Corrosion Fatigue Numerical Model for Austenitic and Lean-Duplex Stainless-Steel Rebars Exposed to Marine Environments

Steel rebars of structures exposed to cyclic loadings and marine environments suffer an accelerated deterioration process by corrosion fatigue, causing catastrophic failure before service life ends. Hence, stainless steel rebars have been emerging as a way of mitigating pitting corrosion contribution to fatigue, despite the increased cost. The present study proposes a corrosion fatigue semiempirical model. Different samples of rebars made of carbon steel, 304L austenitic (ASS), 316L ASS, 2205 duplex (DSS), 2304 lean duplex stainless steels (LDSS), and 2001 LDSS have been embedded in concrete and exposed to a tidal marine environment for 6 months. Corrosion rates of each steel rebar have been obtained from direct measurement and, considering rebar standard requirements for fatigue and fracture mechanics, an iterative numerical model has been developed to derive the cycles to failure for each stress range level. The model resulted in a corrosion pushing factor for each material, able to be used as an accelerating coefficient for the Palmgren-Miner linear rule and as a performance indicator. Carbon steel showed the worst performance, while 2001 LDSS performed 1.5 times better with the best cost-performance ratio, and finally 2205 DSS performed 1.5 times better than 2001 LDSS.


Introduction
Corrosion is one of the main pathologies that deteriorates reinforced concrete structures. For instance, when concrete structures are placed in corrosive environments, rich in CO 2 , the alkalinity of the concrete will tend to decrease, due to carbonation. This process will continue until the pH value decreases below the threshold of nine, according to the Pourbaix diagram for iron at 25 • C [1], losing its protective effect over carbon steel rebars [1,2]. The passive layers surrounding the rebars will then break down and both generalized and pitting corrosion will start to deteriorate them [3]. Many solutions have been proposed to counter such problems, namely: corrosion inhibitors in concrete [4,5], coatings applied to concrete surfaces and reinforcement surfaces, such as hydrophobic coatings, electrochemical rehabilitation and similar interventions [2]. Nevertheless, none of those treatments will definitively where n i is the number of cycles and N fi is the number of cycles to failure of each cycle block i of the same stress range ∆σ.
In this paper, an application for DSS rebars under marine environment is developed [39], based on the same corrosion pushing factor (F CP ) approach [26]. Thus, since F CP is an accelerating coefficient equivalent to corrosion leverage on the fatigue process, it can be used to set a comparative basis between different material alternatives. A marine environment can, for instance, be characterized as the tidal zone in a laboratory, one of the most aggressive environments. Six different steels were studied, namely: one CS (B500SD), two ASS (304L and 316L), one DSS (2205), and two LDSS (2304 and 2001).

Materials
The fatigue corrosion assessment was carried out in the XS3 exposure class (whose full definition is tidal, splash and spray zone) environment, which was simulated under controlled conditions in the laboratory [40]. The reinforcement specimens were embedded in C30/37 concrete with a w/c ratio of 0.5 and exposed to the XS3 environment for six months. As previously mentioned, six steels were studied, CS B500SD, 304L ASS, 316L ASS, 2205 DSS, and 2001 LDSS and 2304 LDSS. All steel-reinforcing specimens were 12 mm in diameter rebars. Table 1 summarizes the chemical composition of each reinforcement in accordance with EN10080:2005 and EN10088-1:2006 [41,42].

Methodology
This paper reports a semi-empirical research study. Accordingly, the corrosion fatigue assessment was performed in two steps; firstly, the corrosion rate (v c ), was measured in specimens exposed to a tidal environment, and all measurements were done in triplicate; then, fracture mechanics theory was considered for corrosion-fatigue life prediction, to build up equivalent S-N curves, as described in a previous study by the authors [22]. First of all, the specimens were exposed to simulated tidal marine environments for 6 months. The chloride content in a seawater tidal zone was assumed to be 3.5 wt.% NaCl, according to the literature [43,44]. The specimens were equally divided and placed in two polyvinyl chloride (PVC) tubs. Then, using automated pumping equipment and a timer, one tub was filled with water for 6 h while the other was left empty, exposed only to the atmosphere in the laboratory at 20 • C. After 6 h, the water was pumped into the empty tub, thereby simulating the conditions of a structure in a tidal zone.
Hence, the corrosion fatigue life prediction was based on the theoretical framework of fracture mechanics. First, the main variables involved in the corrosion fatigue process were defined. Those variables were the mean stress level (σ m ), the ∆σ, the cycle frequencies (f ), the N f and the v c . Once all the variables were defined, the modified, ∆σ-N, curves were plotted.
Finally, the semi-empirical procedure used in this study was a first generalized model, this meant that previously defined variables were only the minimum common ground for every stainless steel required to complete the procedure. Nevertheless, there were other important parameters that were indirectly considered, such as material structure, properties and metallurgical aspects. However, knowing how these metallurgical aspects affected mechanical properties and corrosion kinetics and, by means of this semi-empirical procedure, how to forecast the corrosion fatigue behaviour, it became possible to further develop new materials and processes with improved corrosion fatigue behavior.

