue. 101(1): 9-17. 10.1016/j.ijfatigue.2017.04.005) extended the Sparks and Menzies’ law by integrating it to yield a failure criterion expressed as:

${\displaystyle {ϵ}_c={ϵ}_{max}+{10}^m{N}^{s+1}}$ [2]  

where 𝜖_c is the critical strain at failure and 𝜖_max is the maximum strain recorded in the first cycle. The red curve in Figure 5 represents this criterion on a strain vs. number of cycles plot. Failure of a specimen occurs when the cyclic creep curve intersects this failure curve, as indicated by the black dots in Figure 5.

FIGURE 5. Graphical representation of the failure criterion based on the Sparks & Menzies law by Poveda et al. (14PovedaE, RuizG, CifuentesH, YuRC, ZhangXX. 2017. Influence of the fiber content on the compressive low-cycle fatigue behavior of self-compacting SFRC. Int. J. Fatigue. 101(1): 9-17. 10.1016/j.ijfatigue.2017.04.005). 

While strain-based failure criteria offer a convenient method for assessing fatigue behavior, the fatigue strength of concrete is often represented by S-N curves. In these curves, S represents the normalized stress (S = σ/f_c) and N is the number of cycles to failure, typically shown on a logarithmic scale. For instance, the S-N curves proposed in the 2010 Model Code and developed by Lohaus et al. (53LohausL, OneschkowN, WeferM. 2012. Design model for the fatigue behaviour of normal-strength, high-strength and ultra-high-strength concrete. Struct. Concr. 13(3): 182-192. 10.1002/suco.201100054) (see Figure 6) illustrate the relationship between maximum stress and N, with the minimum stress as a parameter. These curves are generally considered applicable to all types of concrete, as the influence of composition on fatigue behavior is relatively minor.

FIGURE 6. S-N curves for concrete according to the Model Code 2010 (55fib Bulletins 65 & 66. 2012. Model Code 2010, Final Draft. International Federation for Structural Concrete (fib), Lausanne, Switzerland.): (a) 3D representation from the model by Lohaus, Oneschkow, and Wefer (53LohausL, OneschkowN, WeferM. 2012. Design model for the fatigue behaviour of normal-strength, high-strength and ultra-high-strength concrete. Struct. Concr. 13(3): 182-192. 10.1002/suco.201100054), and (b) 2D plot included in the standard. 

A distinction is often made between low-cycle and high-cycle fatigue. Low-cycle fatigue results from a small number of cycles at high stress levels, such as those during an earthquake, whereas high-cycle fatigue occurs over the structure’s service life at lower stress levels. However, both fatigue types are driven by the same damage mechanisms, making the distinction largely formal. According to CEB Committee 188 (54Comité Euro-International du Béton (CEB). 1988. Fatigue of concrete structures. State of the art report. CEB Report188, Lausanne.), the boundary between low-cycle and high-cycle fatigue lies between 10³ and 10⁴ cycles, but no clear discontinuity is observed in the S-N curves (Figure 6b).

Unlike metals, concrete appears to lack an endurance limit, meaning no stress threshold guarantees infinite fatigue life. Tests beyond 10⁷ cycles are uncommon, yet the Model Code curves (Figure 6b) extend up to 10²⁰ cycles. Reaching this extreme number of cycles at a frequency of 1 Hz would take approximately 230 times the time since the Big Bang—about 13.8 billion years, which gives an idea of the absolute nonsense of extrapolating results in the logarithmic scale.

Concrete’s fatigue life is highly sensitive to both the stress level and the loading range, defined by the difference between maximum and minimum stress (∆σ = σ_max−σ_min), or by the stress ratio R = σ_min/σ_max. A smaller stress range or higher stress ratio allows the concrete to withstand a greater maximum cyclic load. This relationship is shown in Figure 6b, where higher minimum stresses correspond to higher maximum stresses in cyclic loading.

2.2. Experimental scatter

 ⌅

It is important to understand that concrete fatigue can vary significantly, meaning that a single average S-N curve may not fully represent all fatigue behaviors. Figure 7 illustrates this by displaying the probability of failure versus the number of cycles for almost 100 identical compression fatigue tests conducted by Ortega et al. (56OrtegaJJ, RuizG, YuRC, Afanador-GarcíaN, TarifaM, PovedaE, ZhangX, EvangelistaF. 2018. Number of tests and corresponding error in concrete fatigue. Int. J. Fatigue. 116: 210-219. 10.1016/j.ijfatigue.2018.06.022). The compressive strength of the SFRC was 58.9 MPa, and it was reinforced with 13 mm straight steel fibers at a 0.2% volumetric ratio. The results were analyzed using a two-parameter Weibull function. By randomly selecting test groups and plotting their failure probability functions, we can estimate the range of error associated with different sample sizes.

FIGURE 7. Effect of sample size (n) on the measurement of fatigue life. The shaded band represents the variability in failure probability for random groups of 25 tests. The right plot shows relative error curves for different sample sizes (56OrtegaJJ, RuizG, YuRC, Afanador-GarcíaN, TarifaM, PovedaE, ZhangX, EvangelistaF. 2018. Number of tests and corresponding error in concrete fatigue. Int. J. Fatigue. 116: 210-219. 10.1016/j.ijfatigue.2018.06.022). 

In Figure 7, the shaded area represents the range covered by random groups of 25 tests. It required half a million random groups to stabilize this band and have it aligned with the overall distribution obtained from all tests, represented by the thick black line, which is considered the correct distribution. From this, we can derive the probability density function for various failure probabilities and calculate confidence margins. For example, the green curve shows the density function for a 40% failure probability, while the dashed blue line indicates the 95% confidence margin. This means there is a 95% likelihood that the distribution function of a random group of 25 tests will fall to the left of this curve.

The right plot in Figure 7 presents error curves for different sample sizes, enabling us to determine the associated error for each group size. For instance, the relative error for a group of 15 fatigue tests (orange error curve) is 5.7% at a 50% failure probability, rising to 10.5% for groups of 5 tests (red curve), and decreasing to 4.1% for groups of 25 tests (blue curve). If the failure probability is 10%, the relative errors increase to 22.2%, 10.9%, and 7.5% for groups of 5, 15, and 25 specimens, respectively.

Even with large sample sizes, there is considerable variability in failure probabilities. In this example (Figure 7), the average fatigue life is approximately 10700 cycles, but at a 10% failure probability, the fatigue life drops to just 1500 cycles. S-N curves are typically constructed using the average values of the failure distribution for each load case, necessitating a statistical approach to ensure adequate safety levels.

