PSI - Issue 52

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Fracture, Damage and Structural Health Monitoring 50 Years of Controversy on Fatigue Crack Closure D. Kujawski a* , A.K. Vasudevan b , R. E. Ricker c and K. Sadananda b a Mechanical and Aerospace Engineering Western Michigan University, Kalamazoo, MI 49008 b TDA Inc, Falls Church, VA 22043 c Univeristy of Maryland, College Park, MD 20742 Fracture, Damage and Structural Health Monitoring 50 Years of Controversy on Fatigue Crack Closure D. Kujawski a* , A.K. Vasudevan b , R. E. Ricker c and K. Sadananda b a Mechanical and Aerospace Engineering Western Michigan University, Kalamazoo, MI 49008 b TDA Inc, Falls Church, VA 22043 c Univeristy of Maryland, College Park, MD 20742

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Professor Ferri Aliabadi 10.1016/j.prostr.2023.12.030 2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Professor Ferri Aliabadi Keywords: plasticity induced crack closure; oxide closure; roughness closure; vacuum experiments; crack tip chemistry. * Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000 . E-mail address: daniel.kujawski@wmich.edu 2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Professor Ferri Aliabadi Abstract In this article the 50 years of observations, implications and debates related to fatigue crack closure are discussed. New insights related to plasticity, oxide, and roughness induced crack closures and their role in shielding effects of the fatigue crack tip are re examined. Supporting evidence for these insights comes from the lack of  K th dependence on R in a high vacuum (with partial pressure of 10 -5 Pa or less). The presented new critical chemical-mechanical analyses are based on experimental results reported in the literature that demonstrate the marginal R-ratio effect on  K th of long cracks in vacuum for both planar/wavy slip alloys but show R-dependence in the lab air and in chemical environment. The latter is due to the formation of viscous nature of the oxide, which forms in humid air at the newly expose fresh fracture surfaces. It is demonstrated that the dominant factor related to the experimentally observed R-ratio effects on fatigue crack growth (FCG) behavior (on several alloys) in not related to crack closure but the access of the environment to the crack tip region that affects fatigue damage. In chemical environments, our viewpoint is supported by a critical analysis of corrosion processes that found that there is insufficient time for most metallic species to form ions, hydrolyze, and transform into hard phases at the crack tip before closure. Therefore, when crack flanks contact occurs, most of the oxidized metallic species will exist as aquo-complexes, gel, or colloids, that have insufficient shear strength to wedge crack faces during unloading. Dislocation-based models have indicated that the crack tip shielding effect from a single asperity is small. The roughness induced crack closure has been suggested as a mechanical obstruction in the wake of the crack during cyclic unloading for planar slip alloys at the threshold region, the emphasis on the access of the environment to the crack tip for an environmental damage was not considered. Keywords: plasticity induced crack closure; oxide closure; roughness closure; vacuum experiments; crack tip chemistry. * Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000 . E-mail address: daniel.kujawski@wmich.edu © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Professor Ferri Aliabadi Abstract In this article the 50 years of observations, implications and debates related to fatigue crack closure are discussed. New insights related to plasticity, oxide, and roughness induced crack closures and their role in shielding effects of the fatigue crack tip are re examined. Supporting evidence for these insights comes from the lack of  K th dependence on R in a high vacuum (with partial pressure of 10 -5 Pa or less). The presented new critical chemical-mechanical analyses are based on experimental results reported in the literature that demonstrate the marginal R-ratio effect on  K th of long cracks in vacuum for both planar/wavy slip alloys but show R-dependence in the lab air and in chemical environment. The latter is due to the formation of viscous nature of the oxide, which forms in humid air at the newly expose fresh fracture surfaces. It is demonstrated that the dominant factor related to the experimentally observed R-ratio effects on fatigue crack growth (FCG) behavior (on several alloys) in not related to crack closure but the access of the environment to the crack tip region that affects fatigue damage. In chemical environments, our viewpoint is supported by a critical analysis of corrosion processes that found that there is insufficient time for most metallic species to form ions, hydrolyze, and transform into hard phases at the crack tip before closure. Therefore, when crack flanks contact occurs, most of the oxidized metallic species will exist as aquo-complexes, gel, or colloids, that have insufficient shear strength to wedge crack faces during unloading. Dislocation-based models have indicated that the crack tip shielding effect from a single asperity is small. The roughness induced crack closure has been suggested as a mechanical obstruction in the wake of the crack during cyclic unloading for planar slip alloys at the threshold region, the emphasis on the access of the environment to the crack tip for an environmental damage was not considered.

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1. Introduction In structural materials that are subjected to cyclic loading, fatigue cracks typically initiate on the surface and propagate until failure for a smooth tensile sample. For a pre-cracked sample, the cracks begin to grow at low applied stress intensity. For both cases, during this process, a significant portion of the component's fatigue life is consumed as cracks propagate in the threshold regime. The threshold regime is affected by the simultaneous influence of mechanical and chemical effects on damage, where long-time exposure is necessary for this to occur. Figure 1 schematically illustrates the stages to failure in terms of defect/crack size versus time/cycles of service. Key features of this figure show that by the time non-destruction inspection (NDI) finds a crack, the component has lost 80% of its life. Above the NDI limit, cracks enter Paris region where only 20% of the life remains.

Fig. 1 Schematic illustration of the stages to failure in terms of defect/crack size versus time/cycles of service.