Fatigue Parameters
Eurocode 2 (EC2) and the seminal European Convention for Constructional Steelwork (ECCS) document [32,45] define the fatigue resistance conditions for carbon steel reinforcements. As there is no specific directive for stainless steel reinforcements, their study would have to be based on EC2 requirements. Accordingly, a fatigue test must be performed according to UNE-EN ISO 15630-1 [46]. Table 2 summarizes the minimum requirements of rebars under cyclic loading like the N f , superior/maximum stress (σ sup ), ∆σ and f. EC2 defines two service-life periods depending on the infrastructure type and importance: 50 years and 100 years [32]. Considering N f as 2 × 10 6 cycles during such service-life periods, based on the constant amplitude fatigue limit (CAFL) approach at EC2, two frequencies can be derived: 1.27 × 10 −3 Hz and 6.34 × 10 −4 Hz, simply by dividing the expected amount of cycles N f by the corresponding service-life period in seconds. Another thing to consider is that the lower the f of the process, the higher the weight of the corrosion within the corrosion fatigue process [21,22,[26][27][28]. Accordingly, 6.34 × 10 −4 Hz was considered in this study, corresponding to a service-life period of 100 years.
Most structures are designed to withstand permanent and variable loads, but when corrosion is also involved, a doubt could arise over the development of stress corrosion cracking (SCC) process, which severely reduces the service lifetime of the structure. However, the development of corrosion fatigue and/or SCC depends on the magnitude of the variable and the permanent loads; since SCC requires σ m above 80% of the yield stress (σ y ), it is more unlikely to happen under normal working conditions. However, considering that EC2 [32] defines conservative safety factors for ultimate limit states (ULS), namely 1.35 for permanent load majoration (γ G ) and 1.15 for rebar resistance minoration (γ S ), even if all loads were permanent, the maximum stress at the rebars would be circa 64% σ y , if any such structure were in fact designed according to the standards, see Equation (2), therefore no SCC development would be expected in the rebars, unless the fatigue crack reached a large enough size for stress concentrations to get above 80% σ y . Nevertheless, according to Paris's law [35], when this occurs the fatigue propagation is so quick that the weight of any other processes is minimized and therefore negligible when compared to the mechanical fatigue.
where σ sup is the superior (maximum) stress.

Corrosion Parameters
As previously mentioned, all the specimens were exposed to simulated tidal marine environments [39] and a corrosion current density (i corr ) was measured after 6 months exposure with a Gecor8 Corrosimeter from GEOCISA (Madrid, Spain). Applying Faraday's law (2) defined in UNE 112072-2011 [48], the mass loss (m) in mdd, defined as mg/(dm 2 × day), is obtained.
where F is the Faraday constant (96,485 C/mol), P at is the atomic weight and n is the number of electrons. Then, a generalized corrosion rate (v corr.g ), was defined by considering the material density (ρ) in g/cm 3 (see Equation (3)). v corr.g = 36.5 × m ρ (4)