2.3. Effect of frequency

 ⌅

The frequency of cyclic loading significantly affects fatigue life, especially at low frequencies (1 Hz and below) and high peak stresses (above 75% of static strength). Figure 8 presents fatigue life results at varying frequencies (4, 1, 1/4, and 1/16 Hz) for compression tests with a peak load of 85% of the static strength (79 MPa for the C1 concrete, tested when it was approximately one year old) and a stress ratio of 0.3 (17MedeirosA, ZhangXX, RuizG, YuRC, VelascoMSL. 2015. Effect of the loading frequency on the compressive fatigue behavior of plain and fiber reinforced concrete. Int. J. Fatigue. 70: 342-350. 10.1016/j.ijfatigue.2014.08.005.). A notable drop in fatigue life occurs when shifting from 4 Hz to 1 Hz, where log N decreases from 4.3 to 2.8, indicating a reduction in fatigue life by more than an order of magnitude. This reduction results from lower strain rates at reduced frequencies, which diminish static strength. Additionally, the interaction between the detrimental effects of compressive cycles and the beneficial self-healing due to fatigue may play a role in this behavior (discussed later in Section 5).

FIGURE 8. Effect of frequency on fatigue life for three concretes with the same matrix (17MedeirosA, ZhangXX, RuizG, YuRC, VelascoMSL. 2015. Effect of the loading frequency on the compressive fatigue behavior of plain and fiber reinforced concrete. Int. J. Fatigue. 70: 342-350. 10.1016/j.ijfatigue.2014.08.005.): (a) plain concrete (C1), (b) polypropylene fiber reinforced concrete (C2), and (c) steel fiber reinforced concrete (C3). 

This sensitivity to frequency is particularly relevant for structures like wind towers, which experience low oscillation frequencies. To improve fatigue life in such applications, reinforcing concrete with steel fibers is advisable, as they help extend fatigue life. For instance, concrete C3 (steel fiber reinforced) in Figure 8 shows consistent performance across all studied frequencies, unlike C2 (polypropylene fiber reinforced), which is more frequency sensitive.

2.4. Effect of fiber content

 ⌅

We investigated the effect of fiber content on the mechanical behavior of steel-fiber reinforced concrete (SFRC) under low-cycle fatigue (14PovedaE, RuizG, CifuentesH, YuRC, ZhangXX. 2017. Influence of the fiber content on the compressive low-cycle fatigue behavior of self-compacting SFRC. Int. J. Fatigue. 101(1): 9-17. 10.1016/j.ijfatigue.2017.04.005). Five self-compacting concrete mixtures were designed with fiber contents ranging from 0 to 0.8% by volume. All mixtures shared the same concrete matrix, ensuring flowability and fiber-matrix adhesion. The base concrete had a compressive strength of 33.4 MPa. The steel fibers used were hook-ended, 35 mm in length, and 0.55 mm in diameter. The specimens were 100 mm cubes.

In terms of static behavior, the fibers did not significantly enhance compressive strength. However, the presence of fibers resulted in a more gradual post-peak softening after maximum load. Regarding flexural behavior, increasing fiber content improved post-peak performance, though certain fiber amounts produced similar residual tensile strength. For example, concrete with 0.6% fiber content exhibited post-peak behavior similar to that with 0.4%.

The addition of steel fibers markedly improved compressive fatigue performance, as fatigue life increased with fiber content (see Figure 9a). Low fiber contents did not substantially improve fatigue properties compared to plain concrete. However, intermediate fiber contents —from 0.4% to 0.6%— extended fatigue life up to five times longer than that of plain concrete (equivalent to a 31% increase in the logarithmic average of resisted cycles). The optimal fiber dosage was found to be 45 kg/m³ in this particular study. Interestingly, higher fiber contents reduced fatigue life, likely due to increased matrix imperfections and pore formation, which promote crack initiation.

FIGURE 9. Effect of fiber content on fatigue life of SFRC (14PovedaE, RuizG, CifuentesH, YuRC, ZhangXX. 2017. Influence of the fiber content on the compressive low-cycle fatigue behavior of self-compacting SFRC. Int. J. Fatigue. 101(1): 9-17. 10.1016/j.ijfatigue.2017.04.005): (a) Fatigue life as a function of fiber dosage; (b) Secondary strain rate vs. fatigue life following Sparks and Menzies’ model. 

Notably, the Sparks and Menzies’ plot of secondary strain rate versus fatigue life showed a consistent linear relationship (Figure 9b), confirming that this relationship is independent of fiber content and primarily depends on the base concrete.

The improvement in fatigue life with fiber content may be explained by the interplay between microcrack propagation and autogenous self-healing due to fatigue, which will be discussed further in Section 5.

2.5. Effect of load eccentricity

 ⌅

We also examined how load positioning uncertainty affects the fatigue life and result variability of SFRC (57TarifaM, RuizG, PovedaE, ZhangXX, VicenteMA, GonzálezDC. 2018. Effect of uncertainty on load position in the fatigue life of steel-fiber reinforced concrete under compression. Mater. Struct. 51(1). 10.1617/s11527-018-1155-6), a factor often blamed on compressive strength scatter (22CachimPB, FigueirasJA, PereiraPAA. 2002. Fatigue behavior of fiber-reinforced concrete in compression. Cem. Concr. Compos. 24(2): 211-217. 10.1016/S0958-9465(01)00019-1, 45HolmenJO. 1979. Fatigue of concrete by constant and variable amplitude loading [Ph.D. thesis]. University of Trondheim, Trondheim, Norway.). While inhomogeneity at the mesoscale undoubtedly contributes to scatter, there may be experimental factors influencing fatigue life as well.

To reduce unintended loading eccentricity, we developed an individualized ball-and-socket joint (i-BSJ) (see Figure 10a). Two types of tests were conducted: first, we used an instrumented aluminum cube to measure eccentricity under two conditions—using the i-BSJ and without it. Second, two series of 15 monotonic compressive tests and two series of 15 cyclic compressive tests were performed on 40 mm SFRC cubes (using 15 kg/m³ of straight, 13 mm long, and 0.20 mm in diameter fibers), again with and without the i-BSJ.

FIGURE 10. Impact of eccentricity on fatigue life (57TarifaM, RuizG, PovedaE, ZhangXX, VicenteMA, GonzálezDC. 2018. Effect of uncertainty on load position in the fatigue life of steel-fiber reinforced concrete under compression. Mater. Struct. 51(1). 10.1617/s11527-018-1155-6): (a) 40-mm SFRC cube tested with an individualized ball-and-socket joint (i-BSJ); (b) Probability density function for load eccentricity in two test series (90% of results without i-BSJ fall within the grey area, while 90% of results with i-BSJ fall within the red area); (c) Fatigue results for identical SFRC specimens tested with and without the i-BSJ. 

The results revealed that while average eccentricity values remained similar, their standard deviation was reduced by an order of magnitude when using the i-BSJ. Figure 10b illustrates the two-dimensional probability distribution of load position. Without the i-BSJ, load eccentricity had wider variation (grey-shaded area in Figure 10b), whereas with the i-BSJ, it was significantly more concentrated (red-shaded area). Thus, while some variability remained due to equipment tolerances and specimen geometry, the i-BSJ effectively minimized misalignments from manual specimen centering and other geometrical factors.