Based on the elastic materials behavior, a tension stress applied to a cracked specimen would cause the crack to open and the crack opening displacement would return to zero once the applied load is removed. However, when an elastic plastic material is loaded, Elber (1970) observed that the fatigue crack was closed during unloading, even before the tension stress returned to zero. This unexpected observation, Elber called it as plasticity-induced crack closure (PICC). He postulated that for the PICC an effective range of stress intensity factor (SIF),  K eff , can be calculated as: ∆ = − (1) where K max is the maximum stress intensity factor (SIF) during a load cycle and K op is the SIF when the crack tip is just fully open. Often, K op is replaced or used interchangeably with K cl , which relates to the instance when the crack flanks come into contact. According to PICC model (illustrated in Fig. 2a), any load below K op (or K cl ) is not damaging therefore, the fatigue crack growth rate depends solely on  K eff . Thus, using Eq. (1) one can express the normalized effective SIF range or ‘crack driving force’ as following: ∆ =1− (2)

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This postulate expressed by Eq. (2) assumes that a globally measured K op is equal to the local crack tip K op,tip . So, Eq. (2) can be generalized as: ( ) =1− (3) where m = K op,tip /K op ranges from 0 to 1 and represents the effectiveness of a globally measured K op with respect to the local K op,tip . In Elber’s model m = 1. On the other hand, due to the inadequacy of PICC to explain the results at the near-threshold regime, other types of crack closure mechanisms, such as oxide-induced (OICC) and roughness-induced (RICC) crack closure, have been proposed, Suresh (1991). 2. General Experimental and Theoretical Observations The first experimental attempt aims to assess Elber’s postulate given by Eq. (2) was conducted by Vicchio et al. (1986). They investigated the effect of crack closure on FCG behavior in a 2024 Al alloy by inserting an artificial asperity (e.g., a needle tip) in the wake of the crack and measured closure using load-displacement data and found little influence on the crack growth rate. They concluded that such artificial single asperity closure had little effect in the reduction of FCG rate observed. In addition to artificially induced closure, Vicchio et al. (1986) also conducted FCG tests at R=0.5 on Astroloy (planar slip nickel-based superalloy) using the same specimen geometry and identical load reduction procedures. Although the corresponding (da/dN) vs.  K appl for these two tests showed perfect agreement while the closure measurements for these tests were significantly different and didn’t correlate with their data. A similar conclusion was also drawn by Bowman et al. (1988) that most measurements of closure tend to grossly overestimate its effect on FCG. They observed inconsistency in P OP measured for a single specimen for which testing specifications were satisfied. In addition, they disputed the notion “ that all of the load is transmitted through the plastic wedge is not physically reasonable ”. Very recently two other articles pertaining to K OP and  K eff were published by Tong et al. (2019) and Gonzales et al. (2020). Tong et al. (2019) demonstrated that the variations in K OP do not notably affect the cyclic crack-tip deformation, even when the crack was visually closed at P min . They used both digital image correlation (DIC) and compliance curves to determine the “crack opening” levels. They concluded that crack closure, although observed in the compliance curves and DIC technique, does not appear to impact the global crack driving force, such as J integral, or measured strains field ahead of the fatigue crack tip, thus it could be a misconception. Gonzáles et al. (2020), on the other hand, conducted well controlled tests under constant ΔK and K max loading conditions using 3 different methods to quantify the opening loads K OP along the whole crack path at each test. Their records show that the FCG rates remained constant along the whole crack path in all tests, whereas the measured opening loads K OP gradually decreased as the crack grew. They concluded that the  K eff given by Eq. (1) is not a suitable driving force for FCG analysis. We will now examine the implications of some theoretical approaches related to crack flanks contacts/closure that include a single or multiple rigid asperities, a rigid wedge, and energy release rate. Without losing generality, these approaches will be illustrated for R=0 where applied values of K max,appl =  K appl . and assuming that K op or K cl is equal to about 0.5K max . In general, it is observed that K op /K max at R=0 varies between 0 and 0.5 depending on specimen’s geometry, loading conditions (plane stress or plane strain), measurement techniques used or numerical simulations. The first analytical assessment on the effect of crack flanks contact, using dislocation analysis, was given by Vasudevan et al. (1992) They have shown that premature crack flanks contact due to the presence of oxides/asperities exists, but the magnitude of the closure contributions to the crack tip shielding are about 0.25K cl determined from the experimental load-displacement measurements. Five years later, a similar dislocation analysis was conducted by Riemelmoser and Pippan (1997) for a linear elastic crack with a rigid single asperity and multi asperities. Their

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analysis showed that the contribution from a single asperity (SA) contact (illustrated in Fig. 1b) contributes only about 0.21K cl . This results in m =0.21 in Eq. (2) and was in good agreement with the conclusion drawn earlier by Vasudevan et al. (1992) analysis where m was 0.25. Recently, Pippan and Hohenwarter (2017) have provided a review on the PICC, OICC and RICC phenomena. By using multiple rigid asperities contacting the crack flanks at the same K cl (similar to a rigid wedge) the calculated crack tip shielding contribution was about 0.7K cl (corresponding to m =0.7), which is very similar to Paris et al. (1999) analysis of a rigid wedge contact (illustrated in Fig. 1c), called (2/  or 2/pi) partial crack closure, suggesting its effectiveness at the crack tip of (2/  )K cl = 0.64K cl or m =0.64 in Eq. (2). Depending on the contact types (single or multiple rigid asperities) the m varied from 0.21 to 0.7. It is worth noting that roughness induced asperities are not rigid material but exhibit elastic-plastic behavior since they are ductile and deformable, therefore the m-value may significantly reduce the actual effect of theoretically calculated crack tip shielding.