Calculation Procedure Overview
Now, for demonstrative purposes of the semi-empirical proposed approach, the generalized calculating procedure is applied to a single rebar of 25 mm diameter (circa-1 inch), made of carbon steel CS, with a yield limit of 500 MPa. According to current standards, any commercial rebar should survive after 2 × 10 6 cycles, whose ∆σ = 300 MPa, and σ m = 150 MPa, with a survival rate of 95% and a reliability of 75% [32,41,46,47]. Such rebars should remain under service for 50-100 years and are typically designed with reference to the CAFL, meaning mean design frequencies spanning from 6.34 × 10 −4 Hz to 1.27 × 10 −3 Hz, being the first finally chosen since they would maximize the corrosion contribution in the corrosion-fatigue process.
Besides, from the safety side, the Paris Law constants can be taken as A = 2 × 10 −13 and m = 3 for any steel [22,26,49,50]. Additionally, the plane strain fracture toughness, or critical stress intensity factor (K IC ) at mode I, is K IC = 3000 Nmm −3/2 , a parameter related with resilience, which barely changes between steel grades [22,26,50,51]. Thus, the critical crack size (a cr ) determining the failure point, can be derived from Equation (4), where Y cr = Y(a cr ) is the geometrical factor for rebars [52] and particularized for this critical crack size and σ sup = 300 MPa is the maximum stress reached at each cycle. Substituting these values into Equation (4) leads to a critical crack size a cr = 12.28 mm.
Nevertheless, the procedure to derive the initial crack size (a 0 ) is more sensible. For instance, some approaches can be based on the threshold stress intensity factor (Kth) [22,26,50], but factors potentially relevant for a fatigue failure, such as the surface finishing or the presence of corrugation, could be then obviated. Thus, the best approach is to start from the a cr and go backwards for 2 × 10 6 cycles until reaching the initial crack size. However, since the presence of a geometrical factor depending on crack size impedes any direct calculation, an iterative calculation method is developed, i.e., discretizing the geometrical factor making it constant for smaller cycle blocks or amounts and converging to a value. The final value of the equivalent a 0 now including surface finishing, etc. is then defined as a 0 = 0.012 mm and the amount of cycles until failure depending on the smoothness of discretization is presented in Table 3, with a final error of less than 0.037%. Table 3. Cycles to failure (N f ) from initial crack size (a 0 ) of 0.012 mm to critical crack sizes (a cr ) of 12.38 mm discretizing geometrical factor (Y cr = Y (a)) for an amount of mechanical cycles (∆N m ).
∆N m N f 100,000 2,000,000 50,000 2,000,000 10,000 2,020,000 5000 2,020,000 2000 2,020,000 1000 2,019,000 750 2,019,000 This makes the difference because in such processes corrosion is critical mainly at the first stages, where the corrosion pits initiate the cracks and the anodic zone at the very crack tip corrodes faster than the mechanical contribution to crack growth, up to a point where the mechanical process causes a faster crack growth and the situation is reversed, even going to a point where the corrosion contribution is neglectable. This can be seen in Equation (5), where the impact of increasing the v c of a certain cycle block is highly reduced by increasing the a 0 . Second, at the end of the day a continuous time-dependent process can be modelled by discretizing it in several steps, keeping its value constant within each step and varying it step by step. Such steps can be reduced after each iteration, letting the result to converge up to a cut-off point where it is not necessary to continue. Therefore, a constant corrosion rate approach can be generalized and applied too for any varying corrosion rate, while a varying corrosion rate model could not be suitable to another variable scenario.
Nevertheless, the best way to show how the v c variation after the initial stage affects the corrosion fatigue process is to directly calculate it. Therefore, an initial scenario of no corrosion rate is defined, equivalent to an inert environment XS0, and three more are defined under an XS3 scenario of tidal and splash marine environments, each starting at 200 µm/year, the first one v c1 parabolically decreasing until 100 µm/year at 50 years, the second one v c2 parabolically increasing until 300 µm/year at 50 years, and the third one v c3 constant. See Figure 1.  Hence, with these three corrosion rates, the crack growth can be calculated with calculation steps or iterations of 1000 mechanical cycles. Each mechanical cycle block ΔNm at certain f lasts a certain time increment Δt = f × ΔNm. Therefore, with the initial crack size of any iteration (a0,i), its particular time (ti), time increment (Δti), vc,i, and amount of mechanical cycles (ΔNm), it is possible to derive the final crack size of such an iteration (af,i) simply by adding the mechanical and corrosion effect at each calculation step, then the a0,i will be exactly the final one of the previous one a0,i + 1 = af,i, and so on continuing until reaching the acr. Figure 2 shows the a0,i in mm with time expressed in years of a 25 mm rebar subjected to corrosion-fatigue, with cycles of Δσ = 300 MPa and σm = 150 MPa at a f = 6.34 × 10 −4 Hz. It can be seen that this rebar, in an inert environment (vc0) could reach the service life of 100 years stated for infrastructures, but this is because it takes almost the whole service life (95 years) to reach 1 mm size, the point where the crack starts to grow faster and reaches its critical size in a few years more. The same mechanical fatigue but in a corrosive environments (vc1, vc2, vc3) rapidly loses this "truce" period of 95 years because the pitting corrosion pushes forward the fatigue process, reaching the crack size of 1 mm in the first lustrum, then the crack growth with time is almost parallel at the end stage for corrosive and non-corrosive environments. Hence, with these three corrosion rates, the crack growth can be calculated with calculation steps or iterations of 1000 mechanical cycles. Each mechanical cycle block ∆N m at certain f lasts a certain time increment ∆t = f × ∆N m . Therefore, with the initial crack size of any iteration (a 0,i ), its particular time (t i ), time increment (∆t i ), v c,i , and amount of mechanical cycles (∆N m ), it is possible to derive the final crack size of such an iteration (a f,i ) simply by adding the mechanical and corrosion effect at each calculation step, then the a 0,i will be exactly the final one of the previous one a 0,i + 1 = a f,i , and so on continuing until reaching the a cr . Figure 2 shows the a 0,i in mm with time expressed in years of a 25 mm rebar subjected to corrosion-fatigue, with cycles of ∆σ = 300 MPa and σ m = 150 MPa at a f = 6.34 × 10 −4 Hz. It can be seen that this rebar, in an inert environment (v c0 ) could reach the service life of 100 years stated for infrastructures, but this is because it takes almost the whole service life (95 years) to reach 1 mm size, the point where the crack starts to grow faster and reaches its critical size in a few years more. The same mechanical fatigue but in a corrosive environments (v c1 , v c2 , v c3 ) rapidly loses this "truce" period of 95 years because the pitting corrosion pushes forward the fatigue process, reaching the crack size of 1 mm in the first lustrum, then the crack growth with time is almost parallel at the end stage for corrosive and non-corrosive environments. Finally, conducting a zoom on the final stage for corrosive environments, see magnification of Figure 2, it can be seen that there is not so much difference between a constant, decreasing and increasing vc scenario, because at some point when the mechanical fatigue becomes dominant on coupled corrosion-fatigue, the progress in terms of crack growth caused by pitting corrosion is neglectable.
Therefore, this evidence validates the procedure on a constant corrosion rate basis for hedge applications under corrosive environments, like marine environments in coastal areas, at least when the corrosion intensity is high enough.
As a last comment, normally fatigue at rebars occurs at the most tensioned parts of the reinforced concrete structure, where the concrete cover suffers cracks that facilitate the transport of electrolytes to the rebar surface, tending to the development of localized attacks. Thus, from the safety side, in this manuscript the pitting corrosion is assumed during all service life, neglecting diffusion time or repassivation periods. Nevertheless, the cyclic polarization technique can be used to verify whether all of them are constantly subject to pitting corrosion or only for reduced periods.