In addition, when the i-BSJ was used in the SFRC cube tests, the average compressive strength increased by 10% (from 53.6 MPa to 58.9 MPa). The average fatigue life also saw a significant increase, multiplying by six in the second series (refer to the red curve in Figure 10c). Furthermore, the standard deviation of fatigue life decreased, as indicated by the steeper slope of the red curve in Figure 10c. These results show that undesired loading eccentricity reduces fatigue life and increases result variation, beyond what can be explained by variations in compressive strength alone (22CachimPB, FigueirasJA, PereiraPAA. 2002. Fatigue behavior of fiber-reinforced concrete in compression. Cem. Concr. Compos. 24(2): 211-217. 10.1016/S0958-9465(01)00019-1, 45HolmenJO. 1979. Fatigue of concrete by constant and variable amplitude loading [Ph.D. thesis]. University of Trondheim, Trondheim, Norway.).

In conclusion, when testing specimens in compressive fatigue, it is highly recommended to use an i-BSJ for minimizing the uncertainties caused by manual centering misalignments in both monotonic and cyclic compressive tests.

3. Size effect in fatigue of SFRC

As anticipated in the introduction, we also studied the size effect on fatigue in self-compacting steel-fiber reinforced concrete specimens. Cubes of three sizes (40 mm, 80 mm, and 150 mm edge lengths) (26OrtegaJJ, RuizG, PovedaE, GonzálezDC, TarifaM, ZhangX, YuRC, VicenteMÁ, De La RosaÁ. 2022. Size effect on the compressive fatigue of fibre-reinforced concrete. Constr. Build. Mater.322: 126238. 10.1016/j.conbuildmat.2021.126238) and cylinders of three heights (75 mm, 200 mm, and 300 mm) were tested, with slenderness 2 for the cylinders (Figure 11) (27GonzálezDC, MenaA, RuizG, OrtegaJJ, PovedaE, MínguezJ, YuR, De La RosaÁ, VicenteMÁ. 2023. Size effect of steel fiber–reinforced concrete cylinders under compressive fatigue loading: Influence of the mesostructure. Int. J. Fatigue. 167(B): 107353. 10.1016/j.ijfatigue.2022.107353). All the specimens were made from the same self-compacting steel fiber reinforced concrete. The composition is detailed in Table 1. The fiber content was 0.3% by volume, and the type used was straight fibers, specifically Dramix OL 13/.20 from Bekaert, measuring 13 mm in length with an aspect ratio of 65. This length was selected to ensure that the fibers remained sufficiently short relative to the smallest specimen size, maintaining the homogeneity of mechanical properties throughout the material.

FIGURE 11. (a) SFRC cubes of three sizes: 40 mm, 80 mm, and 150 mm; (b) individualized ball and socket joints for the three cubes (26OrtegaJJ, RuizG, PovedaE, GonzálezDC, TarifaM, ZhangX, YuRC, VicenteMÁ, De La RosaÁ. 2022. Size effect on the compressive fatigue of fibre-reinforced concrete. Constr. Build. Mater.322: 126238. 10.1016/j.conbuildmat.2021.126238); (c) SFRC cylinders of three heights: 150 mm, 200 mm, and 300 mm; (d) fatigue test of one of the medium cylinders (27GonzálezDC, MenaA, RuizG, OrtegaJJ, PovedaE, MínguezJ, YuR, De La RosaÁ, VicenteMÁ. 2023. Size effect of steel fiber–reinforced concrete cylinders under compressive fatigue loading: Influence of the mesostructure. Int. J. Fatigue. 167(B): 107353. 10.1016/j.ijfatigue.2022.107353). 

Cubes and cylinders were stored in a humid chamber maintained at 20°C and 97% relative humidity for at least one year after demolding. The mean compressive strength of the hardened concrete was measured at 28 days. The results showed an average of 26.4 MPa from cylinders with a diameter of 75 mm and a height of 150 mm, and 28.2 MPa from cubes with an edge length of 100 mm. Additionally, the compressive strength of cylinders with the same dimensions was tested at 56 days and yielded an average of 30.6 MPa. The properties exhibited a high degree of homogeneity, as indicated by their small standard deviations (26OrtegaJJ, RuizG, PovedaE, GonzálezDC, TarifaM, ZhangX, YuRC, VicenteMÁ, De La RosaÁ. 2022. Size effect on the compressive fatigue of fibre-reinforced concrete. Constr. Build. Mater.322: 126238. 10.1016/j.conbuildmat.2021.126238, 27GonzálezDC, MenaA, RuizG, OrtegaJJ, PovedaE, MínguezJ, YuR, De La RosaÁ, VicenteMÁ. 2023. Size effect of steel fiber–reinforced concrete cylinders under compressive fatigue loading: Influence of the mesostructure. Int. J. Fatigue. 167(B): 107353. 10.1016/j.ijfatigue.2022.107353).

For each size, 10 quasi-static tests were conducted right before fatigue testing to determine their reference compressive strength, followed by 15 fatigue tests with cyclic loads ranging from 0.20 to 0.85 of the compressive strength. As strain along the loading direction in cubes is not uniform due to the confining effect of friction in contact with the platens, we preferred taking the average strain, i.e. total deformation over the cube’s edge length, measured as the average between the readings of two LVDTs as a representative value. As for the cylinders, they have a slenderness of 2, so the influence of the contact with the platens in the central section is negligible, and thus we took the strain in the central point of the height, actually the average between the recordings of three strain gauges at 120^◦.

A key strength of this methodology lies in the use of diverse equipment to control tests and assess internal changes, including ball-and-socket joints to minimize misalignment and devices like a computed tomography scanner and mercury porosimeter to analyze internal structure and failure modes. Additionally, the number of specimens allowed for a more accurate representation of the probabilistic nature of both quasi-static and fatigue results.

For cubes, the reference compressive strength was consistent across all sizes, with means of 30.5 MPa, 30.6 MPa, and 30.7 MPa for small, intermediate, and large cubes, respectively. This lack of size effect reflects the material’s high post-peak ductility from fiber reinforcement, leading to a failure closer to plastic behavior than quasi-brittle. This may change for larger sizes (1Bažant ZP, Planas J. 1998. Fracture and size effect in concrete and other quasibrittle materials. CRC Press, Boca Raton, Florida, USA., 2Del VisoJR, CarmonaJR, RuizG. 2008. Shape and size effects on the compressive strength of high-strength concrete. Cem. Concr. Res. 38(3): 386-395. 10.1016/j.cemconres.2007.09.020, 13CarmonaJR, Cortés-BuitragoR, Rey-ReyJ, RuizG. 2022. Planar crack approach to evaluate the flexural strength of fiber-reinforced concrete sections. Materials. 15(17): 5821. 10.3390/ma15175821). In contrast, the quasi-static reference compressive strength of cylinders displayed a slight inverse size effect: the mean compressive strength increased from 26.3 MPa for small cylinders to 32.5 MPa for large ones. This was attributed to a reduction in porosity near the cylinder surfaces as size increased, as detected by the computed tomography study (27GonzálezDC, MenaA, RuizG, OrtegaJJ, PovedaE, MínguezJ, YuR, De La RosaÁ, VicenteMÁ. 2023. Size effect of steel fiber–reinforced concrete cylinders under compressive fatigue loading: Influence of the mesostructure. Int. J. Fatigue. 167(B): 107353. 10.1016/j.ijfatigue.2022.107353, 58GonzálezDC, Mena-AlonsoÁ, PovedaE, MínguezJ, De La RosaÁ, YuRC, RuizG, VicenteMA. 2024. Study of the edge effect in fiber-reinforced concrete cylindrical molded specimens using computed tomography. In: MechtcherineV, SignoriniC, JungerD, editors. Transforming Construction: Advances in Fiber Reinforced Concrete, BEFIB 2024. RILEM Bookseries54. p. 27-34.). The inverse size effect impacts fatigue behavior: smaller cylinders experience lower maximum and minimum loads, contributing to longer fatigue lives, while larger cylinders benefit from higher static strength. However, other size-dependent factors may counteract these effects.