Fig. 2 Illustrations of (a) Elber’s model, (b) single asperity (SA) model, (c) partial closure model ( 2/p) due to rigid wedge, and (d) potential energy released (dU) model Another form of crack driving force, equivalent to the Griffith model, is an energy release rate, , which is the energy available for an incremental crack extension. For a load control loading, as it is illustrated in Fig.1d, the energy release rate is expressed as: = 1 ( ) = 2 ( Δ ) (4)

D. Kujawski et al. / Procedia Structural Integrity 52 (2024) 293–308 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 5 where B is the specimen’s thickness, = ( 2 Δ ) is the change in potential energy when crack increases from ( a) to ( a+da) . Assuming that crack flanks begin to contact at P cl = nP max the corresponding energy is = 2 ( 2 Δ ) (5) Thus, the effective amount of the potential energy, dU eff , available for crack extension is: =(1− 2 )( 2 Δ ) (6) where n = P cl /P max . The corresponding normalized effective potential energy released is: ( ) =1− 2 (7) Figure 3 illustrates the relation for normalized (K eff /K max ) tip and ( / ) tip versus P cl /P max for Elber’s model (Elber), single asperity (SA), partial crack closure (2/pi), and potential energy release (dU). 297

Fig.3 Dependence of normalized crack tip driving force on normalized crack closure. An inspection of Fig. 3 indicates that the Elber’s model shields the crack tip the most, followed by (2/p) model, then the potential energy rate and single asperity model. It is seen from Fig. 3 that the contribution from (dU) and (SA) models on the crack tip shielding is very small if (P cl /P max) < 0.3. It worth noting that for the last 50 years, it was inferred that a single parameter driving force  K eff , given by Eq. (1) is sufficient to analyze fatigue damage and to model fatigue crack growth (FCG) behavior via the classical Paris-Erdogan relationship (often referred as the Paris law): = (∆ ) (8) where C and m are the fitting parameters. Thus, according to crack closure hypothesis, FCG data from different R-

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ratios can be collapsed into a single curve in terms of  K eff . It can be noted that forty years after Elber (1970) published his paper on crack closure, Wei (2010) wrote a book on ‘Fracture Mechanics’ with no reference to crack closure and the analyses rested entirely on the role of environment in fatigue damage. The access of the environment to the crack tip is related to K max applied. Similarly, Hertzberg et al. (2022) in their book on ‘Deformation and Fracture Mechanics of Engineering Materials’ in all editions including the latest 6 th edition also has no reference to crack closure. It is not disputed that two loading parameters (  K and K max ) are needed to characterize fatigue crack growth driving force. The interaction between these parameters can be complex, and the specific mechanisms responsible may require detailed experimental, computational or simulation studies to be fully understood. Additionally, different mechanisms may be dominant under different environmental conditions or loading scenarios. In the present article we discuss our latest analyses Vasudevan et al. (2022), (2022) and Vasudevan and Kujawski (2023) related to PICC, OICC, and RICC and their relevant implications in shielding effects on FCG. We present some insights on these 3 types of closure and the importance on the role of crack tip chemistry on FCG behavior, in particular, at the threshold region. Analysis points out that the common theme in the reduction in  K th with increasing R-ratio in lab air/aqueous solutions is less related to shielding effects of closure and more to the access of environment to the crack tip. A key role in fatigue damage is due to the chemistry and its effect on deformation at the crack tip. 3.0 Review of Key Points for PICC, OICC and RICC 3.1 On Plasticity-Induced Crack Closure [Vasudevan et al. (2022a)] The PICC hypothesis proposes that fracture growth data from various R-ratios can be collapsed into a single curve based on effective stress intensity range, ΔK eff , using equations (1) and (9). This hypothesis has been widely accepted for the past 50 years and is commonly used in the research and analysis of FCG behavior. However, experimental studies have encountered difficulties in determining the K OP (or K CL ) values at near-threshold conditions, primarily due to low sensitivity in the compliance curves used to measure these parameters. Researchers such as Pippan and Hohenwater (2017), Newman (2000), Riddell et al. (1999), and Pippan (1987) have noted that near-threshold K OP values are often simulated or calculated due to the challenges in measuring them directly. Figures 4a and 4b present FCG data for two different alloys, AA7075-T7351 overaged (OA) with wavy slip from Kirby and Beevers (1979) and single crystal PWA1480 with planar slip from Holtz and Sadananda (1998), respectively. The AA7075-T7351 OA alloy was tested in a vacuum of ~10 -3 Pa while the single crystal PWA1480 alloy was tested at 10 -5 Pa. Despite having different slip characteristics, both alloys exhibited a weak dependence of FCG behavior on R-ratio, over several orders of magnitude in FCG rate. The inserts in Figs. 4a and 4b show crack path profiles taken from publications of Petit (2008) and Busse (2019), respectively. The crack profile for the AA7075 T7351 OA alloy was flat, while that for the PWA1480 alloy was zigzag. The overall scatter range in terms of the applied SIF range, ΔK, for the PWA1480 alloy was less than 2 MPa √ , which may be attributed to the experimental resolution of measurements of its rough fracture surface. Trend lines depicted in Fig. 4 represent the average FCG behavior with scatter less than ± 1 MPa√m for the PWA1480 alloy. On the other hand, Fig. 4a sho ws that the scatter in ΔK was much smaller (~0.3 MPa √ ) for the AA7075-T7351 OA alloy with a linear crack path.

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Fig. 4 FCG data for (a) AA7075-T7351 alloy (wavy slip), and (b) a single crystal PWA-1480 alloy, (planar slip) tested in high vacuum are revealing small variation with R-ratio. Short cracks tested in laboratory air often observed to propagate considerably faster than their long crack counterparts under the same applied effective stress intensity range, ΔK. This phenomenon is commonly explained using the PICC argument, which suggests that short cracks have less closure than long cracks due to an undeveloped crack wake. Figure 5a shows the FCG behavior of small cracks (SC) and long cracks (LC) for three different alloys, AA7075-T651, Ti-6Al-4V, and EN460, tested under vacuum (~10 -5 Pa), as reported by Petit and his group, Petite (1998) and Petite et al. (2000). The experimental data for SC and LC at R=0.1 are well within the experimental scatter, as observed in Fig. 5a. It should be noted that these three alloys have significant differences in terms of microstructures, yield strengths, elastic moduli, and slip modes. These differences may contribute to the observed scatter due to variations in local grain deformation at the vicinity of the crack tip or may affect the measurement resolutions. Recently, Yoshinaka et al. (2016) observed a similar trend in Ti-6Al-4V alloy (Fig. 5b) in an ultra-high vacuum (UHV of 10 -7 Pa) between small and long cracks FCG behavior. Experimental data for FCG rates below 10 10 m/cycle often exhibit a slightly larger scatter compared to higher FCG rates. These results demonstrate again that in UHV both small and long cracks behave very similarly in terms of  K applied following the same scatter.