Corrosion Test
Four specimens were tested for each steel grade rebar. Table 4 summarizes the icorr mean values and vcorr,g, according to Faraday's law, Equation (3). Therefore, the pitting corrosion rate, vcorr,p, can be Finally, conducting a zoom on the final stage for corrosive environments, see magnification of Figure 2, it can be seen that there is not so much difference between a constant, decreasing and increasing v c scenario, because at some point when the mechanical fatigue becomes dominant on coupled corrosion-fatigue, the progress in terms of crack growth caused by pitting corrosion is neglectable.
Therefore, this evidence validates the procedure on a constant corrosion rate basis for hedge applications under corrosive environments, like marine environments in coastal areas, at least when the corrosion intensity is high enough.
As a last comment, normally fatigue at rebars occurs at the most tensioned parts of the reinforced concrete structure, where the concrete cover suffers cracks that facilitate the transport of electrolytes to the rebar surface, tending to the development of localized attacks. Thus, from the safety side, in this manuscript the pitting corrosion is assumed during all service life, neglecting diffusion time or repassivation periods. Nevertheless, the cyclic polarization technique can be used to verify whether all of them are constantly subject to pitting corrosion or only for reduced periods.

Results and Discussion
Corrosion Test Four specimens were tested for each steel grade rebar. Table 4 summarizes the i corr mean values and v corr,g , according to Faraday's law, Equation (3). Therefore, the pitting corrosion rate, v corr,p , can be derived from v corr,g [23,[53][54][55], in this study a ratio of 10 was considered from the safety side for generalized penetration to pit depth according to the literature [54][55][56]. Even though structures are designed for 50 or 100 years of service life, the directives define a generalized corrosion rate of 50 µm/year over such periods for B500SD reinforcements in concrete structures placed in a tidal zone (XS3). Considering a factor of 10, a threshold value of 500 µm/year is then defined for pitting corrosion. Dividing 500 µm/year by the v corr,p of B500SD, a scale factor was defined to extrapolate measured values after 6 months exposure to those expected for service-life. Besides, the pitting corrosion rate obtained by the linear polarization corrosion rate, when applied in concrete with possible oxygen depolarization and SS with passive layers, is better to be compared with some known values, since these are not be the ideal conditions for this technique. Accordingly, Table 5 summarizes the v corr,p , and the pitting corrosion rate for a whole service-life period (v corr,p *). Corrosion potential (E corr ) versus time for the different steel rebars embedded in ordinary Portland cement (OPC ) concrete is shown in Figure 3a. After 6 months exposure, it was observed that B500SD rebars exposed to a chloride-free environment exhibited an E corr value of around −300 mV CSE , thus showing an intermediate risk of corrosion. In Figure 3, there are four points at each abscissa value, the two values in Figure 3a correspond to E corr measurements under an inert or non-corrosive environment (black point), and also after 6 months of tidal marine environment exposure (white point), while the values in Figure 3b show the monitored i corr . This shows the relative E corr drop and corresponding i corr increment of each material, the performance of each steel when compared together and the scattering of the results.  However, B500SD rebars submitted to the tidal zone suffered high corrosion risk as their E corr values were lower than −400 mV CSE . Critical E corr threshold values for carbon steel rebar in OPC concrete, measured versus copper/copper sulfate reference electrode (CSE), are as follows: E corr > −200 mV CSE (low corrosion, <10% risk), −200 mV CSE < E corr < −350 mV CSE (intermediate corrosion risk, 10-90% risk), E corr < −350 mV CSE (high corrosion, >90% risk), E corr < −500 mV CSE (severe corrosion). These changes in the B500SD E corr may be attributed to the moisture saturation and wetness of the pore network in the concrete specimens. The best corrosion performance was exhibited by 316 ASS rebars; more cathodic corrosion potential values close to E corr ≈ −200 mV CSE were registered. This outstanding corrosion behavior of 316 ASS is attributed to the high pitting resistance equivalent number (PREN). Following the 316 ASS, the 304 ASS has similar E corr values in the tidal zone; however, under non-corrosive conditions it has E corr values of −250 mV CSE , higher than the 316 ASS.
After exposure to the tidal zone XS3 environment, a very active ≈ −440 mV CSE was registered for the 2001 LDSS, showing similar trend to the carbon steel rebar BS500SD. In spite of the low E corr in the corrosive environment, 2001 LDSS has a more noble potential compared to the BS500SD when subjected to the non-corrosion condition. Both 2304 LDSS and 2205 DSS show a similar trend in the XS3 environment, having an E corr value of approximately −350 mV CSE ± 30 and 285 mV CSE , respectively; but the 2205 DSS still has a less active behavior. Comparing both SS in the non-corrosive environment, both have an E corr value of −280 mV CSE .
Furthermore, the i corr monitoring (Figure 3b) for the non-corrosive and tidal zone environments shows that 2304 LDSS, 2205 LDSS, 316 ASS and 304 ASS have a similar behavior, having current density values of 0.04 µA/cm 2 and 0.06 µA/cm 2 , for the non-corrosive and tidal zone, respectively. The 2001 LDSS behaves similarly in the non-corrosive environment but it has a higher i corr value of 0.1 µA/cm 2 for the corrosion environment. The worst behaving steel is CS B500SD, which shows i corr values of 0.08 µA/cm 2 and 0.2 µA/cm 2 , for the non-corrosive and tidal zone environments, respectively (see Figure 3b).
The ∆σ-N curves [31] corresponding to a certain stress range ∆σ will determine the number of cycles until failure, N f , due to mechanical processes. Besides, when the structure is exposed to a corrosive environment, the structural failure will occur with fewer cycles and both ∆σ-N and S-N curves for inert environments become inappropriate for safe predictions. Thus, corrosion amplifies the number of equivalent load cycles during exposure; in other words, corrosion adds virtual cycles, reducing the number of actual mechanical cycles needed to reach the same crack size, a, compared to mechanical fatigue alone. The total cycle amount needed to reach the crack size when mechanical and corrosion processes act simultaneously can be estimated with Equation (5) (for more details, see [22]).