In terms of fatigue life, 80 mm and 150 mm cubes showed similar Weibull distributions, while 40 mm cubes exhibited much higher scatter, spanning over two orders of magnitude at 50% failure probability (Figure 12). Many smaller cubes survived longer cycles, indicating an autogenous self-healing effect. Post-fatigue monotonic tests on run-out cubes revealed a 42% strength increase compared to initial values (43.2 MPa compared with the initial 30.5 MPa). This could be due to cyclic loading inducing microcracks, allowing water to react with unhydrated cement and thus improving strength. Cylinders followed a similar pattern: small cylinders had longer lives compared to larger ones, suggesting self-healing, though not as pronounced as with cubes. No runouts occurred in the cylinder tests.

FIGURE 12. Plot representing the cumulative probability of failure vs. fatigue life of cubes and cylinders of three sizes made of the same SFRC (26OrtegaJJ, RuizG, PovedaE, GonzálezDC, TarifaM, ZhangX, YuRC, VicenteMÁ, De La RosaÁ. 2022. Size effect on the compressive fatigue of fibre-reinforced concrete. Constr. Build. Mater.322: 126238. 10.1016/j.conbuildmat.2021.126238, 27GonzálezDC, MenaA, RuizG, OrtegaJJ, PovedaE, MínguezJ, YuR, De La RosaÁ, VicenteMÁ. 2023. Size effect of steel fiber–reinforced concrete cylinders under compressive fatigue loading: Influence of the mesostructure. Int. J. Fatigue. 167(B): 107353. 10.1016/j.ijfatigue.2022.107353). 

It is important to emphasize that all the cylinder and cube specimens, which results are shown in Figure 12 covering seven orders of magnitude in N, were made of the same material, and all tests were conducted at 20% to 85% of the reference compressive strength of the respective series. The findings indicate that the size of the specimen has a significant impact on fatigue life. Additionally, the specimen’s shape has a considerable effect on fatigue behavior, with cylinders being more conservative than cubes. We believe that cubes provide more confinement to the material, facilitating the extension of micro-damage in a stable manner and allowing self-healing to happen in all these damaged regions, as discussed in the next section. These results highlight the significant size effect in fatigue testing, cautioning against using small specimen data to characterize structural element behavior.

4. Autogenous self-healing due to fatigue

During our study of fatigue aspects of SFRC, we observed significant improvement in the material after compressive fatigue loading. However, there were various hypotheses to explain this behavior that lacked sufficient evidence. To address this, we initiated further research to identify the changes in the material’s microstructure caused by compressive fatigue and to determine if there is an increase in the production of hydrated compounds due to the fatigue mechanical actions.

As we had found that cubes, not cylinders, experienced runouts (see previous section), we suspected that the confinement by friction against the platens was causing extensive microdamage due to fatigue, which we believe is associated with the autogenous self-healing phenomenon. Cylinders with a slenderness of 2 have less volume affected by friction with the platens. So, microdamage extension is smaller than in cubes, making them less likely to develop autogenous self-healing. Note that this does not mean cylinders do not experience self-healing since small cylinders endure longer than taller ones, but it is less noticeable than in cubes. Thus, we used only cubes for our new experimental campaign, subjecting specimens to programmed fatigue loading.

For this research on autogenous self-healing induced by fatigue, we collaborated with experts in the chemistry of cement and concrete at the Institute for Construction Sciences “Eduardo Torroja”. They designed and fabricated the SFCR (see the composition in Table 2) and conducted and analyzed physical tests that complemented our fatigue tests. The result was a publication on the autogenous self-healing of SFRC (28De La RosaÁ, OrtegaJJ, RuizG, García CalvoJL, Rubiano SánchezFJ, CastilloÁ. 2023. Autogenous self-healing induced by compressive fatigue in self-compacting steel-fiber reinforced concrete. Cem. Concr. Res. 173: 107278. 10.1016/j.cemconres.2023.107278), which we will describe in this section.

4.1. Mechanical tests

 ⌅

4.1.1. Compressive tests

 ⌅

Compressive tests were conducted on six 100 mm cubes, all made with the corresponding ball-and-socket joint, to determine their reference compressive strength just before the beginning of fatigue tests, when the SFRC was four months old. The cubes were kept in a humid chamber until then. In Figure 13a, one of the cubes is shown immediately after testing. The average value obtained was 77.6 MPa, with a small standard deviation of 2.4 MPa, resulting in a coefficient of variation of only 3%. This indicates that the mechanical behavior of this concrete for this specimen size is very consistent. Using the ball-and-socket joint contributed to this result by eliminating the source of scatter caused by load eccentricity, which is unrelated to the material (as shown in Subsection 2.5).

FIGURE 13. (a) 80 mm cube of a series of high strength SFRC tested monotonically; (b) fatigue results of this material for four stress levels (28De La RosaÁ, OrtegaJJ, RuizG, García CalvoJL, Rubiano SánchezFJ, CastilloÁ. 2023. Autogenous self-healing induced by compressive fatigue in self-compacting steel-fiber reinforced concrete. Cem. Concr. Res. 173: 107278. 10.1016/j.cemconres.2023.107278); the dashed black straight line is the S-N curve given by the Model Code 2010 (54Comité Euro-International du Béton (CEB). 1988. Fatigue of concrete structures. State of the art report. CEB Report188, Lausanne.) and the dashed grey line is given by the model of Saucedo et al. (59SaucedoL, YuRC, MedeirosA, ZhangXX, RuizG. 2013. A probabilistic fatigue model based on the initial distribution to consider frequency effect in plain and fiber reinforced concrete. Int. J. Fatigue. 48: 308-318. 10.1016/j.ijfatigue.2012.11.013) for a probability of failure of 50%. 