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Fig. 5 Long and short cracks FCG data in vacuum for (a) Ti64, AA7075, and E460 alloys, Petite (1998) and Petite et al. (2000), and (b) for Ti 6Al-4V alloy, Yoshinaka et al. (2016). It should be noted that the FCG behavior of both small and long cracks, as shown in Figs. 5a and 5b, is plotted in terms of the applied  K without any reference or modification to crack closure. As small cracks have limited crack wake and are not affected by crack closure, this implies that long crack behavior should also not be affected by any type of closure. Therefore, the similitude concept associated with  K applicability is preserved for both small and long cracks. However, it should be noted that these interesting data were limited only to a single R =0.1. The lack of an effect of R-ratios on the threshold  K value (  K TH ) for many alloys tested in vacuum, as shown in Fig. 6, suggests that crack closure may not play a significant role in determining  K TH in these conditions. This is noteworthy as crack closure is often considered to be a key factor in determining  K TH , and the absence of closure in vacuum conditions implies that other factors may be at play. It should be noted that the alloys shown in Fig. 6, Vasudevan et al. (2005) exhibit a wide range of properties, including variations in microstructure, slip modes, chemical compositions, and heat treatments. This suggests that the lack of an R-ratio effect on  K TH may be a general phenomenon that is not strongly influenced by the specific properties of a given alloy.

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Fig. 6 A lack of or minimal dependency of ΔK th threshold with R-ratio in vacuum for several engineering alloys, Vasudevan et al. (2005).

3.2 On oxide-induced crack closure [Vasudevan et al. (2022b)] Recently, we reviewed the chemical reactions that occur at the tip of a fatigue crack in Al and Fe systems exposed to humid air and aqueous solutions under ambient conditions, Vasudevan et al. (2022b). This study carefully reviewed literature on the reactions, intermediate phases, and kinetics for the transition of metal ions into hard phases and considered in recent publications. Here we will briefly summarize the findings of that study and their impact on OICC, Vasudevan et al. (2022b). When a bare metallic alloy is exposed to an environment at ambient pressures (gaseous or aqueous), an adsorbed layer of environmental species forms quickly Roth (1976). In non-inert environments, this adsorbed layer will contain oxidizing species that are readily reduced by accepting electrons from active constituents of the alloy to form chemisorbed and reaction product layers Oudar (1995), Nguyen et al. (2018). Oxides nucleate rapidly and spread laterally to form a continuous thin film. This results in a diffusion couple with cations being generated on the metal side of the insulating oxide and anions on the environment side. For oxide growth to occur, electrons, and at least one of the ion types, must transport through the insulating oxide. Mott (1939), (1940), (1947) examined this situation and proposed that oxide growth would arrest at a thickness determined by the tunneling limit for electrons Cabrera (2010). Cabrera and Mott (1949) presented gas phase measurements supporting this model and showed that the limiting thickness for pure Al was ≈ 1.8 nm at 20 °C. Furthermore, they showed that the oxide thickness remained at this value

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over 4 additional hours of exposure Cabrera (2010). Since this seminal work, numerous authors have studied thin oxide film growth in gaseous and aqueous environments, and while modifications have been proposed, they essentially build on the Mott (1939), (1940) and Cabrera-Mott (1949), (2010) model. The Cabrera-Mott (1949) model applies to the growth of thin, flaw free, oxide films, but in both gaseous and aqueous environments oxide films frequently grow thicker than this model predicts. This is the result of a much slower growth mechanism that deposits metal ions in hard solid phases on the environment side of the initial film as illustrated in Fig 6. This change from oxide growth to oxide deposition results in a change in the structure and morphology of the phases present and the outer (precipitated) layer is very heterogeneous with fissures that in some cases, extend all the way down to the barrier layer as illustrated in Fig. 6. The structure and morphology of the phases present in this layer indicates that it is formed by the precipitation and growth of solid phases from metal ions in aqueous solution. At a crack tip, metal ions would be solvated by the formation of aquo-metal ion complexes Vasudevan et al. (2022). These complexes typically have 6 water molecules bound to a central metal ion and this increase in size and results in an increase in viscosity. As these aquo-metal ion complexes transport from the crack tip, they begin to decompose forming hydroxide ion complexes (hydrolysis). The linking of the hydroxide ions in these complexes (olation) further increases the size and viscosity of these complexes. The end-product phases must nucleate and grow in these gels. The time required for the kinetics of these reactions will depend on the particulars of alloy and environment chemistry, but our survey of the literature found that these processes tend to be slow compared to fatigue testing frequencies Vasudevan et al. (2022). Therefore, when a crack closes during a fatigue experiment, the outer, precipitated, layer shown in Fig. 6 will not yet have formed, and only the barrier layer and solution rich in hydroxide ion complexes will be present in the region of the crack tip Vasudevan et al. (2022).

Fig. 7 Schematic of a typical passivating film on a metallic alloy boldly exposed to humid air or water long enough to reach steady state film thicknesses. It appears that the oxides in the barrier layer may fail both of the requirements above for causing OICC. First, its growth at room temperature is limited by solid state diffusion and the Carbera-Mott limit to values on the order of 1 to 20 nm. These factors will not allow oxide volumes to grow to the required values for OICC during a load cycle. Second, recent research indicates that these thin films can be ductile. For example, Yang et al. (2018) used an environmental transmission electron microscope (ETEM) and found that at room temperature thin oxide films of this type can be quite ductile and deform in a viscous manner. This behavior was attributed to the suppression of the glass transition temperature, but they also reported less ductile behavior for thicker films and higher strain rates Yang et al. (2018). Therefore, at this time it is unclear how these layers will behave during a fatigue cycle. They may deform in a ductile manner, like that reported by Yang et al., (2018) or they may fracture in a brittle manner. In either case, bare metal will

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be exposed at the crack tip resulting in the production of aquo-metal ions that eventually cause the growth of the passive film outer layer, but it is concluded that the inner, or barrier layer, will be too thin to contribute to OICC. The above discussion is supported by an interesting experiment on fatigue threshold with an ASTM A542 Class-3 steel in distilled water Suresh and Ritchie (1982). Dry oxide thickness and  K th was measured at three R-ratios, Fig. 7.

