Experimental, Analytical and Numerical Studies of Interfacial Bonding Properties between Silane-Coated Steel Fibres and Mortar

: A systematic investigation of the effects of silane coatings on steel ﬁbre–mortar interfacial bond properties was conducted, combining pullout tests, analytical solutions, and meso-scale FE simulations. Nine silane coatings were tested, and their effects were evaluated by 30 single ﬁbre pullout tests. They were found to increase the peak force and energy consumption up to 5.75 times and 2.48 times, respectively. Closed-form analytical solutions for pullout load, displacement, and interfacial stress distribution during the whole pullout process were derived based on a tri-linear bond-slip model, whose parameters were calibrated against the pullout tests. Finally, the calibrated bond-slip models were used to simulate the pullout tests and complex failure of multi-ﬁbre specimens in mesoscale ﬁnite element models. Such an approach of combining pullout tests, analytical solutions, and mesoscale modelling provides a reliable way to investigate the effects of ﬁbre–mortar interfacial properties on the mechanical behaviour of steel ﬁbre reinforced concrete members in terms of structural strength, stiffness, ductility, and failure mechanisms.


Introduction
Plain concrete consists of aggregates, mortar, and pores, and its low tensile strength makes concrete structures susceptible to fracture and brittle failure. To overcome this weakness, fibres made from steel [1], polymer [2,3], carbon [4], glass [5] etc., are often added to reinforce the concrete. Steel fibres are the most widely used fibres to reinforce concrete in structural applications due to its easy access, recyclability, and reinforcement efficiency [6]. Steel fibre reinforced concrete (SFRC) has higher tensile/flexural/compressive/fatigue strengths, fracture toughness, and ductility than plain concrete [7][8][9][10] due to the crack-bridging effect of steel fibres, which delays crack initiation and propagation in concrete [11,12]. Such effect is mainly dependent on the interfacial bonding properties and energy dissipation between fibres and mortar during fibre pullout. Improving the interfacial properties between these two materials is, therefore, important for SFRC to achieve better macroscopic mechanical performances.
To date, substantial efforts have been made to enhance the interfacial properties between steel fibres and mortar, mainly adopting two techniques, i.e., using deformed fibres and modifying fibre-mortar interfaces. The deformed fibres include hooked-end, corrugated, waved, and twisted fibres and those with irregular shapes. Extra mechanical anchor forces are created as the deformed fibres are gradually pulled out from the mortar, compared with straight circular fibres. For example, hooked-end and corrugated steel fibres can increase the

Coating Procedure and Specimen Fabrication
Silane coating of the steel fibres involves three steps: (1) Hydrolysis [37,38]: mix one portion of methanol, six portions of deionized water, and one portion of silane together and stir the solution at room temperature for 20 min, and then add a proper amount of acetic acid to adjust the PH value of the solution in the range of 4-6; wait for 24 h until the silane completely hydrolysed. The four silane coupling agents (Z6011, Z6020, Z6030, and Z6040) in group Z were produced first, and Zr(NO 3 ) 4 with concentration of 0.01 mol/L was then added to make the four solutions in Group ZZ. Group ZH2 was produced by adding H 2 ZrF 6 (with 8% mass fraction of the final solution) into the Z6020 agent. (2) Pretreatment of fibres: fibres were polished using sandpapers with fineness of 600 and 1500 to remove the oxide layer and original metal coating and were further cleaned using alkaline detergent to remove other impurities on the surface. They were subsequently soaked in sodium hydroxide solution of 20% (wt) for 30 min, rinsed by deionized water to remove the residual lye, and then immersed in anhydrous ethanol to remove the residual water film on the fibres' surfaces and left to dry naturally. (3) Coating: the steel fibres were immersed in the silane-based coupling solutions and stirred gently for 2 min to ensure solution to be adsorbed onto the fibre surfaces. The fibres were then taken out and dried at a constant temperature of 110 • C for 30 min until the surface modification were completed. Untreated steel fibres of the same lengths were also prepared as the reference group.
The fine sand, cement, and water were prepared according to the designed mix proportion. After the cement and fine sand were dry-mixed for 2 min, half the water was added and mixed for 4 min, followed by addition of another half water and 8 min mixing. The mixture was then moved into a cylinder mould with a diameter of 68 mm and a height of 136 mm. A steel fibre was embedded in the cylinder centre with an embedment length of 40 mm, as shown in Figure 1. The steel fibre and the mould were fixed and vibrated for 1 min with a vibration frequency of 50 Hz. The specimen was then moved into a curing chamber with a temperature of 20 ± 1 • C and a relative humidity of 90% and were cured for 28 days. Three repeat specimens for each group, i.e., 30 specimens in total, were prepared. Meanwhile, three 70.7 mm mortar cubes were prepared under the same condition and tested according to the standard JCJ/T70-2009 [40]. The obtained average compressive strength at 28 days was 43.6 MPa with a standard deviation (SD) of 2.31 MPa. The sketch.

Single Fibre Pullout Tests
The test setup of single fibre pullout is similar to those used in [14,16,24]. As shown in Figure 1c, the exposed end of the steel fibre was bent at 45° and buried with a length of 50 mm in a copper pipe (12 mm in outer diameter, 10 mm in inner diameter and 100 mm in length) filled with epoxy resin to ensure that the pullout occurred in the mortar. After the epoxy resin was cured for 24 h, pullout tests were performed on an Instron 8802 test machine as shown in Figure 2. The specimen was fixed by a jack set between the specimen and the customized steel frame with the copper pipe passing through a reserved hole in the middle bottom of the steel frame. A force transducer with a measuring range/accuracy of 10 kN/0.5 N was connected with the copper pipe. The loading was controlled by displacement with a rate of 0.5 mm/min.