4.1.2. Fatigue tests

 ⌅

After determining the mean strength, we carried out fatigue tests at four stress ranges, all starting from minimum stress of 5% the minimum reference strength to 70%, 75%, 80%, and 90% the reference strength. The number of tests for each stress level was determined based on the expected chances of specimen failure. In Figure 13b, the S-N curves show the relationship between the maximum stress level S_c,max = σ_max/f_c and the logarithm of the number of cycles N. As expected, the mean number of cycles and the number of runout specimens increase as the maximum stress decreases. Runout specimens are denoted by markers with an arrow that mainly overlap, as they represent points with similar coordinates (for instance, there were 3 runouts for S_c,max = 0.8, and 5 for S_c,max = 0.75, which are almost overlapped in Figure 13b). The distribution of the results is compared with the fatigue life prediction model included in the Model Code 2010 (54Comité Euro-International du Béton (CEB). 1988. Fatigue of concrete structures. State of the art report. CEB Report188, Lausanne.) for the minimum stress level used. Most of the points lie to the right of the model, so its prediction can be considered conservative, at least for this type of fiber reinforced concrete.

To accurately describe the material’s behavior, the probabilistic model by Saucedo et al. (59SaucedoL, YuRC, MedeirosA, ZhangXX, RuizG. 2013. A probabilistic fatigue model based on the initial distribution to consider frequency effect in plain and fiber reinforced concrete. Int. J. Fatigue. 48: 308-318. 10.1016/j.ijfatigue.2012.11.013) has been applied to fit the experimental data. This model incorporates the initial distribution, corresponding to quasi-static tests as the failure limit for one cycle while accounting for the dynamic effects of loading frequency. Consequently, the extrapolation for log N = 0 exceeds S_c,max = 1. The fatigue data fitting was performed for S_c,max = 0.9, a case with no runouts, providing complete statistical information. The model line in Figure 13b represents a 50% failure probability, illustrating the estimated mean behavior for other stress levels, assuming the life estimate given by the Sparks and Menzies’ law (52SparksPR, MenziesJB. 1973. Effect of rate of loading upon static and fatigue strengths of plain concrete in compression. Mag. Concr. Res. 25(83): 73-80. 10.1680/macr.1973.25.83.73) for the runouts.

4.1.3. Fatigue strain

 ⌅

During each fatigue test, strain evolution relative to the number of loading cycles n was recorded. As illustrated in Figure 3, the strain evolution until failure typically follows three stages: primary strain, characterized by rapid growth in the initial cycle interval; secondary strain, exhibiting uniform and stable behavior for most of the test; and tertiary strain, marked by accelerated strain before failure. The stability and duration of the secondary strain make it representative of the material’s fatigue creep behavior. The average slope of this phase, or secondary strain rate per cycle d𝜖/dn, is related to the cycles to failure N. For all failed specimens, the data align along a straight line, confirming Sparks and Menzies’ law (52SparksPR, MenziesJB. 1973. Effect of rate of loading upon static and fatigue strengths of plain concrete in compression. Mag. Concr. Res. 25(83): 73-80. 10.1680/macr.1973.25.83.73) (see Figure 4). Fitting the strain-based failure model of Eq. 1 to the experimental data yields parameters s = -1.103 and m = -2.466, with a determination coefficient of 0.984. Previous studies (14PovedaE, RuizG, CifuentesH, YuRC, ZhangXX. 2017. Influence of the fiber content on the compressive low-cycle fatigue behavior of self-compacting SFRC. Int. J. Fatigue. 101(1): 9-17. 10.1016/j.ijfatigue.2017.04.005, 26OrtegaJJ, RuizG, PovedaE, GonzálezDC, TarifaM, ZhangX, YuRC, VicenteMÁ, De La RosaÁ. 2022. Size effect on the compressive fatigue of fibre-reinforced concrete. Constr. Build. Mater.322: 126238. 10.1016/j.conbuildmat.2021.126238, 52SparksPR, MenziesJB. 1973. Effect of rate of loading upon static and fatigue strengths of plain concrete in compression. Mag. Concr. Res. 25(83): 73-80. 10.1680/macr.1973.25.83.73) demonstrated this relationship remains constant across variables such as loading frequency (17MedeirosA, ZhangXX, RuizG, YuRC, VelascoMSL. 2015. Effect of the loading frequency on the compressive fatigue behavior of plain and fiber reinforced concrete. Int. J. Fatigue. 70: 342-350. 10.1016/j.ijfatigue.2014.08.005.), specimen size (26OrtegaJJ, RuizG, PovedaE, GonzálezDC, TarifaM, ZhangX, YuRC, VicenteMÁ, De La RosaÁ. 2022. Size effect on the compressive fatigue of fibre-reinforced concrete. Constr. Build. Mater.322: 126238. 10.1016/j.conbuildmat.2021.126238, 27GonzálezDC, MenaA, RuizG, OrtegaJJ, PovedaE, MínguezJ, YuR, De La RosaÁ, VicenteMÁ. 2023. Size effect of steel fiber–reinforced concrete cylinders under compressive fatigue loading: Influence of the mesostructure. Int. J. Fatigue. 167(B): 107353. 10.1016/j.ijfatigue.2022.107353), and fiber content (14PovedaE, RuizG, CifuentesH, YuRC, ZhangXX. 2017. Influence of the fiber content on the compressive low-cycle fatigue behavior of self-compacting SFRC. Int. J. Fatigue. 101(1): 9-17. 10.1016/j.ijfatigue.2017.04.005) for fiber reinforced concrete.

While prior work focused on a single stress range, this study extended theanalysis to three stress ranges, considering only fatigue-failed specimens. The results confirm the relationship between dE/dn and N remains constant across stress levels. Higher fatigue loads correspond to higher strain rates and shorter fatigue lives (points shift left), while lower loads result in lower strain rates and longer fatigue lives. However, all points follow the same linear trend, suggesting this relationship is a material property. As mentioned above, for runout specimens, the fatigue life can be estimated from the secondary strain rate given by the second stretch of their cyclic creep curves (14PovedaE, RuizG, CifuentesH, YuRC, ZhangXX. 2017. Influence of the fiber content on the compressive low-cycle fatigue behavior of self-compacting SFRC. Int. J. Fatigue. 101(1): 9-17. 10.1016/j.ijfatigue.2017.04.005), or also using the master cyclic curve developed by Blasón et al. (60BlasónS, PovedaE, RuizG, CifuentesH, Fernández CanteliA. 2019. Twofold normalization of the cyclic creep curve of plain and steel-fiber reinforced concrete and its application to predict fatigue failure. Int. J. Fatigue. 120: 215-227. 10.1016/j.ijfatigue.2018.11.021, 61BlasónS, Fernández CanteliA, PovedaE, RuizG, YuRC, CastilloE. 2022. Damage evolution and probabilistic strain-lifetime assessment of plain and fiber-reinforced concrete under compressive fatigue loading: Dual and integral phenomenological model. Int. J. Fatigue. 158: 106739. 10.1016/j.ijfatigue.2022.106739).