Fig. 8 Dry oxide thicknesses and  K th variation with R in A542 steel in distilled water.

The dry oxide thickness was constant at ≈ 0.8 mm with increasing R, from R = 0.1-0.75. This experiment suggests that oxide formation and its thickness is influenced more by chemistry than cyclic loading, but  K th decreased monotonically with R. Their results were interpreted by the authors as a complex combination of both OICC and environmental effects on  K th . In contrast, we now can interpret the same data as being due mostly to the crack tip environmental access with increasing R to result in damage, with little contribution from OICC.

Fig. 9 Effects of silicone oil viscosity on fatigue crack growth rate in (a) 2.25Cr-1Mo steel (50 Hz) and (b) 316 stainless steel (4 Hz). Results compared with dry He and air in (a) and (b), respectively.

Figure 9a shows the results of Tzou et al. (1985) have studied the effect of viscous fluids (silicon oils) of

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various viscosities, in the wake of a crack on 2.25Cr-1Mo steels show that crack growth threshold  K th changed less than 1.0 MPa(m) 1/2 and the data lies close to the inert He. Bae and Conrad (1989) conducted similar experiments on AISI 316 stainless steel FCG using silicone oil and found that the overall FCG results were not affected by viscous oil in the wake of a fatigue crack (Fig. 8 b). AISI 316 stainless steel is a planar slip alloy with a non-linear branched crack. Trend line in both Figs. 8a and 8b shows the average trend of the data. The above observations are similar to both Fe and Al-hydroxides on a FCG in Fe and Al-alloys. Such observations suggest that the overall effect of viscous oxides play a small role at the FCG thresholds. The cause that reduces  K th with increasing R is mainly the access of environment to the crack tip and the possible hydrogen assisted damage. 3.3 On roughness-induced crack closure [Vasudevan and Kujawski (2023)] The RICC is considered to be more common in a planar slip alloy where Mode-II displacements are prevalent. Suresh and Ritchie (1982) developed a 2-D model to describe the effect of RICC on crack tip shielding for lab air conditions. In their model fracture surface roughness is idealized to be of triangular cross section and roughly equal in size in terms of height ( ℎ ) and width ( ) (Fig. 9). Based on their geometrical considerations they derived the following relationship for the first contact of fracture surfaces during unloading = 2ℎ +2ℎ (9) where = is the ratio of the Mode II displacement, , over the Mode I displacement, . Then, the magnitude of the closure effect, was expressed as the ratio of the closure (K cl ) to maximum stress intensity (K max ) as =√ =√ 2ℎ +2ℎ (10) or in a nondimensional form as =√ 2 1+2 (11) where = ℎ/ is a nondimensional fracture surface roughness factor where can be related to the average grain size by the relation = .

Fig. 10 Schematic illustration of a near-threshold fatigue crack displacement at (a) K=K max and the corresponding d max and (b) after unloading (by Mode I and Mode II) to the first contact between fracture surfaces at K=K cl and corresponding d cl (After Suresh and Ritchie (1982)). The viscous oxide layer forms at humidity level greater than about 40%.

D. Kujawski et al. / Procedia Structural Integrity 52 (2024) 293–308 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 13 Equation (11) indicates that the magnitude of the RICC depends on the size of the asperity ( ) and to the extent of the shear displacements ( ). The value of =0 if one or both variables equal to zero. In other words, under pure Mode I loading, RICC is not present since =0 and/or the crack fracture surface is smooth with =0 . It should be noted that in this analysis and corespond to the first contact of fracture surfaces during unloading. This first contact of crack surfaces doesn’t constitute complete “locking” nor “fixing” the crack -tip displacement since the inclined fracture surfaces in contact can slide against each other for a slip line fracture. Such sliding is more prevalent in humid air (or aqueous solution) due to a viscous layer of oxides that is formed with metal ions in various states of the transition from aqueous ions to stable end-product phases Vasudevan et al. (2022) (shown in Fig. 9 in blue). Such viscous oxide layer can work as a lubricant and promotes sliding of the inclined fracture surfaces resulting in crack tip displacement during additional unloading below . The observed effect of reduction of  K th (or an increase in (da/dN) rate) with increasing R-ratio is not due to RICC but due to more easily accessible access of the environment to the crack tip. Roughness develops in a planar slip alloy due to the crystallographic crack path. Thus, roughness is intrinsic to such alloys and exists in vacuum as well as in the environment. Therefore, in vacuum surface roughness should not affect  K th or (da/dN) in any significant way near the threshold region. However, at very high vacuum (>10 -7 Pa) the contact between fracture surfaces may cause rewelding 67 due to lack of oxygen to form a monolayer of oxide to prevent welding, which in turn may affect  K th and (da/dN) by bridging the fracture surfaces at low R-ratios. The above geometric model is presented in terms of Eqs. (10-12) that do not include environmental components. It is a purely a 2D geometrical model related to alloy systems with planar slip deformation which results in crack deflection. This deflected crack gives rise to asperity roughness that was related to RICC. As a result, the model should validate the results in vacuum that has very little environmental component. However, the model was used to validate the fatigue threshold data (  K th -R) in lab air (30-50%RH) and in aqueous solutions like water or dilute NaCl. Limited examples on planar slip alloys in vacuum is shown in Fig. 10 which indicates  K th independent of R relationship in vacuum for several planar slip alloys. In the case of 7075-UA, at high R=0.85, DK th drops due to K max approaching K Ic (~25 MPa.m 0.5 ). In addition, Ni-base alloys show the same trend. In the case of PW1480 single crystal alloy (shown in Fig. 2) indicates that RICC is nearly absent in vacuum even though the asperity roughness is present due to the crystallographic nature of the alloy. 305

Fig. 11 Effect of R-ratio on  K th in vacuum for planar slip alloys.