Equally, Tables 6 and 7 summarize the number of cycles until failure according to Equation (6) for 12 mm and 25 mm diameter rebars, respectively, for different stress ranges, ∆σ, considering a constant mean stress of σ m = 150 MPa. As a clarification, the values are obtained for a class XS3 environment, taking into account the pitting corrosion rate values for a service life shown in Table 5 for all specimens; besides, for comparative purposes, a control specimen under an inert environment was also considered. Thus, the N f for each rebar is compared to the equivalent data for an inert environment, the control value shared by all steel types, since being stainless means no difference to fatigue behavior under corrosion-free environments; this was plotted in double logarithm scale, following Basquin's Law [57], in Figures 4 and 5. To clarify, these ∆σ-N curves were derived from the crack growth obtained by fracture mechanics, adding iteratively the corrosion crack growth effect at each calculation step. Once the amount of cycles until failure for each stress range and for each material, with their corresponding corrosion rates, were known, then the S-N curve was derived.       Hence, regardless of the reinforcement diameter, CS B500SD underwent the greatest fatigue life reduction, due to coupled corrosion. Analyzing the LDSS rebars, they presented a greatly improved corrosion fatigue behavior compared to the CS, but within the two LDSS, 2205 showed the best corrosion fatigue behavior, while 2001 was slightly worse. Moreover, Figure 4 refers to 12 mm rebar diameter, where 304L ASS and 316L ASS were also studied, showing quite a similar performance to 2205 DSS surrounding the ASS reinforcements. Comparing Figures 4 and 5, it can be appreciated how, in general, the 12 mm diameter rebars are moved to the left, which demonstrates that the higher the diameter, the better the reinforcement performance in fatigue and corrosion fatigue. Figure 4 and Figure 5 also show the effect of the Δσ, on corrosion fatigue. The gap between the inert curve and any specimen in an XS3 environment increased as the Δσ decreased; so, the effect of the environment under corrosion fatigue was more notorious at lower Δσ.
In a previous work, [22,39] a corrosion pushing factor was defined, enabling the use of Δσ-N curves and the Palmgren-Miner linear rule for damage accumulation for coupled corrosion fatigue scenarios. The FCP increased the number of cycles considering only a purely mechanical process, simulating an increment in crack size. The FCP is therefore defined in Equation (6), where ΔNm is number of cycles until failure due to purely mechanical processes, and ΔNt is the total number of cycles needed to reach acr when mechanical and corrosion processes act simultaneously, calculated as Equation (6) (for more details see [22]). Tables 8 and 9 show the FCP, applying Equation (7) to the data from Tables 6 and 7, respectively. = (7) Hence, regardless of the reinforcement diameter, CS B500SD underwent the greatest fatigue life reduction, due to coupled corrosion. Analyzing the LDSS rebars, they presented a greatly improved corrosion fatigue behavior compared to the CS, but within the two LDSS, 2205 showed the best corrosion fatigue behavior, while 2001 was slightly worse. Moreover, Figure 4 refers to 12 mm rebar diameter, where 304L ASS and 316L ASS were also studied, showing quite a similar performance to 2205 DSS surrounding the ASS reinforcements. Comparing Figures 4 and 5, it can be appreciated how, in general, the 12 mm diameter rebars are moved to the left, which demonstrates that the higher the diameter, the better the reinforcement performance in fatigue and corrosion fatigue. Figures 4 and 5 also show the effect of the ∆σ, on corrosion fatigue. The gap between the inert curve and any specimen in an XS3 environment increased as the ∆σ decreased; so, the effect of the environment under corrosion fatigue was more notorious at lower ∆σ.
In a previous work, [22,39] a corrosion pushing factor was defined, enabling the use of ∆σ-N curves and the Palmgren-Miner linear rule for damage accumulation for coupled corrosion fatigue scenarios. The F CP increased the number of cycles considering only a purely mechanical process, simulating an increment in crack size. The F CP is therefore defined in Equation (6), where ∆N m is number of cycles until failure due to purely mechanical processes, and ∆N t is the total number of cycles needed to reach a cr when mechanical and corrosion processes act simultaneously, calculated as Equation (6) (for more details see [22]). Tables 8 and 9 show the F CP , applying Equation (7) to the data from Tables 6 and 7, respectively. Data from Tables 8 and 9 are plotted in Figures 6 and 7, respectively. As can be appreciated, the F CP increased in an exponential way as the ∆σ decreased for all of the specimens. This fact implies that, under corrosion fatigue, where mechanical and corrosive processes take part simultaneously, the lower ∆σ, the higher the weight of the corrosion within the coupled corrosion fatigue process. Comparing both figures (Figures 6 and 7), the corrosion pushing factor was higher in the φ 12 mm rebar specimens, so smaller diameter bars were more susceptible to corrosion fatigue.  Data from Tables 8 and 9 are plotted in Figures 6 and 7, respectively. As can be appreciated, the FCP increased in an exponential way as the Δσ decreased for all of the specimens. This fact implies that, under corrosion fatigue, where mechanical and corrosive processes take part simultaneously, the lower Δσ, the higher the weight of the corrosion within the coupled corrosion fatigue process. Comparing both figures (Figures 6 and 7), the corrosion pushing factor was higher in the φ 12 mm rebar specimens, so smaller diameter bars were more susceptible to corrosion fatigue.   Figures 6 and 7 show the good corrosion fatigue performance of stainless-steel reinforcements. The use of stainless-steel reinforcements to counter corrosion problems is upheld in many studies [7,12,13,39,58]. Figure 8 presents the FCP of B500SD related to each stainless-steel reinforcement. The performance of stainless-steel reinforcements is in all cases clearly better, with all values higher than one. Examining each one, although 2205 DSS behaved better, it hardly differed from 304L ASS and 316L ASS. 2001 LDSS and 2304 LDSS showed a slightly weaker improvement when compared to the behavior of the CS rebar; an observation in concordance with other corrosion research results [18]; nevertheless, due to their lower nickel content, they are cost-effective solutions. Focusing on FCP with a lower Δσ where corrosion has a greater impact, and analyzing LDSS, 2001 LDSS FCP values compared with 2205 DSS were 1.6 higher and compared with B500SD were 1.5 lower; 2304 LDSS increased its corrosion fatigue life by up to 2 points when compared to B500SD and were quite similar to 2205 LDSS. Even for high Δσ where the corrosion effect impact is insignificant, SS reinforcements withstood more cycles than B500SD, and 2205 LDSS even withstood twice as many cycles. The use of stainless-steel reinforcements to counter corrosion problems is upheld in many studies [7,12,13,39,58]. Figure 8 presents the F CP of B500SD related to each stainless-steel reinforcement. The performance of stainless-steel reinforcements is in all cases clearly better, with all values higher than one. Examining each one, although 2205 DSS behaved better, it hardly differed from 304L ASS and 316L ASS. 2001 LDSS and 2304 LDSS showed a slightly weaker improvement when compared to the behavior of the CS rebar; an observation in concordance with other corrosion research results [18]; nevertheless, due to their lower nickel content, they are cost-effective solutions. Focusing on F CP with a lower ∆σ where corrosion has a greater impact, and analyzing LDSS, 2001 LDSS F CP values compared with 2205 DSS were 1.6 higher and compared with B500SD were 1.5 lower; 2304 LDSS increased its corrosion fatigue life by up to 2 points when compared to B500SD and were quite similar to 2205 LDSS. Even for high ∆σ where the corrosion effect impact is insignificant, SS reinforcements withstood more cycles than B500SD, and 2205 LDSS even withstood twice as many cycles.