4.1.4. Compressive strength of runout specimens

 ⌅

Runout specimens were immediately tested under quasi-static conditions until failure after the fatigue tests. Nine runout specimens were tested, yielding a new mean strength of 95.4 MPa, 23% higher than the original strength. The low standard deviation, with a coefficient of variation around 2%, indicates that the strength increase occurred consistently across all specimens. While strength gain due to aging could be a factor, the specimens were tested at different ages over three months, yet no significant strength evolution was observed. Additionally, the compressive strength of the material at the end of the fatigue tests was checked using the last remaining specimen, yielding compressive strength = 87.8 MPa. Although this single value reflects a 13% natural strength increase due to aging, it is still lower than the strength observed in the fatigue-tested specimens. Thus, a significant part of the strength increase in the runout specimens is likely attributable to the cyclic loading.

The following subsections present analyses of microstructural and compositional changes in concrete subjected to cyclic loads compared to a control specimen.

4.2. Physical tests

 ⌅

Physical tests were performed on one of the runouts and on one intact cube serving as a control specimen. The selected runout corresponds to the sixth specimen from the maximum stress level of 0.75 compressive strength, representing the last fatigue test performed. Both runout and control specimens were preserved under the same conditions. Core samples were taken from them and dried in an oven at 100°C for 24 hours to remove moisture and slow down the hydration process as much as possible. Ideally, the samples should have been immersed in ethanol or propanol to completely stop hydration, but since both samples underwent the same treatment, this fact is not highly relevant when comparing their results to observe the effect of fatigue loads on the self-healing process. After the 24-hour drying period, the samples were analyzed, and physical tests (X-ray diffraction, thermogravimetric analysis, etc.) were performed immediately, with all analyses completed within the following 24 hours.

4.2.1. X-Ray Diffraction (XRD)

 ⌅

The X-ray diffraction (XRD) results for the control and runout samples are presented in Figure 14a. No significant differences were observed between the two samples, either in the types of crystalline hydrates formed or their relative amounts. Therefore, if the self-healing process that occurred during the fatigue tests is associated with the formation of hydrated phases, it is likely related to the less crystalline ones, primarily calcium-silicate-hydrate (C-S-H) gels.

FIGURE 14. Results of physical tests for one the runouts and the non-fatigued control specimen: (a) X-Ray diffractogram; (b) thermogravimetric analysis; and (c) differential thermal analysis (28De La RosaÁ, OrtegaJJ, RuizG, García CalvoJL, Rubiano SánchezFJ, CastilloÁ. 2023. Autogenous self-healing induced by compressive fatigue in self-compacting steel-fiber reinforced concrete. Cem. Concr. Res. 173: 107278. 10.1016/j.cemconres.2023.107278). 

4.2.2. Thermogravimetric Analysis (TGA) and Differential Thermal Analysis (DTA)

 ⌅

Figure 14b illustrates the endothermic heat flow, while Figure 14c depicts the weight loss of the samples during the decomposition process as the temperature rises. The typical peaks observed in cementitious materials are present: the first peak corresponds primarily to the dehydration of ettringite and calcium silicate hydrate (C-S-H) gels; the second peak is associated with the dehydration of portlandite; and the third peak relates to carbonated phases (62ScrivenerK, SnellingsR, LothenbachB, editors. 2016. A practical guide to microstructural analysis of cementitious materials. CRC Press, Taylor & Francis Group. ISBN-13: 978-1-4987-3867-5.).

Although both concretes show similar results, a key difference arises when comparing hydrate contents from TGA results. The portlandite content is similar in both concretes (Table 3), but the bound water content is higher in the runout sample, indicating an increase in hydrates due to fatigue exposure. This increase could be attributed to further hydration of anhydrous phases or pozzolanic reactions between portlandite and silica fume. Since portlandite content is unchanged, cement hydration seems to be the primary mechanism, although pozzolanic reactions cannot be ruled out.

TABLE 3. Portlandite and bound water measured by DTA/TG (in weight percentage). 

Sample	Portlandite	Bound water
Runout	2.0	9.7
Control	2.0	9.4

The additional 3% of hydrates formed in the runout sample, though modest, may contribute to healing microcracks generated during fatigue cycles or at least to a partial self-healing process. Detachment of small particles during fatigue may also fill these microcracks, enhancing compressive strength. However, backscattered scanning electron microscopy (BSEM) evaluations in the next section do not clearly detect such detached particles, though hydrated phases are observed growing within microcracks.

These findings underscore the formation of new C-S-H gel after cyclic compressive loading, crucial for early-age strength gains. Notably, no increase in carbonate phases is observed in the runout sample, suggesting carbonation is not the primary driver of self-healing. However, this does not rule out its occurrence, as discussed in the next section. The movement of free water within the concrete matrix, promoted by cyclic loading, likely facilitates hydrate formation, healing incipient cracks generated by the load.

4.2.3. Backscattered Scanning Electron Microscopy (BSEM)

 ⌅

Figure 15a presents an BSEM image of the surface of the runout sample, highlighting the fatigue-induced damage. Cracks are visible in the aggregate particle and cementitious matrix, particularly in the C-S-H gel phases, surrounding nonhydrated cement particles, portlandite crystals, or near pores, which act as stress concentration zones. Some cracks stop propagating and fade out.

FIGURE 15. (a) BSEM image of a runout specimen, and (b) its corresponding spectral analysis (28De La RosaÁ, OrtegaJJ, RuizG, García CalvoJL, Rubiano SánchezFJ, CastilloÁ. 2023. Autogenous self-healing induced by compressive fatigue in self-compacting steel-fiber reinforced concrete. Cem. Concr. Res. 173: 107278. 10.1016/j.cemconres.2023.107278). 

Figure 15b shows the spectral analysis of the minerals within the rectangular area in Figure 15a, focusing on a smooth surface identified as a siliceous aggregate particle through spectral analysis. Additional images and analyses, particularly of the interfacial transition zone (ITZ), reveal new C-S-H gel phases overlapping cracks in the ITZ and new hydrated C-S-H phases near a silica fume particle, likely formed by pozzolanic reactions, halting crack propagation (see Figures 10-12 in (28De La RosaÁ, OrtegaJJ, RuizG, García CalvoJL, Rubiano SánchezFJ, CastilloÁ. 2023. Autogenous self-healing induced by compressive fatigue in self-compacting steel-fiber reinforced concrete. Cem. Concr. Res. 173: 107278. 10.1016/j.cemconres.2023.107278)). A significant aluminum peak suggests the formation of C-A-S-H gels, as previously reported (63–65FernándezA, AlonsoMC, García-CalvoJL, LothenbachB. 2016. Influence of the synergy between mineral additions and Portland cement in the physical-mechanical properties of ternary binders. Mater. Construcc. 66(324). 10.3989/mc.2016.10815), which consist of longer silica chains and play a role in halting crack propagation. Additionally, potential portlandite carbonation may form calcite, releasing water into the system (66Von Greve-DierfeldS, LothenbachB, VollprachtA, WuB, HuetB, AndradeC, MedinaC, ThielC, GruyaertE, VanoutriveH, et al. 2020. Understanding the carbonation of concrete with supplementary cementitious materials: A critical review by RILEM TC 281-CCC. Mater. Struct. 53(6): 136. 10.1617/s11527-020-01558-w, 67MorandeauA, TognazziC, DanglaP. 2014. Investigation of the carbonation mechanism of CH and C-S-H in terms of kinetics, microstructure changes and moisture properties. Cem. Concr. Res. 56: 153-170. 10.1016/j.cemconres.2013.11.015), further promoting hydration under cyclic compressive loads. Thus, SEM analysis suggests that self-healing under compressive cyclic loading is driven by both hydration of anhydrous phases and pozzolanic reactions forming C-A-S-H gels.