In addition, Petit’s (2008) shows interesting crack path profiles in planar slip AL -4.5Zn-1.25Mg alloys (single and polycrystal) with contrasting images, Fig. 11 a, b. In vacuum (at 4x10 -4 Pa) polycrystal Al-4.5Zn-1.25Mg

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alloy at R=+0.1 show perfect matching of fracture surfaces, Fig. 11a at P min with a zig-zag crack profile with no observable oxide layer. While Fig. 11b the single crystal alloy of same composition, shows some dry oxide debris in the Mode-II-part (on (111) slip plane) due to oxide induced abrasion at R= -1.0 but not in the Mode I part of the deflected crack. But both micrographs in Fig. 11 show good matching of fracture surfaces at P min , even though there is some oxide debris in the Mode II section of the crack path at R= -1.0. This crack surface oxide is a dry oxide observed after the experiment. During the experiment it was a viscous oxide that can impart some sliding to push out the oxide that dried.

Fig. 12 Crack path profiles in planar slip AL-4.5Zn-1.25Mg alloys in vacuum Petite (2008) at P min for (a) R = 0.1 and (b) R = -1. Schematic Fig. 9b show the gaps between the unmatched fracture surfaces at that allow for the environment to enter to the crack tip and induce environmentally assisted fatigue damage. Available examples of  K th data in vacuum suggest that RICC contributions is not significant enough to reduce  K th with R in planar slip alloys as well as in non-planar slip alloys. The factor of environmental access dominates the roughness effect affecting near threshold region for fatigue damage. Vacuum results question the importance of PICC, OICC, and RICC concepts on FCG behavior at threshold, if there are no other deformation processes like creep, void growth, or cleavage at high R contributing when K max → K IC . We suggest that for modelling and analysis direct use of applied  K and K max at a given environment is sufficient to use than any corrected version of  K to  K eff . Conclusions The following conclusions are drawn from our analysis that addresses issues related to PICC, OICC and RICC on FCG behavior at the near threshold region in both lab air and in vacuum environments. 1. Supported by analytical and experimental data in vacuum, one can deduce that PICC, OICC and RICC effect on FCG are overestimated in lab air and in aqueous environments. 2. The PICC and RICC models do not include the environmental component and do not seem to validate the data in vacuum results. 3. Analysis of corrosion processes at room temperature and transition toward the end-product phases in the wake of the crack result in a negligible reduction of  K applied with R by PICC due to deformable viscous nature of the oxide layer. 4. The OICC mechanism is possible at high temperatures where dry oxide may form behind the crack-tip to the size relevant to the cyclic crack tip opening displacement (CTOD). In summary, the experimentally observed R-ratio effects on FCG and threshold behavior at room temperature in humid air are less linked to crack closure behind the crack tip but to the access of the environment to the crack tip region that

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affects fatigue damage ahead of the crack tip. Quantifying the crack tip chemistry and its relation to local stresses is a challenging research problem.

ACKNOWLEDGEMENT We thank Professor Reinhard Pippan for his comments and online discussion (with DK) regarding measurements and implementations of crack closure for FCG analysis. Contribution statement D. Kujawski: conceptualization and writing the original draft, A.K. Vasudevan: conceptualization and editing, R.E. Ricker: detail analysis about crack tip chemistry, K. Sadananda: final analysis and editing. References Bae, K., Conrad, H., 1989. Effect of viscosity of an oil environment on fatigue crack growth rate in AISI 316 Stainless Steel, Proceedings of the 7th International Conference on Fracture (ICF7), 1737-1745. Bowman, R., Antolovich. S.D. and Brown, R.C., 1988. A demonstration of problems associated with crack closure measurements techniques, Eng Fract Mech.. 31(4), 703-712. Busse, C., 2019. Modelling of crack growth in single crystal Ni-base superalloys. PhD Thesis, Linkoping University. Cabrera, N., Mott, N.F., 1949. Theory of the oxidation of metals, Reports on Progress in Physics. 12(1), 163 184. Cabrera, N., 2010. On the oxidation of metals at low temperatures and the influence of light, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 40(301), 175-188. Elber, W., 1970. Fatigue crack closure under cyclic tension loading. Eng Fract Mech. 2(1), 37-45 . Gonzales, J.A.O., Castro, J.T.P., Meggiolaro, M.A., Gonzales, G.L.G., Freire, J.L.F., 2020. Challenging the “ΔK eff is the driving force for fatigue crack growth” hypothesis. Int J Fatigue 136, 105577. Holtz, R.L., Sadananda, K., 1998. Fatigue threshold maps of PWA 1480 superalloy single crystal in air and vacuum at room temperature. In: Srivatsan T.S. Soboyejo W.O. eds . High Cycle Fatigue of Structural Materials. The Minerals, Metals, and Materials Society . TMS Publications, 299-304. Kirby, B.R. and Beevers, C.J., 1979. Slow fatigue crack growth and threshold behavior in air and vacuum of commercial alloys. Fatigue Engng Mater Struct. 1, 203-215. Mott, N.F., 1939. A theory of the formation of protective oxide films on metals, Transactions of the Faraday Society 35. Mott, N.F., 1940. The theory of the formation of protective oxide films on metals, II, Transactions of the Faraday Society 35. Mott, N.F., 1947. The theory of the formation of protective oxide films on metals, III, Trans. Faraday Soc. 43, 429-434. Nguyen, L., Hashimoto, T., Zakharov, D.N., Stach, E.A., Rooney, A.P., Berkels, B., Thompson, G.E., Haigh, S.J., Burnett, T.L., 2018. Atomic-scale insights into the oxidation of aluminum, ACS Appl Mater Interfaces 10(3), 2230-2235. Newman, J.C., 2000. Analysis of fatigue crack growth and closure near threshold conditions. ASTM STP-1372, 227 – 251. Oudar, J., 1995. Introduction to Surface Reactions: Adsorption from Gas Phase in: P. Marcus, J. Oudar (Eds.), Corrosion Mechanisms in Theory and Practice , Marcel Dekker, Inc., New York, 19-54. Paris, P.C., Tada, H. and Donald, J.K., 1999. Service load fatigue damage – A historical perspective. Int J Fatigue 21, S35-S46. Pippan, R. and Hohenwater, A., 2017. Fatigue crack closure: a review of the physical phenomena. Fatigue Fract Engng Mater Struc. 40, 471-495. Petit, J., 1998. Influence of environment on small fatigue crack growth. In: Ravichandran KS, Ritchie RO,