SEM Tests
The surfaces of steel fibres were scanned and analysed using a field emission environmental scanning electron microscopy (SEM: FEG650) and an energy spectrometer (EDS: EDAX), both supplied by FEI, USA. The scanned samples were 15 mm cubes that were cut perpendicular to the fibre axis in groups U and Z1 before the pullout tests. The

Single Fibre Pullout Tests
The test setup of single fibre pullout is similar to those used in [14,16,24]. As shown in Figure 1c, the exposed end of the steel fibre was bent at 45 • and buried with a length of 50 mm in a copper pipe (12 mm in outer diameter, 10 mm in inner diameter and 100 mm in length) filled with epoxy resin to ensure that the pullout occurred in the mortar. After the epoxy resin was cured for 24 h, pullout tests were performed on an Instron 8802 test machine as shown in Figure 2. The specimen was fixed by a jack set between the specimen and the customized steel frame with the copper pipe passing through a reserved hole in the middle bottom of the steel frame. A force transducer with a measuring range/accuracy of 10 kN/0.5 N was connected with the copper pipe. The loading was controlled by displacement with a rate of 0.5 mm/min.

Single Fibre Pullout Tests
The test setup of single fibre pullout is similar to those used in [14,16,24]. As in Figure 1c, the exposed end of the steel fibre was bent at 45° and buried with a len 50 mm in a copper pipe (12 mm in outer diameter, 10 mm in inner diameter and 1 in length) filled with epoxy resin to ensure that the pullout occurred in the mortar the epoxy resin was cured for 24 h, pullout tests were performed on an Instron 88 machine as shown in Figure 2. The specimen was fixed by a jack set between the men and the customized steel frame with the copper pipe passing through a re hole in the middle bottom of the steel frame. A force transducer with a mea range/accuracy of 10 kN/0.5 N was connected with the copper pipe. The loadin controlled by displacement with a rate of 0.5 mm/min.

SEM Tests
The surfaces of steel fibres were scanned and analysed using a field emission ronmental scanning electron microscopy (SEM: FEG650) and an energy spectro (EDS: EDAX), both supplied by FEI, USA. The scanned samples were 15 mm cub were cut perpendicular to the fibre axis in groups U and Z1 before the pullout tes

SEM Tests
The surfaces of steel fibres were scanned and analysed using a field emission environmental scanning electron microscopy (SEM: FEG650) and an energy spectrometer (EDS: EDAX), both supplied by FEI, USA. The scanned samples were 15 mm cubes that were cut perpendicular to the fibre axis in groups U and Z1 before the pullout tests. The sample surfaces were polished by a diamond grinding wheel and encapsulated in vacuum in transparent resin. The interfacial transition zones (ITZ) of fibre-mortar interfaces were also scanned and analysed using SEM.

Experimental Results and Discussion
3.1. Pullout Load-Displacement (P-∆) Curves and the Bond-Slip Process Figures 3-6 show the pullout load-displacement (P-∆) curves, the mean-value curves, and the detailed curves around the peak load, for all the groups (Z, ZZ, ZH2, and U (untreated fibres)) of specimens. It should be noted that the scales of Y axis (loading) are very different. All these curves are typical to single fibre pullout tests and similar in shape to those reported in [14,41].
Buildings 2021, 11, x FOR PEER REVIEW 5 of 30 sample surfaces were polished by a diamond grinding wheel and encapsulated in vacuum in transparent resin. The interfacial transition zones (ITZ) of fibre-mortar interfaces were also scanned and analysed using SEM.

Pullout Load-Displacement (P-Δ) Curves and the Bond-Slip Process
Figures 3-6 show the pullout load-displacement (P-Δ) curves, the mean-value curves, and the detailed curves around the peak load, for all the groups (Z, ZZ, ZH2, and U (untreated fibres)) of specimens. It should be noted that the scales of Y axis (loading) are very different. All these curves are typical to single fibre pullout tests and similar in shape to those reported in [14,41].

Parameters of Interfacial Bond Properties
To assess the effects of silane coatings on the interfacial behaviour, the follo characteristic parameters are defined and extracted from the experiment P-Δ data: where L and rf are the initial embedment length and radius of fibres, respectively and Ps are the peak load and the load when the whole fibre is debonded from the m (only friction remains afterwards), respectively. τmax is the interfacial bond strength

Parameters of Interfacial Bond Properties
To assess the effects of silane coatings on the interfacial behaviour, the following characteristic parameters are defined and extracted from the experiment P-∆ data: where L and r f are the initial embedment length and radius of fibres, respectively. P max and P s are the peak load and the load when the whole fibre is debonded from the mortar (only friction remains afterwards), respectively. τ max is the interfacial bond strength corresponding to P max based on the assumption of a uniform distribution of interfacial shear stress along the fibre [24]. τ s is the residual frictional bonding strength at the end of the debonding process, without taking the fibre surface roughness into account. τ c denotes the chemical bond strength, and G is the area under the P-∆ curve, representing the energy dissipated during the pullout process. The mean values of the above parameters of three specimens for each coating groups, together with the load at the end of elastic stage P e , are summarized in Table 1, where values in parentheses represent the increase percentages over the results in the U group. The data in Table 1 and their SDs are also presented in  creases in τmax with the increased percentages Δτc > Δτmax > Δτs, ind agents mainly contributed to the increase in chemical bonding situation can be seen for the latter two groups with Z6040 and Z creased the interfacial friction. Furthermore, adding Zr(NO3)4 (g interfacial properties (compared with group Z), while adding H ZH2) performed better than Z6020 alone (group Z2).