Future Work
This semi-empirical procedure to model corrosion fatigue was done under several conservative assumptions: the corrosion during all service life without interruptions, the mechanical fatigue behaviour considered as its lower boundary according to standards and the perfect coupling between corrosion and fatigue. For instance, a crack generates an anodic zone at the tip, and a pit produces a stress concentration zone at the bottom, but corrosion can blunt the crack tip relaxing stresses, and the crack size and thickness can affect electrolyte availability because of transport mechanisms or metal residues closing the path. Therefore, at present, it can only be stated that the modeled effects of corrosion fatigue under this procedure are correct in the sense of being safe, but probably too conservative. Accordingly, future works are required to optimize the model on an empirical basis, based on direct corrosion fatigue testing able to validate some assumptions or relax others in order to improve results.
Regarding how to perform such corrosion fatigue tests properly, this is a key issue. It is better to do fatigue tests under corrosive environments, albeit such tests need to be representative of a whole service life process (50-100 years). Then, keeping in mind Equation (5), the effect of corrosion is time dependent and related to mechanical fatigue by the frequency of the process. If this balance was not kept properly, then corrosion or fatigue could have been neglected or completely dominated the process. Therefore, the scale of corrosion rate to frequency (namely, quotient vc/f) needs to be kept constant. This means that, to keep fatigue test duration under a few weeks, the tests need to be done at high frequencies, but accordingly, corrosion rate needs to be scaled properly by electrical means, and such stimulation has limits. Besides, even in an inert environment, the fatigue results are so scattered that we use the 5% percentile with a confidence of 75%, meaning multiple testing at each stress range level and for each material.