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Murakami Y, eds. Small Cracks: Mechanics, Mechanisms and Applications . Elsevier Sci. Ltd., 167-186. Petit, J., 2008. Some critical aspects of fatigue crack propagation in metallic materials. 17th European Conf. on Fracture, Brno, Czech Republic, Sept. 2008, 54-77. Petit, J., Henaff, G., Sarrazin-Baudoux, C., 2000. Mechanisms and modelling of near threshold fatigue crack propagation. In: Newman JC Jr, Piascik RS, eds. ASTM STP 1372. ASTM Pub ., 3-30. Pippan, R., 1987. The growth of short cracks under cyclic compression. Fatigue Fract Eng Mater Struct. 9, 319 – 28. Riemelmoser, F.O. and Pippan, R., 1997. Crack closure: a concept of fatigue crack growth under examination. Fatigue Fract Engng Mater Struc. 20, 1529 – 1540. Riddell, W.T., Piascik, R.S., Sutton, M.A., Zhao, W., McNeill, S.R. and Helm, J.D., 1999. Determining fatigue crack opening loads from near-crack-tip displacement measurements. In: McClung RC, Newman Jr. JC, editors. Advances in fatigue crack closure measurement and analysis: second volume , ASTM STP1343. West Conshohocken (PA): American Society for Testing and Materials, 157 – 174. Suresh, S., Ritchie, R.O., 1982. Near-threshold fatigue crack propagation: A Perspective on the role of crack closure, in: D.L. Davidson, S. Suresh (Eds.), Fatigue Crack Growth Threshold Concepts , TMS, Warrendale, PA. Suresh, S. and Ritchie, R.O.,1982. A model for fatigue crack closure induced by fracture surface roughness. Met. Trans. 13A, 1627-1631. Tzou, J.L., Suresh, S., Ritchie, R.O., 1985. Fatigue crack propagation in oil environments-I. Crack growth behavior in silicone and paraffin oils, Acta Metall. 33(1), 105-116. Tong, J., Alshammrei, S., Lin, B., Wigger, T., Marrow, T., 2019. Fatigue Crack closure: A myth or a misconception? Fatigue Fract Engg Mater Struc. 42, 2747-2763. Vasudevan, A.K., Sadananda, K., Louat, N., 1992. Reconsideration of fatigue crack closure, Scripta Metall. et Materialia 27, 1673-1678. Vasudevan, A.K., Sadananda, K., Holtz, R.L., 2005. Analysis of vacuum fatigue crack growth and its implications. Int J Fatigue 27(10-12), 1519-1539. Vasudevan, A.K., Kujawski, D., 2022a. Implications of  K-R ratio in vacuum, Fatigue Fract Eng Mater Struct..1 – 12. Vasudevan, A.K., Ricker, R.E., Miller, A.C. and Kujawski, D., 2022b. Fatigue crack tip corrosion processes and oxide induce closure. Mater Science Eng A.861, 144383. Vasudevan, A.K. and Kujawski, D., 2023. Roughness induced crack closure: A review of key points. Theoretical and Applied Fracture Mechanics 125, 103897. Vecchio, R.S., Crompton, J.S., Hertzberg, R.W., 1986. Anomalous aspects of crack closure. Inter. J. Fracture 31, R29-R33. Wei, R.P., 2010. Fracture Mechanics: Integration of Mechanics, Materials Science and Chemistry, Cambridge University Press. Yang, Y., Kushima, A., Han, W., Xin, H., Li, J., 2018. Liquid-like, self-healing aluminum oxide during deformation at room temperature, Nano Letters 18(4), 2492-2497. Yoshinaka, F., Nakamura, T., Takaku, K., 2016. Effects of vacuum environment on small fatigue crack propagation in Ti-6Al-4 V. Int J Fatigue 91, 29-38. Roth, A., 1976. Vacuum Technology, North Holland, Amsterdam. Suresh, S., 1991. Fatigue of Materials. Cambridge University Press, UK.