The Effects of Silane Coatings on the Interfacial Properties
It can be seen from Table 1 that all of the interfacial bonding properties of coated fibres were improved compared with the untreated fibres. The peak pullout load P max and the bond strength τ max of specimens in Z1 and ZZ1 groups were increased by 575% and 514%, respectively; the increase percentage ranges from 368 to 488% for the groups of Z2, ZZ2, and ZH2, and is 161% and 87% for Z4 and ZZ4, respectively. The least improvement happened to the group Z3 (41%) and ZZ3 (11%), respectively. Similar trends can be found in the improvements of P e , P s (or τ s ), τ c and G, ranging from 37 to 706%, 20 to 279%, 0.1 to 789%, and 2 to 248%, respectively.
As shown in Figure 8, the ratio of τ c/ τ max was 0.59 for the untreated fibres. This ratio increased to 0.71-0.76 for the groups with Z6011 and Z6020 coatings (Z1, Z2, ZZ1, ZZ2, ZH2) but decreased to 0.41-0.55 for those with Z6030 and Z6040 coatings (Z3, Z4, ZZ3, ZZ4), indicating the former ones mainly enhanced the chemical bonding strength τ c and the latter mainly increased the residual frictional bonding strength τ s . Figure 9 compares the total pullout energy consumption G. Comparison of Figures 7 and 9 indicates a strong correlation between G and P max . The increased percentages over the untreated fibres for τ max , τ s , τ c and G are plotted in Figure 10. In general, Z6011 (Z1 and ZZ1) performed the best, followed by Z6020 (Z2, ZZ2, and ZH2) and Z6040 (Z4 and ZZ4), while Z6030 (Z3 and ZZ3) was not effective. The first two groups with Z6011 and Z6020 had high increases in τ max with the increased percentages ∆τ c > ∆τ max > ∆τ s , indicating these two silane agents mainly contributed to the increase in chemical bonding properties. The reverse situation can be seen for the latter two groups with Z6040 and Z6030, which mainly increased the interfacial friction. Furthermore, adding Zr(NO 3 ) 4 (group ZZ) weakened the interfacial properties (compared with group Z), while adding H 2 ZrF 6 to Z6020 (group ZH2) performed better than Z6020 alone (group Z2). Figure 11 presents the SEM micrographs and EDS energy spectra of steel fibres from the groups of U, Z1, and ZZ1 before the pullout tests. The fibre surface looks brighter than the silane coatings due to the steel's higher reflectivity. The surface of the untreated fibre is smooth ( Figure 11a) while those treated fibres are much rougher (Figure 11b,c). Figure 11d-f show that only Fe and C elements exist on the surface of untreated fibres, while additional elements of Si, O, and Ca were detected on the Z1-treated fibre surface and extra Zr and Ca elements were found on ZZ1-treated fibre surface. This suggests the existence of silane coatings and the effectiveness of the coating technique used.

Surface Observation of Steel Fibres
The coating mechanism can be explained in chemical reactions. The chemical structure of silane molecules is represented by R-Si(OX) 3 , where R represents carbon-chain functional groups that can react with organic compounds, such as carbonyl, amino, and epoxy; OX is a group that can react with H 2 O and generate Si-OH, and X is commonly -CH 3 , -C 2 H 5 , etc. According to the chemical bond theory [42], forming of silane coating on fibres involves four step chemical reactions: (1) hydrolysis: the Si-OX group is hydrolysed to Si-OH; (2) condensation: Si-OH is dehydrated and condensed into a multi-molecular structure; (3) hydrogen bonding: Si-OH in the oligomer forms a hydrogen bond with -OH on the surface of steel fibres; and (4) coating formation: a covalent bond is formed with the fibre along with the dehydration reaction during heating and curing. The rough silane coatings increase the contact area and form an anchorage effect between the steel fibres and the mortar matrix, leading to enhanced pullout responses in the treated fibres over the untreated (Figures 3-6).
the groups of U, Z1, and ZZ1 before the pullout tests. The fibre surface looks brighter than the silane coatings due to the steel's higher reflectivity. The surface of the untreated fibre is smooth (Figure 11a) while those treated fibres are much rougher (Figure 11b,c). Figure 11d-f show that only Fe and C elements exist on the surface of untreated fibres, while additional elements of Si, O, and Ca were detected on the Z1-treated fibre surface and extra Zr and Ca elements were found on ZZ1-treated fibre surface. This suggests the existence of silane coatings and the effectiveness of the coating technique used. The coating mechanism can be explained in chemical reactions. The chemical structure of silane molecules is represented by R-Si(OX)3, where R represents car-

ITZ Observation
The SEM images of fibre-mortar interface transitional zones (ITZ) are shown in Figure 12a,b, for the untreated group U and treated group Z1, respectively. It is clearly visible that the interfacial microstructure was relatively loose and there existed some microcracks for the untreated specimen, probably caused by shrinkage from the highly different elastic moduli and thermal expansion coefficients of steel fibres and the mortar. On the contrary, the ITZ microstructure of the sample in the group Z1 is much denser with fewer micro-cracks of smaller width, probably due to enhanced hydration reactions by the silane coating and, thus, stronger and denser C-H-S materials around the ITZ, as reported in [37]. In addition to the anchorage effect described above, the densification of the ITZ due to the fibre treatment leads to higher bonding performances. micro-cracks for the untreated specimen, probably caused by shrinkage from the highly different elastic moduli and thermal expansion coefficients of steel fibres and the mortar. On the contrary, the ITZ microstructure of the sample in the group Z1 is much denser with fewer micro-cracks of smaller width, probably due to enhanced hydration reactions by the silane coating and, thus, stronger and denser C-H-S materials around the ITZ, as reported in [37]. In addition to the anchorage effect described above, the densification of the ITZ due to the fibre treatment leads to higher bonding performances.