Future Work
This semi-empirical procedure to model corrosion fatigue was done under several conservative assumptions: the corrosion during all service life without interruptions, the mechanical fatigue behaviour considered as its lower boundary according to standards and the perfect coupling between corrosion and fatigue. For instance, a crack generates an anodic zone at the tip, and a pit produces a stress concentration zone at the bottom, but corrosion can blunt the crack tip relaxing stresses, and the crack size and thickness can affect electrolyte availability because of transport mechanisms or metal residues closing the path. Therefore, at present, it can only be stated that the modeled effects of corrosion fatigue under this procedure are correct in the sense of being safe, but probably too conservative. Accordingly, future works are required to optimize the model on an empirical basis, based on direct corrosion fatigue testing able to validate some assumptions or relax others in order to improve results.
Regarding how to perform such corrosion fatigue tests properly, this is a key issue. It is better to do fatigue tests under corrosive environments, albeit such tests need to be representative of a whole service life process (50-100 years). Then, keeping in mind Equation (5), the effect of corrosion is time dependent and related to mechanical fatigue by the frequency of the process. If this balance was not kept properly, then corrosion or fatigue could have been neglected or completely dominated the process. Therefore, the scale of corrosion rate to frequency (namely, quotient v c /f ) needs to be kept constant. This means that, to keep fatigue test duration under a few weeks, the tests need to be done at high frequencies, but accordingly, corrosion rate needs to be scaled properly by electrical means, and such stimulation has limits. Besides, even in an inert environment, the fatigue results are so scattered that we use the 5% percentile with a confidence of 75%, meaning multiple testing at each stress range level and for each material.
In summary, this semiempirical model is a relevant step, albeit still too conservative, in the research line on corrosion fatigue.