4.2.4. Mercury Intrusion Porosimetry (MIP)

 ⌅

Figure 16 shows the pore size distribution for both specimens, with peaks corresponding to predominant pore sizes that account for higher void volume. Peaks below 1 µm are similarly distributed, but beyond this size, differences emerge. Around 10 µm, the runout specimen exhibits a peak above the control specimen, indicating greater void volume, likely due to crack openings caused during compressive testing. These openings would depend on the damage level of each specimen. Conversely, smaller size differences are more indicative of the concrete matrix’s microstructure. Notably, the runout specimen shows a clear reduction in pore volume around 0.04 µm, and between 1 µm and 5 µm, as seen in the logarithmic plot of injected mercury per sample mass (Figure 16).

FIGURE 16. Porosimetries of both runout and control specimens (28De La RosaÁ, OrtegaJJ, RuizG, García CalvoJL, Rubiano SánchezFJ, CastilloÁ. 2023. Autogenous self-healing induced by compressive fatigue in self-compacting steel-fiber reinforced concrete. Cem. Concr. Res. 173: 107278. 10.1016/j.cemconres.2023.107278). 

Table 4 presents additional MIP analysis results. The runout specimen shows a lower total pore area and average pore diameter compared to the control specimen. Tortuosity, defined as the ratio of actual flow path length to the straight distance between the path ends, is also reduced, with the runout specimen at 56% of the control’s value. These results suggest a reduction in pore volume and closure of microcracking paths after the application of cyclic compressive loads.

TABLE 4. Parameters obtained by MIP. 

Parameter	Runout	Control
Total pore area (m2/g)	2.23	2.80
Average pore diameter (nm)	55.6	68.4
Tortuosity	5.17	9.18

5. Discussion

The results support the notion that cyclic compressive loads induce an autogenous self-healing process in the material. The primary causes of this phenomenon are analyzed below.

Considering the types of water and voids within the hydrated cement paste is crucial to understanding autogenous self-healing in concrete. Capillary water, which is not bound to the solid surface, plays a vital role in the self-healing process. Capillary voids—spaces not filled by hydration products, see Table 5 —depend on the degree of hydration (68YeG. 2003. Experimental study and numerical simulation of the development of the microstructure and permeability of cementitious materials [Ph.D. thesis]. T.U. Delft, Delft University Press.). For autogenous self-healing to take place, there must be sufficient free water available. The movement of water into cracks is influenced by factors such as porosity, pore structure, and connectivity (Figure 16, Table 4). Although the concrete examined in the previous section has a low water-binder ratio, its short age and humid curing conditions are conducive to potential self-healing.

Cyclic compressive loads exert simultaneous damage and healing effects. Whether healing or damage dominates influences the residual compressive strength and fatigue life (as sketched in Figure 17). As cyclic loads generate microcracks, water moves through them (70WangL, ZhangW. 2021. Investigation on water absorption in concrete after subjected to compressive fatigue loading. Constr. Build. Mater. 299: 123897. 10.1016/j.conbuildmat.2021.123897, 71LeiB, LiW, LiZ, WangG, SunZ. 2018. Effect of cyclic loading deterioration on concrete durability: Water absorption, freeze-thaw, and carbonation. J. Mater. Civ. Eng. 30(9): 04018220. 10.1061/(ASCE)MT.1943-5533.0002450), reaching other areas via flow and diffusion (70WangL, ZhangW. 2021. Investigation on water absorption in concrete after subjected to compressive fatigue loading. Constr. Build. Mater. 299: 123897. 10.1016/j.conbuildmat.2021.123897), as schematically represented in Figure 18. The dynamic pressure from water at crack tips induces stress, potentially deepening cracks (72SunX, TianY, YinW, WangH. 2022. Effect of free water on fatigue performance of concrete subjected to compressive cyclic load. Constr. Build. Mater. 318: 125995. 10.1016/j.conbuildmat.2021.125995). Higher pressure and larger crack openings accelerate microcrack growth, reducing fatigue life if no self-healing occurs (73ZhangY, ZhangS, WeiG, WeiX, JinL, XuK. 2019. Water transport in unsaturated cracked concrete under pressure. Adv. Civ. Eng. 2019(1): 4504892. 10.1155/2019/4504892).

FIGURE 18. Plausible autogenous self-healing mechanisms of cracks in SFRC [28De La RosaÁ, OrtegaJJ, RuizG, García CalvoJL, Rubiano SánchezFJ, CastilloÁ. 2023. Autogenous self-healing induced by compressive fatigue in self-compacting steel-fiber reinforced concrete. Cem. Concr. Res. 173: 107278. 10.1016/j.cemconres.2023.107278]. 

However, self-healing comes about through the hydration of non-hydrated particles in young concrete, while carbonation dominates in older concrete (74Van TittelboomK, De BelieN. 2013. Self-healing in cementitious materials—A review. Materials. 6(6): 2182-2217. 10.3390/ma6062182). Water movement through cracks (Figure 18) promotes ongoing hydration, carbonation, and dissolution reactions, creating complex chemical processes that influence self-healing (75YangS, YangY, CaggianoA, UkrainczykN, KoendersE. 2022. A phase-field approach for portlandite carbonation and application to self-healing cementitious materials. Mater. Struct. 55(46). 10.1617/s11527-022-01887-y). Crack morphology affects healing, with faster product formation at crack tips (76YangS, AldakheelF, CaggianoA, WriggersP, KoendersE. 2020. A review on cementitious self-healing and the potential of phase-field methods for modeling crack-closing and fracture recovery. Materials. 13(22): 5265. 10.3390/ma13225265).