Analytical Solutions for the Full-Range Pullout Process
The experimental pullout force-displacement curves and the calculated interfacial properties from single fibre pullout tests are very useful to compare the effectiveness of different silane coatings. However, these curves are geometry and boundary dependent and are not material properties. Therefore, they cannot be used as the constitutive laws of the interface that are needed for the analysis and design of SFRC structures with many randomly distributed fibres. A direct measurement of the interfacial constitutive laws, namely, the interfacial stress-slip relations, involves careful setup of strain gauges along the fibres and is very difficult to conduct for such narrow steel fibres in mm or tenth of mm (e.g., for UHPFRC). Here, we developed an indirect method to characterize a simplified tri-linear softening interfacial bond-slip constitutive law, combining analytical solutions and parameter calibrations against the experimental pullout curves. Closed-form analytical solutions have been derived in our previous study [43] for the whole pullout process of rockbolts. Although mechanically very similar, the solutions in [43] are applicable for pullout problems with the embedment length of fibres/bars/rockbolts being longer than the effective embedment length (see Section 4.3), which is typically not the case for SFRC or UHPFRC with short fibres. A new set of analytical solutions are, thus, derived as below. Figure 13 shows the idealized model of the single fibre pullout tests. The experimental results indicate that the ultimate failure in the pullout tests is debonding and pullout of a very thin layer of mortar attached in the critical interfacial zones. The deformation in this zone is assumed to be lumped into the zero-thickness fibre-matrix interface, thus, the mortar deformation outside this zone is neglected, due to the massive volume and stiffness of the surrounding mortar. The fibre-matrix interface is assumed to be under pure shear and the fibre is assumed to be under uniaxial tension. It is also assumed that the pullout force P is horizontal so that the stress in the protruding length of the fibre is uniform. formation in this zone is assumed to be lumped into the zero-thickness fibre-m terface, thus, the mortar deformation outside this zone is neglected, due to the volume and stiffness of the surrounding mortar. The fibre-matrix interface is ass be under pure shear and the fibre is assumed to be under uniaxial tension. It is sumed that the pullout force P is horizontal so that the stress in the protruding l the fibre is uniform. Based on the assumptions stated above, the governing equations can be esta from force equilibrium considerations:

The Idealized Model and Assumptions
where τ is the shear stress at the interface, σf the axial stress in the fibre, x t coordinate along the fibre's length with its origin at the embedment end and rf t radius. Assuming that the fibre remains linear elastic throughout the pullout the constitutive equation for the fibre is Based on the assumptions stated above, the governing equations can be established from force equilibrium considerations: where τ is the shear stress at the interface, σ f the axial stress in the fibre, x the axial coordinate along the fibre's length with its origin at the embedment end and r f the fibre radius. Assuming that the fibre remains linear elastic throughout the pullout process, the constitutive equation for the fibre is where E f is the Young's modulus of the fibre, u f is the axial displacement of the fibre and δ is the shear slip between the fibre and the mortar. Substituting Equation (6) into Equation (5), the governing equation and the axial stress of the fibre are expressed as: where Equation (7) can be solved once the bond-slip model represented by τ(δ) is defined.

The Tri-Linear Bond-Slip Model
The same tri-linear bond-slip model as used for grout-rockbolt interfaces [43] and fibre reinforced concrete joints [44] was assumed as the interfacial constitutive law for the fibre-mortar interface. As illustrated in Figure 14, the model consists of an ascending linear elastic branch (I) up to the peak stress or bond strength at (δ 1 , τ f ), followed by a softening branch (II) down to (δ f, τ r ), and finally a horizontal branch (III) representing the non-zero residual frictional strength τ r after complete debonding. the fibre-mortar interface. As illustrated in Figure 14, the model consis linear elastic branch (I) up to the peak stress or bond strength at (δ1, softening branch (II) down to (δf, τr), and finally a horizontal branch (III non-zero residual frictional strength τr after complete debonding.
is defined as the ratio of the residual shear strengt strength τf. The tri-linear bond-slip model is assumed as a material pr parameters can be calibrated from pullout tests, as discussed later.

Analysis of the Full-Range Behaviour and Derivation of Analytical Soluti
Ⅰ Ⅱ Figure 14. The tri-linear bond-slip model.
The equation for the bond-slip model is expressed as where k (0 ≤ k ≤ 1) is defined as the ratio of the residual shear strength τ r and the bond strength τ f . The tri-linear bond-slip model is assumed as a material property and all the parameters can be calibrated from pullout tests, as discussed later.