Conclusions
Different steel rebars embedded in OPC concrete have been analyzed after exposure to a marine environment (XS3): carbon steel, austenitic stainless-steel (ASS), and lean-duplex stainless-steels (LDSS), considering both the diameter and the stress range effects.

1.
A semi-empirical numerical model has been developed to consider the contribution of corrosion during the fatigue process. This model is based on fracture mechanics and it is an iterative process adding the corrosion contribution to crack growth step by step. The inputs for this model came from direct measurement of corrosion rate, chemical composition, mechanical properties and standard fatigue requirements.

2.
The corrosion pushing factor F CP has been defined to consider the synergistic deterioration caused by simultaneous mechanical fatigue and pitting corrosion processes. It increases the number of effective loading cycles during the exposure time to compute fatigue damage as if it was due only to a mechanical process. Besides, it serves as an indicator of corrosion fatigue performance.

3.
Corrosion fatigue life decreased as the reinforcement diameter decreased. It can be due to a higher perimeter to cross-section area relationship, increasing the relative surface under exposure. 4.
The influence of corrosion was clearly higher at lower stress ranges, where the pitting corrosion became more dominant. Figures 4 and 5 show the gap between the specimens at inert X0 and at tidal marine XS3 environments in terms of cycles to failure loss at each stress range level. The lower the stress range, the higher the cycle loss by coupled pitting corrosion.

5.
Stainless-steel reinforcements had better corrosion fatigue resistance than carbon steel reinforcements. The values of the corrosion pushing factor endorse the use of stainless-steel reinforcements in marine environments. 6.
Comparing 2001 LDSS with B500SD, the performance of 2001 LDSS was 1.5 times better. So, attending to the nickel content, 2001 LDSS is the best cost-performing and it can be a cost-effective alternative for reinforced concrete structures exposed to marine environments, considering material optimization at the design stage and maintenance savings during infrastructure exploitation. 7.
The 2001 LDSS presented the lowest corrosion fatigue resistance increment compared to 304L ASS, 316L ASS, 2304 LDSS, and 2205 DSS. Besides, 2205 DSS showed the best improvement, 1.5 times better than 2001 LDSS, which in turn was 1.5 times better than CS B500SD carbon steel. 8.
The F CP defined the fatigue service life and damage to structural elements in an easy way. It also served as a performance indicator for comparative purposes.