The primary physical cause of self-healing is the expansion of hydration products near microcracks when exposed to water (77YaoC, ShenA, GuoY, LyuZ, HeZ, WuH. 2022. A review on autogenous self‐healing behavior of ultra‐high performance fiber reinforced concrete (UHPFRC). Arch. Civ. Mech. Eng. 22(145). 10.1007/s43452-022-00462-0). The main product is CaCO₃ (77YaoC, ShenA, GuoY, LyuZ, HeZ, WuH. 2022. A review on autogenous self‐healing behavior of ultra‐high performance fiber reinforced concrete (UHPFRC). Arch. Civ. Mech. Eng. 22(145). 10.1007/s43452-022-00462-0), and non-hydrated particles also form portlandite and C-S-H gel. However, C-S-H gel primarily accumulates on crack surfaces, not within cracks (77YaoC, ShenA, GuoY, LyuZ, HeZ, WuH. 2022. A review on autogenous self‐healing behavior of ultra‐high performance fiber reinforced concrete (UHPFRC). Arch. Civ. Mech. Eng. 22(145). 10.1007/s43452-022-00462-0). Self-healing near crack surfaces is driven by rapid crystalline precipitation. Continuous hydration enhances early-age self-healing due to higher portlandite and moisture content (36Roig-FloresM, FormaginiS, SernaP. 2021. Self-healing concrete: What is it good for?Mater. Construcc. 71(341): e237. 10.3989/mc.2021.07320), although carbonation may play a greater role (36Roig-FloresM, FormaginiS, SernaP. 2021. Self-healing concrete: What is it good for?Mater. Construcc. 71(341): e237. 10.3989/mc.2021.07320). These reactions alter concrete’s microstructure, affecting porosity, permeability, and hydration (78JavierreE, GasparF, RodrigoC. 2019. Modelling the carbonation reactions in self-healing concrete. In: 10th International Conference on Fracture Mechanics of Concrete and Concrete Structures, FraMCoS-X. 10.21012/FC10.235637, 79ChitezA, JeffersonA. 2016. A coupled thermo-hygro-chemical model for characterising autogenous healing in ordinary cementitious materials. Cem. Concr. Res. 88: 184-197. 10.1016/j.cemconres.2016.07.002).

5.4. Effect of the damage distribution and size effect

 ⌅

The self-healing process competes with fatigue-induced damage, and understanding their interaction is crucial, as the material response depends on which mechanism dominates (Figure 17). Fatigue damage in cementitious materials primarily arises from the initiation and propagation of microcracks (105GanY, ZhangH, ZhangY, XuY, SchlangenE, van BreugelK, SavijaB. 2021. Experimental study of flexural fatigue behaviour of cement paste at the microscale. Int. J. Fatigue. 151: 106378. 10.1016/j.ijfatigue.2021.106378). To interpret overall fatigue behavior, including the interfacial transition zone (ITZ), it is important to examine fatigue behavior at different scales.

FIGURE 17. Effect of the autogenous self-healing due to fatigue on the cyclic creep curves. Blue curves schematically represent specimens experiencing significant self-healing, thus enduring longer —solid line— or surviving —dashed line— than another specimen of the same material with no or minimal self-healing —orange curve—. 

At the nanoscale, crack propagation is governed by the breaking of nanoparticle connections, particularly the rupture of C-S-H gel, which propagates until merging into the main crack (105GanY, ZhangH, ZhangY, XuY, SchlangenE, van BreugelK, SavijaB. 2021. Experimental study of flexural fatigue behaviour of cement paste at the microscale. Int. J. Fatigue. 151: 106378. 10.1016/j.ijfatigue.2021.106378). Weakening localized stress concentrations through pore structure refinement or connecting nano-cracks allows a broader distribution of microcracks, promoting energy dissipation. Nano-crack propagation can be slowed by the bonding effect of nano-cracks and by the formation of denser C-S-H gel with higher polymerization, along with smaller calcium hydroxide crystals (70WangL, ZhangW. 2021. Investigation on water absorption in concrete after subjected to compressive fatigue loading. Constr. Build. Mater. 299: 123897. 10.1016/j.conbuildmat.2021.123897, 80LiL, XuL, HuangL, XuF, HuangY, CuiK, ZengY, ChiY. 2022. Compressive fatigue behaviors of ultra-high performance concrete containing coarse aggregate. Cem. Concr. Compos. 128: 104425. 10.1016/j.cemconcomp.2022.104425, 106CuiX, HanB, ZhengQ, YuX, DongS, ZhangL, OuJ. 2017. Mechanical properties and reinforcing mechanisms of cementitious composites with different types of multiwalled carbon nanotubes. Compos. Part A: Appl. Sci. Manuf. 103: 131-147. 10.1016/j.compositesa.2017.10.001).

At the microscale, existing cracks, especially in the ITZ, coalesce with fatigue cracks in the cement matrix (105GanY, ZhangH, ZhangY, XuY, SchlangenE, van BreugelK, SavijaB. 2021. Experimental study of flexural fatigue behaviour of cement paste at the microscale. Int. J. Fatigue. 151: 106378. 10.1016/j.ijfatigue.2021.106378). Due to its porosity, the ITZ is highly vulnerable to fatigue, exhibiting a higher likelihood of crack initiation and propagation compared to the matrix (105, 107–109). Faster crack growth in the ITZ accelerates damage, while increased damage distribution enhances autogenous self-healing by generating more microcracks and facilitating energy dissipation, thus strengthening the material.

The internal stress distribution, influenced by specimen size, further affects self-healing potential, as previously noted in Section 3 (26OrtegaJJ, RuizG, PovedaE, GonzálezDC, TarifaM, ZhangX, YuRC, VicenteMÁ, De La RosaÁ. 2022. Size effect on the compressive fatigue of fibre-reinforced concrete. Constr. Build. Mater.322: 126238. 10.1016/j.conbuildmat.2021.126238). Scale effects on crack propagation and fatigue life can be analyzed from the micro- to the macroscale (110BažantZP, XuKM. 1991. Size effect in fatigue fracture of concrete. ACI Mater. J. 88(4): 390-399., 111KiraneK, BažantZP. 2016. Size effect in Paris law and fatigue lifetimes for quasibrittle materials: Modified theory, experiments and micro-modeling. Int. J. Fatigue. 83(2): 209-220. 10.1016/j.ijfatigue.2015.10.015). From this perspective, smaller specimens result in a more distributed arrangement of microcracks, making them more susceptible to significant self-healing compared to larger specimens of the same material (26OrtegaJJ, RuizG, PovedaE, GonzálezDC, TarifaM, ZhangX, YuRC, VicenteMÁ, De La RosaÁ. 2022. Size effect on the compressive fatigue of fibre-reinforced concrete. Constr. Build. Mater.322: 126238. 10.1016/j.conbuildmat.2021.126238, 27GonzálezDC, MenaA, RuizG, OrtegaJJ, PovedaE, MínguezJ, YuR, De La RosaÁ, VicenteMÁ. 2023. Size effect of steel fiber–reinforced concrete cylinders under compressive fatigue loading: Influence of the mesostructure. Int. J. Fatigue. 167(B): 107353. 10.1016/j.ijfatigue.2022.107353).

6. Conclusions

This article examined the effect of size on the compressive fatigue of steel fiber reinforced concrete, described the autogenous self-healing of this material resulting from compressive fatigue, and the interactions between them. Additionally, it included a review of fatigue phenomenology to familiarize the reader with the topic, drawing mainly from the authors’ contributions to the state of the art. The research outlined in the paper yielded several relevant conclusions:

Overall, this research highlights the complex interaction between fatigue loading, size effect, fiber reinforcement, and autogenous self-healing in SFRC. It underscores the need for careful consideration of these phenomena in both material design and the scaling of experimental results to full-scale structural elements.