Analysis of the Full-Range Behaviour and Derivation of Analytical Solutions
Using the tri-linear bond-slip model defined above, the shear slip δ and shear stress τ distributions along the interface, the axial stress σ f in the fibre, and the relation between load P-pulling end displacement ∆ can be obtained by solving the governing equation Equation (7) for every loading stage. Figure 15 illustrates the five-stage evolution of interfacial shear stress distribution when the fibre embedment length is shorter than the effective embedment length l e . It is significantly different from the evolution when the embedment length is much longer than l e as in [43]. This leads to very different analytical solutions as derived below. Figure 16 shows the analytical load-displacement curve with the different stages in Figure 15 highlighted. It is not plotted up to scale for clarity. equation Equation (7) for every loading stage. Figure 15 illustrates the five-stage evolution of interfacial shear stress distribution when the fibre embedment length is shorter than the effective embedment length le. It is significantly different from the evolution when the embedment length is much longer than le as in [43]. This leads to very different analytical solutions as derived below. Figure 16 shows the analytical load-displacement curve with the different stages in Figure 15 highlighted. It is not plotted up to scale for clarity.
τ r <τ <τ f Point A in Fig. 16 Point C in Fig. 16 Point B in Fig. 16 (e) Fig. 16 Point E in Fig. 16    τ =τ r , ∆ Ⅲ τ =τ r Ⅱ Ⅲ τ r <τ < Point A in Fig. 16 Point C in Fig. 16 Point B in Fig. 16 (e) In this stage, the pullout force is low and the entire interface remains linear elastic (state I) until the interfacial shear stress at x = L reaches τ f (Figure 15a,b and segment OA in Figure 16). Substituting Equation (10a) for 0 ≤ δ ≤ δ 1 into Equation (7), the differential equation for this stage can be obtained as where Considering the boundary conditions in this stage: the interfacial slip, interfacial shear stress and axial stress in the fibre are obtained by solving the governing equation, Equation (11), as The slip at the loading end with x = L is defined as the fibre displacement, denoted as ∆. The following load-displacement expression can be obtained from Equation (15) As in [43], the effective bond length is defined as the bond length over which the shear stresses offer a total resistance at least 97% of the applied load for an infinite bond length. Based on this definition and considering that tanh(2) ≈ 0.97, the effective bond length in the elastic stage is given by l e = 2/λ 1 . The embedment length is generally less than l e for the steel fibres in SFRC, as will be shown later (Table 2). Equation (18) indicates that P increases proportionally with ∆ in the elastic stage, which ends when ∆ = δ 1 (Point A in Figure 16 and τ = τ f in Figure 15b).

Elastic Softening Stage (ABC)
As the pullout force continues to increase, the shear stress at the loaded end (x = L) starts to decrease and the softening stage begins. The peak shear stress τ f moves towards the embedment end, and a part of the interface near the loaded end enters the softening state (state II), as shown in Figure 15c. When the shear stress at the embedment end reaches τ f , this stage is complete (Figure 15d and Point C in Figure 16).
During the elastic-softening stage, the following differential equations are obtained by substituting Equation (10a) and Equation (10b) into Equation (7): where The boundary conditions are The solutions for the elastic region of the interface with 0 ≤ δ ≤ δ 1 (or 0 ≤ x ≤ L − a) are: where a is the length of the softening region. The solutions for the softening region with δ 1 ≤ δ ≤ δ f (or L − a ≤ x ≤ L) are: The applied load is obtained as: The displacement at x = L can be obtained as To derive the peak force during this stage, the evolution of shear stress in Figure 15b-d are redrawn in Figure 17, where the areas enclosed by τ and the x-axis consist of two segments (a and L − a). They are denoted as A b1 , A b2 , A c1 , A c2 , A d1 , and A d2 , respectively, with A c1 = A b2 and A c2 = A d1 . As the evolution is continuous, there must exist a a short enough to make A c2 > A b1 , so A c > A b with A b = A b1 + A b2 and A c = A c1 + A c2 . Similarly, there must exist a long enough a (a < L) to make A c1 > A d2 , so A c > A d with A d = A d1 + A d2 . Therefore, there must exist an intermediate a = a* (0 < a* < L) that results in the highest area, which corresponds to the peak pullout force at point B in Figure 16. The value of a* can be identified by making the derivative of Equation (32) w spect to a zero: This means either The solution of Equation (35) is a* = L, which is invalid because of a* < L ba the above analysis. However, an analytical solution to Equation (36) is difficult to Define The value of a* can be identified by making the derivative of Equation (32) with respect to a zero: This means either The solution of Equation (35) is a* = L, which is invalid because of a* < L based on the above analysis. However, an analytical solution to Equation (36) is difficult to obtain. Define Equation (36)  The period T of f2(a) can be estimated considering the physical materials used in the experiments. The magnitude (in the internati system) of τf , δf − δ1, Ef and rf is in the order of 10 6 Pa, 10 −3 m, 10 11 Pa tively, so λ2 2 is in the order of 10 2 from Equation (21), that is, 1< λ2 < /( 2 √1 − ) is higher than π/10 = 0.314 m, considering 0 < k < 1. The of steel fibres in this study is L = 0.04 m, which is much lower than T. SFRC and UHPFRC materials, which rarely use steel fibres longer th neering practice. Therefore, there must be only one solution a = a* in for f1(a) = f2(a) or Equation (36).
The value of a* will be obtained using a simple numerical proc The peak force Pu and corresponding displacement Δu at point B in Fig   = 2 This stage ends at Figure 15d or point C in Figure 16 and the ment can be calculated by substituting a = L into Equation (32) and spectively.

Softening Stage (CD)
In this stage, the entire interface enters softening (state II), as il 15e-f and represented by the segment CD in Figure 16. The different  The period T of f 2 (a) can be estimated considering the physical parameters of the materials used in the experiments. The magnitude (in the international standard unit system) of τ f , δ f − δ 1 , E f and r f is in the order of 10 6 Pa, 10 −3 m, 10 11 Pa and 10 −4 m, respectively, so λ 2 2 is in the order of 10 2 from Equation (21), that is, 1 < λ 2 < 10. Therefore, T = π/ λ 2 √ 1 − k is higher than π/10 = 0.314 m, considering 0 < k < 1. The embedment length of steel fibres in this study is L = 0.04 m, which is much lower than T. This also applies to SFRC and UHPFRC materials, which rarely use steel fibres longer than 40 mm in engineering practice. Therefore, there must be only one solution a = a* in the range of (0, L) for f 1 (a) = f 2 (a) or Equation (36).
The value of a* will be obtained using a simple numerical procedure given in 4.4. The peak force P u and corresponding displacement ∆ u at point B in Figure 16 become This stage ends at Figure 15d or point C in Figure 16 and the force and displacement can be calculated by substituting a = L into Equation (32) and Equation (33), respectively.