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© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Professor Ferri Aliabadi Abstract A computational model is stated to study dynamic crack propagation in quasi-brittle materials exposed to time-dependent loading conditions. Under such conditions, inertial e ff ects of structural components play an important role in modelling crack propagation problems. The computational model is proposed within the theory of smeared cracks which use damage-like internal variables. Here, fracture considers phase-field damage which gives rise to a material degradation in a narrow material strip defining the smeared crack. Based on the energy formulation using the Lagrangian of the system, the proposed computational approach in troduces a staggered scheme adopted to solve the coupled system and providing it in a variational form within the time stepping procedure. The numerical data are obtained by quadratic programming algorithms implemented together with a finite element code. Keywords: Phase-field fracture; Dynamic crack propagation; Quadratic programming; Staggered approach Fracture, Damage and Structural Health Monitoring A Computational approach of Dynamic Quasi-Brittle Fracture Using a Phase-Field Model Roman Vodicˇka a a Technical University of Kosˇice, Faculty of Civil Engineering, Vysokosˇkolska´ 4, 042 00 Kosˇice, Slovakia Abstract A computational model is stated to study dynamic crack propagation in quasi-brittle materials exposed to time-dependent loading conditions. Under such conditions, inertial e ff ects of structural components play an important role in modelling crack propagation problems. The computational model is proposed within the theory of smeared cracks which use damage-like internal variables. Here, fracture considers phase-field damage which gives rise to a material degradation in a narrow material strip defining the smeared crack. Based on the energy formulation using the Lagrangian of the system, the proposed computational approach in troduces a staggered scheme adopted to solve the coupled system and providing it in a variational form within the time stepping procedure. The numerical data are obtained by quadratic programming algorithms implemented together with a finite element code. Keywords: Phase-field fracture; Dynamic crack propagation; Quadratic programming; Staggered approach Fracture, Damage and Structural Health Monitoring A Computational approach of Dynamic Quasi-Brittle Fracture Using a Phase-Field Model Roman Vodicˇka a a Technical University of Kosˇice, Faculty of Civil Engineering, Vysokosˇkolska´ 4, 042 00 Kosˇice, Slovakia Abstract A computational model is stated to study dynamic crack propagation in quasi-brittle materials exposed to time-dependent loading conditions. Under such conditions, inertial e ff ects of structural components play an important role in modelling crack propagation problems. The computational model is proposed within the theory of smeared cracks which use damage-like internal variables. Here, fracture considers phase-field damage which gives rise to a material degradation in a narrow material strip defining the smeared crack. Based on the energy formulation using the Lagrangian of the system, the proposed computational approach in troduces a staggered scheme adopted to solve the coupled system and providing it in a variational form within the time stepping procedure. The numerical data are obtained by quadratic programming algorithms implemented together with a finite element code. Keywords: Phase-field fracture; Dynamic crack propagation; Quadratic programming; Staggered approach Damage, degradation and eventually fracture of a material are significant concepts in solid mechanics. Even if these phenomena are solved quasi-statically, they provide satisfactory agreement with real behaviour of structures in such models. Nevertheless, if the processes tend to be fast, the inertial e ff ect may substantially modify the response of the structural components and in such a way also modify cracks and their formation processes. Anyhow, any information, including those of crack propagation, in real structures may be transfered only at a finite speed. Therefore, dynamic crack propagation should be taken into account in more complex computational models of fracture. Many brittle fracture computational approaches suitable for a finite-element implementation, are related or directly founded on the work Francfort and Marigo (1998), which formulated the problem variationally in terms of strain en ergy in domains and surface energy of arisen cracks related to Gri ffi th’s concept of crack propagation. To facilitate the solution of the problem with unknown location of discrete cracks, a rearrangement of the fracture mechanisms was proposed to define a concept of smear cracks which does not define the crack as a place of a material discontinuity. It works with an internal variable instead which di ff uses the flaw locus to have a finite width and the crack is than seen as a narrow band of the material. One of such concepts includes the phase-field model (PFM) of fracture, which can www.elsevier.com / locate / procedia www.elsevier.com / locate / procedia Fracture, Damage and Structural Health Monitoring A Computational approach of Dynamic Quasi-Brittle Fracture Using a Phase-Field Model Roman Vodicˇka a a Technical University of Kosˇice, Faculty of Civil Engineering, Vysokosˇkolska´ 4, 042 00 Kosˇice, Slovakia 1. Introduction 1. Introduction Damage, degradation and eventually fracture of a material are significant concepts in solid mechanics. Even if these phenomena are solved quasi-statically, they provide satisfactory agreement with real behaviour of structures in such models. Nevertheless, if the processes tend to be fast, the inertial e ff ect may substantially modify the response of the structural components and in such a way also modify cracks and their formation processes. Anyhow, any information, including those of crack propagation, in real structures may be transfered only at a finite speed. Therefore, dynamic crack propagation should be taken into account in more complex computational models of fracture. Many brittle fracture computational approaches suitable for a finite-element implementation, are related or directly founded on the work Francfort and Marigo (1998), which formulated the problem variationally in terms of strain en ergy in domains and surface energy of arisen cracks related to Gri ffi th’s concept of crack propagation. To facilitate the solution of the problem with unknown location of discrete cracks, a rearrangement of the fracture mechanisms was proposed to define a concept of smear cracks which does not define the crack as a place of a material discontinuity. It works with an internal variable instead which di ff uses the flaw locus to have a finite width and the crack is than seen as a narrow band of the material. One of such concepts includes the phase-field model (PFM) of fracture, which can Damage, degradation and eventually fracture of a material are significant concepts in solid mechanics. Even if these phenomena are solved quasi-statically, they provide satisfactory agreement with real behaviour of structures in such models. Nevertheless, if the processes tend to be fast, the inertial e ff ect may substantially modify the response of the structural components and in such a way also modify cracks and their formation processes. Anyhow, any information, including those of crack propagation, in real structures may be transfered only at a finite speed. Therefore, dynamic crack propagation should be taken into account in more complex computational models of fracture. Many brittle fracture computational approaches suitable for a finite-element implementation, are related or directly founded on the work Francfort and Marigo (1998), which formulated the problem variationally in terms of strain en ergy in domains and surface energy of arisen cracks related to Gri ffi th’s concept of crack propagation. To facilitate the solution of the problem with unknown location of discrete cracks, a rearrangement of the fracture mechanisms was proposed to define a concept of smear cracks which does not define the crack as a place of a material discontinuity. It works with an internal variable instead which di ff uses the flaw locus to have a finite width and the crack is than seen as a narrow band of the material. One of such concepts includes the phase-field model (PFM) of fracture, which can ∗ Corresponding author. Tel.: + 421-55-602-4388. E-mail address: roman.vodicka@tuke.sk 1. Introduction

∗ Corresponding author. Tel.: + 421-55-602-4388. E-mail address: roman.vodicka@tuke.sk ∗ Corresponding author. Tel.: + 421-55-602-4388. E-mail address: roman.vodicka@tuke.sk

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of Professor Ferri Aliabadi 10.1016/j.prostr.2023.12.025 2210-7843 c 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of Professor Ferri Aliabadi. 2210-7843 c 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of Professor Ferri Aliabadi. 2210-7843 c 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of Professor Ferri Aliabadi.