Softening Stage (CD)
In this stage, the entire interface enters softening (state II), as illustrated in Figure 15e-f and represented by the segment CD in Figure 16. The differential equation can be obtained by substituting Equation (10b) into Equation (7) The general solution of Equation (41) is: Defining the slip at the embedment end as δ 0 , the boundary conditions are: The solutions are: Substituting Equation (45) into Equation (48) gives The displacement at x = L can be obtained from Equation (46) as The range of the variable δ 0 (slip at the embedment end) in Equations (46)-(50) needs to be defined. At the beginning of this stage (Figure 15d or point C in Figure 16), the shear stress at the embedment end reaches τ f with δ 0 = δ 1 . At the end of this stage ( Figure 15f and the point D in Figure 16), the shear stress at the loaded end decreases to τ r with the slip equal to δ f . Substituting τ = τ r and δ = δ f into Equation (46) leads to Therefore, the range of the variable δ 0 in this stage is which now fully defines Equations (46)-(50). Equations (49) and (50) indicate that the pulling force P decreases monotonically with both δ 0 and ∆.

Softening-Debonding Stage (DE)
This stage starts when δ = δ f and τ = τ r at the loaded end (x = L), with the shear stress distribution shown in Figure 15f (Point D in Figure 16). Debonding with residual friction shear strength (τ r ) only then initiates at the loaded end and moves towards the embedment end, until the whole interface is debonded at the point E in Figure 16. In this stage, the softening part is governed by Equation (41) and the debonding part is governed by which is obtained by substituting Equation (10c) into Equation (7). Equation (41) and Equation (53) can be solved considering the following boundary conditions: The solutions for the softening region with 0 ≤ x ≤ a are The solutions for the debonding region with L − a ≤ x ≤ L are The applied load P is obtained from Equation (63) at x = L as The displacement can be obtained from Equation (61) at x = L as Substituting a = 0 into Equations (64) and (65), there are where P d and ∆ d are the load and displacement at the pulling end when the whole interface is debonded, as shown in Figure 15h (point E in Figure 16).

Frictional Stage (EF)
In this stage, the shear resistance is provided by the residual interfacial friction strength τ r only (Figure 15i). The shear stress is a constant The pullout displacement ∆ varies from ∆ d at the beginning of this stage to L when the fibre is completely pulled out. Neglecting the fibre elongation, which is very small compared with ∆, the load-displacement relationship in this stage can be expressed as Equation (69) indicates that P reduces linearly to zero with ∆, as shown by the segment EF in Figure 16.

Calibration of Control Parameters
With the above analytical solutions available, the four parameters δ 1 , τ f , δ f , and τ r (or k) of the tri-linear bond-slip model in Figure 14 can now be calibrated against pullout experimental data of three control points A(∆ A , P A ), B(∆ B , P B ) and E(∆ E , P E ) in Figure 16. These points are chosen because A is the end of the linear elastic stage, B is the peak point, and E is the starting point of the straight line EF, and they can all be easily identified on a pullout experimental curve.
From Figure 15b, it is clear that δ 1 = ∆ A . Substituting δ 1 = ∆ A , P d = P E and ∆ d = ∆ E into Equations (66)-(67), we get: Substituting δ 1 = ∆ A and Equations (70) and (71) into Equation (36) and (39) (P u = P B ) in the second stage (elastic-softening) leads to two equations with two unknown variables τ f and a*, which are difficult to solve analytically. Herein, a simple numerical procedure is used. As seen in Figure 15, the variation of the interface shear stress along short steel fibres is not significant in this stage. It is reasonable (and safe) to assume that it ranges from zero to three times the interfacial bond strength τ max calculated from Equation (1) at the peak point B (P max = P B ). The range of τ f ∈[0,3τ max ] is divided into N number of equal intervals (N = 10,000 used for high accuracy herein). For each τ fi = i(3τ max /N) with i = 1~N, Equation (36) is solved to get a i * using the function fsolve() in MatLab R2019b. τ fi and a i * are then substituted into Equation (39). If the calculated |P u − P B |/P B ≤ 10 −3 , then τ f = τ fi is found as the solution; or otherwise, another loop is carried out with i= i + 1 until the convergence condition is met. Once τ f is solved, k is calculated by Equation (71) and the full tri-linear bond-slip model is calibrated. Table 2 summarises the calibrated parameters for all the ten groups of interfaces and the tri-linear bond-slip models are compared in Figure 19. It can be seen that at the peak load, the corresponding a* is quite close to the embedment length L = 0.04 m, which means the peak point B is close to the point C ( Figure 16). The differences between the analytical peak loads and the experimental data are within 0.05% for all the tests, indicating the high accuracy of the numerical solutions (τ f and a*) to Equation (39). It can also be seen that all types of silane coatings increase the bond strength τ f except the ZZ3 group, with the Z1, ZZ1, and ZH2 groups being the most effective in enhancing both the bond strength τ f and the residual strength τ r . This is highly consistent with the test data analysis in 3.3 and proves that the silane-coating improves the interfacial bond-slip properties, which are indeed the underlying physical/chemical mechanisms for the improved pullout structural behaviour.

Numerical Simulations
The calibrated bond-slip models in Figure 19 can now be used as constitutive laws the interfaces in finite element simulations of SFRC materials and structures. The sin fibre pullout tests are modelled first as a validation of the FE modelling approach and bond-slip laws, followed by simulations of direct tensile tests of an SFRC plate w randomly distributed fibres, as an application. In the FE models, the mortar is simula by the continuum damage plasticity model in ABAQUS 6.13 [45] and fibres by elas plastic beam elements. The fibre-mortar interfaces are modelled as zero-thickness co sive elements (COH2D4), which are inserted between the steel fibres and the mortar. T developed tri-linear bond-slip laws in Figure 19 are used as the constitutive relations the cohesive elements, with the damage index defined by Equation (72). In this way, the deformation/failure modes such as fibre bending and breakage, mortar cracki spalling and crushing, and interfacial bond-slip behaviour, can all be simulated in o model. The details of this discrete continuum coupled modelling approach can be ferred to [46]. The FE simulations are conducted by the ABAQUS/Explicit solver w displacement control at the pulling end.

Modelling of the Single Fibre Pullout Tests
The FE mesh, boundary conditions, and geometry of the single fibre pullout te are shown in Figure 20. The material properties are listed in Table 3. The elemental s is 1 mm along the fibre, and, therefore, there are 40 fibre elements and 40 cohesive int face elements, respectively. Note that the slip distance or embedment length of the co Figure 19. Calibrated tri-linear bond-slip models for all pullout tests (all curves are flat for 10 mm ≤ δ ≤ 40 mm).

Numerical Simulations
The calibrated bond-slip models in Figure 19 can now be used as constitutive laws of the interfaces in finite element simulations of SFRC materials and structures. The single fibre pullout tests are modelled first as a validation of the FE modelling approach and the bond-slip laws, followed by simulations of direct tensile tests of an SFRC plate with randomly distributed fibres, as an application. In the FE models, the mortar is simulated by the continuum damage plasticity model in ABAQUS 6.13 [45] and fibres by elastoplastic beam elements. The fibre-mortar interfaces are modelled as zero-thickness cohesive elements (COH2D4), which are inserted between the steel fibres and the mortar. The developed tri-linear bond-slip laws in Figure 19 are used as the constitutive relations of the cohesive elements, with the damage index defined by Equation (72). In this way, all the deformation/failure modes such as fibre bending and breakage, mortar cracking, spalling and crushing, and interfacial bond-slip behaviour, can all be simulated in one model. The details of this discrete continuum coupled modelling approach can be referred to [46]. The FE simulations are conducted by the ABAQUS/Explicit solver with displacement control at the pulling end.

Modelling of the Single Fibre Pullout Tests
The FE mesh, boundary conditions, and geometry of the single fibre pullout tests are shown in Figure 20. The material properties are listed in Table 3. The elemental size is 1 mm along the fibre, and, therefore, there are 40 fibre elements and 40 cohesive interface elements, respectively. Note that the slip distance or embedment length of the cohesive elements, ordered from the loaded end to the embedment end, is set as (L − δ 1 )/40, 2 (L − δ 1 )/40, 3 (L − δ 1 )/40, . . . , 40(L − δ 1 )/40, respectively. The simulated pullout load-displacement curves are shown in Figures 21-24, with excellent agreement with the analytical solutions and the experimental data for all the ten groups of tests. In particular, the analytical solutions are accurately reproduced by the FE simulations, indicating the effectiveness of FE modelling approach. The slight discrepancies in the last stage may be caused by the damage plasticity of the mortar which is not considered in the analytical solutions but considered in the FE simulations.
solutions but considered in the FE simulations.  Figure 19) The curves in Figure 19 are used for the cohesive elements.   The curves in Figure 19 are used for the cohesive elements.
Buildings 2021, 11, x FOR PEER REVIEW 23 of 30 caused by the damage plasticity of the mortar which is not considered in the analytical solutions but considered in the FE simulations.  Figure 19) The curves in Figure 19 are used for the cohesive elements.    Figure 25 shows the geometry, boundary conditions, and a typical F an assumed SFRC plate with random fibres under uniaxial tension. The size is 0.4 mm. Five mesoscale FE models with untreated fibres and th four types of silane coatings (Z1-Z4), respectively, are simulated. The sa rameters listed in Table 3 and the same distribution of fibres are used Figure 26 shows the final crack patterns. It can be seen that the model main crack, whereas the Z1-Z4 models have two main cracks. In ad    Figure 25 shows the geometry, boundary conditions, and a typical F an assumed SFRC plate with random fibres under uniaxial tension. The size is 0.4 mm. Five mesoscale FE models with untreated fibres and th four types of silane coatings (Z1-Z4), respectively, are simulated. The s rameters listed in Table 3 and the same distribution of fibres are used Figure 26 shows the final crack patterns. It can be seen that the mode main crack, whereas the Z1-Z4 models have two main cracks. In ad more secondary cracks in the models Z1 and Z2 than the others. This is d bonding properties of interfacial cohesive elements in Z1 and Z2 than t  Figure 25 shows the geometry, boundary conditions, and a typical FE mesh used for an assumed SFRC plate with random fibres under uniaxial tension. The mean elemental size is 0.4 mm. Five mesoscale FE models with untreated fibres and those treated with four types of silane coatings (Z1-Z4), respectively, are simulated. The same material parameters listed in Table 3 and the same distribution of fibres are used for comparison. Figure 26 shows the final crack patterns. It can be seen that the model U has only one main crack, whereas the Z1-Z4 models have two main cracks. In addition, there are more secondary cracks in the models Z1 and Z2 than the others. This is due to the higher bonding properties of interfacial cohesive elements in Z1 and Z2 than the others, which have higher constraints in the mortar and, thus, more cracks to dissipate energy. This is also reflected in the simulated cross-sectional average stress-displacement curves, as shown in Figure 27. All of the models with silane coated fibres show improvement on the load-carrying capacities in terms of both structural strength and ductility (i.e., the area or energy consumption under the curve). In particular, the model Z1 exhibits the most significant strain-hardening effect after the first crack appears with the highest ductility, followed by Z2 and Z4 models. This is again consistent with the used bond-slip relations as shown in Figure 19 for different silane treated fibres. These results demonstrate that the technique of silane coating of fibres can not only enhance structural performance, but also can change the structural failure mode from brittle to ductile.

Direct Tensile Tests of SFRC Plates with Random Fibres
Buildings 2021, 11, x FOR PEER REVIEW 26 of 30 ductility, followed by Z2 and Z4 models. This is again consistent with the used bond-slip relations as shown in Figure 19 for different silane treated fibres. These results demonstrate that the technique of silane coating of fibres can not only enhance structural performance, but also can change the structural failure mode from brittle to ductile. It is noteworthy that the simulated failure patterns in Figure 26 can be directly validated only when real internal distributions of fibres are available and taken into account in the FE models. This can be realized by advanced techniques such as in-situ X-ray computed tomography scanning under progressive loadings and high-fidelity image-based FE simulations [47,48].     ductility, followed by Z2 and Z4 models. This is again consistent with the used bond-slip relations as shown in Figure 19 for different silane treated fibres. These results demonstrate that the technique of silane coating of fibres can not only enhance structural performance, but also can change the structural failure mode from brittle to ductile. It is noteworthy that the simulated failure patterns in Figure 26 can be directly validated only when real internal distributions of fibres are available and taken into account in the FE models. This can be realized by advanced techniques such as in-situ X-ray computed tomography scanning under progressive loadings and high-fidelity image-based FE simulations [47,48].

Conclusions
This study has carried out a systematic investigation of the effects o silane coatings on the steel fibre-mortar interfacial bond properties, com tests, analytical solutions, and mesoscale FE simulations. The following ma can be drawn: (1) The 30 single fibre pullout tests show that coating steel fibres by silane solutions leads to 11-575% increase on the bonding strength at peak increase on the chemical bond strength, 20-279% increase on the resi strength, and 2-248% increase on the pullout energy. Overall, the pure Z6011 is the most effective one, followed by Z6020 and Z6040, whereas t tion is not effective. The addition of H2ZrF6 further enhances the bond st addition of Zr(NO3)4 has adverse effects. The Z6020, Z6011, and Z6020 so with H2ZrF6 mainly improve the chemical bond strength, while the Z6030 a mainly improve the residual frictional strength.
(2) Based on a simple tri-linear interfacial bond-slip model, closed-f solutions of the load-displacement curves and the distributions of fibre facial stresses and slips along the embedment length, are derived for fiv whole pullout process for relatively short embedment length of fibres. Th ters δ1, τf, δf, and τr (or k) in the bond-slip model are calibrated against the p for untreated steel fibres and those with the nine types of silane coatings.
(3) The calibrated bond-slip models are then successfully used as int It is noteworthy that the simulated failure patterns in Figure 26 can be directly validated only when real internal distributions of fibres are available and taken into account in the FE models. This can be realized by advanced techniques such as in-situ X-ray computed tomography scanning under progressive loadings and high-fidelity image-based FE simulations [47,48].

Conclusions
This study has carried out a systematic investigation of the effects of nine types of silane coatings on the steel fibre-mortar interfacial bond properties, combining pullout tests, analytical solutions, and mesoscale FE simulations. The following main conclusions can be drawn: (1) The 30 single fibre pullout tests show that coating steel fibres by nine types of silane solutions leads to 11-575% increase on the bonding strength at peak load, 0.1-789% increase on the chemical bond strength, 20-279% increase on the residual frictional strength, and 2-248% increase on the pullout energy. Overall, the pure silane solution Z6011 is the most effective one, followed by Z6020 and Z6040, whereas the Z6030 solution is not effective. The addition of H 2 ZrF 6 further enhances the bond strength but the addition of Zr(NO 3 ) 4 has adverse effects. The Z6020, Z6011, and Z6020 solutions mixed with H 2 ZrF 6 mainly improve the chemical bond strength, while the Z6030 and Z6040 ones mainly improve the residual frictional strength.
(2) Based on a simple tri-linear interfacial bond-slip model, closed-form analytical solutions of the load-displacement curves and the distributions of fibre strength, interfacial stresses and slips along the embedment length, are derived for five stages of the whole pullout process for relatively short embedment length of fibres. The four parameters δ 1 , τ f , δ f , and τ r (or k) in the bond-slip model are calibrated against the pullout test data for untreated steel fibres and those with the nine types of silane coatings.
(3) The calibrated bond-slip models are then successfully used as interfacial constitutive laws to simulate the single fibre pullout tests and more complicated damage and fracture behaviour of multi-fibre tensile models, using a discrete-continuum coupled mesoscale finite element approach. The numerical simulations demonstrate that the technique of silane coating of fibres can not only enhance the structural performances in terms of higher strength and ductility, but also can change the structural failure mode from brittle to ductile.
(4) The developed approach of combining single fibre/bar pullout tests, analytical solutions and numerical modelling, provides a systematic and feasible way to investigate the effects of interfacial properties between fibres/bars and mortar/concrete, which are difficult to measure directly from pullout tests (e.g., using conventional strain gauges), on the complicated deformation and failure behaviour of SFRC, UHPFRC, and RC structural members.
(5) Experimental and numerical studies of silane-coated SFRC structural members need to be carried out under complicated loading conditions before this technique can be confidently used in engineering practice. Thinner fibres should also be studied to investigate potential scale effects of the fibre diameter.  Data Availability Statement: Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.