When exposed to a magnetic field, the steel fibers become magnetized with N and S poles at each end. The steel fiber rotates and turns in a direction that coincides with the magnetic induction lines of the magnetic field. Therefore, forming a magnetic field with a fixed direction of the magnetic induction lines is essential for aligning steel fibers using a magnetic field. The bar magnet with a rectangular cross-section (bar magnet for short in this study) creates a zone where the magnetic induction lines are almost straight, as shown in Figure 1. That means the intensity of the magnetic field components in the Y- and X-directions is much smaller than in the Z-direction. These components are so small that they can be ignored. However, due to the limitation of the dimension, a single bar magnet cannot provide sufficient size and intensity of the magnetic field to rotate the steel fibers in fresh cement-based materials. Then, the connection of a series of bar magnets is required to ensure that the magnetic field is sufficiently strong and big. In addition, using multiple magnets creates a more flexible magnetic field that can easily adapt to the size of engineering structures. When bar magnets are connected in series, the magnetic fields of the magnets superpose, and the magnetic induction lines become flat and dense. In addition, wavy magnetic induction lines are formed between the gaps of two adjacent magnetic poles (i.e., the magnetic leakage), as shown in Figure 1. The wavy magnetic induction lines can be neglected when the gap spacing between two adjacent magnetic poles is small enough. The magnetic field with quasi-straight magnetic induction lines at the lower side of the assembled magnets can then be used to align the fibers to prepare ASFRCs.

Liu (Li et al., 2008) derived an analytical equation to describe the magnetic field distribution around a single magnet based on Biot-Savart’s law and Ampere’s molecular circulation hypothesis. This study derives an analytical expression of the magnetic field distribution around the assembled bar magnets based on Liu’s work. Figure 2 illustrates a rectangular-cross section bar magnet with dimensions a, b, and h.

According to Ampere’s molecular circulation hypothesis (Young et al., 1996; Tang et al., 2021), the closed current loop on the side surface of the bar magnet excites the magnetic field outside the bar magnet. The magnetic induction at any point P(x, y, z) outside the bar magnet can be expressed as Eqs 1–3 (Li et al., 2008): B x = K 2 − Γ a − x , b − y , z − Γ a − x , y , z + Γ x , b − y , z + Γ x , y , z (1) B y = K 2 − Γ b − y , a − x , z − Γ b − y , x , z + Γ y , a − x , z + Γ y , x , z (2) B z = K Ψ b − y , a − x , z − Ψ y , a − x , z − Ψ a − x , b − y , z − Ψ x , b − y , z − Ψ b − y , x , z − Ψ y , x , z − Ψ a − x , y , z − Ψ x , y , z (3)

The auxiliary functions' expressions are expressed by Eqs 4–6: Γ γ 1 , γ 2 , γ 3 = ln γ 1 2 + γ 2 2 + γ 3 − z 2 − γ 2 γ 1 2 + γ 2 2 + γ 3 − z 2 + γ 2 z = h z = 0 (4) Ψ ψ 1 , ψ 2 , ψ 3 = arctg ψ 1 ψ 2 − ψ 3 − z ψ 1 2 + ψ 2 2 + ψ 3 − z 2 z = h z = 0 (5) K = μ 0 J s 4 π (6) where μ ₀ is the vacuum permeability, equals to 4π×10^–7(H/m), and J _s is the magnetization current density (A·m) in the loop, which is related to the magnetic properties of permanent magnet materials.

Assuming that the assembled magnets are arranged in m rows and n columns. The superimposed magnetic induction intensity at any point P (x, y, z) outside the assembled magnets can be expressed using the Eqs 7-18: B x t = ∑ i = 1 m B x a i + ∑ j = 1 m B x b j + ∑ k = 1 m B x c k . . . (7) B x a i = B x x , y − b , − z − i − 1 h (8) B x b j = B x x − a + s d , y − b , − z − j − 1 h (9) B x c k = B x x − 2 a + s d , y − b , − z − k − 1 h (10) B y t = ∑ i = 1 m B y a i + ∑ j = 1 m B y b j + ∑ k = 1 m B y c k + ⋯ (11) B y a i = B y x , y − b , − z − i − 1 h (12) B y b j = B y x − a + s d , y − b , − z − j − 1 h (13) B y c k = B y x − 2 a + s d , y − b , − z − k − 1 h (14) B y t = ∑ i = 1 m B y a i + ∑ j = 1 m B y b j + ∑ k = 1 m B y c k + ⋯ (15) B y a i = B y x , y − b , − z − i − 1 h (16) B y b j = B y x − a + s d , y − b , − z − j − 1 h (17) B y c k = B y x − 2 a + s d , y − b , − z − k − 1 h (18) where B _xt , B _yt , and B _zt are the magnetic induction components of the assembled magnets along the X-, Y-, and Z-direction, respectively, at any point (T); B _xai , B _xbj , and B _xck are the magnetic induction components of any square magnet in the assembled magnets along the X-direction at any point (T); B _yai , B _ybj , and B _yck are the magnetic induction components of any square magnet in the assembled magnets along the Y-direction at any point (T); B _zai , B _zbj , and B _zck are the magnetic induction components of any square magnet in the assembled magnets along the Z-direction at any point (T); and s _d is the distance between the combined bar magnets (mm).

When a sufficiently large external magnetic field created by the assembled magnets is applied to fresh steel fiber-reinforced cement-based materials, the fiber overcomes the yield stress of the cement to rotate. At this stage, the steel fiber is subjected to magnetic force, flow resistance, gravity, and buoyancy, as shown in Figure 3. Gravity and buoyancy are in the vertical direction and have no influence on the rotation of the steel fibers, but only on the motions of sinking or floating.

Ordinary Portland cement type P∙O 42.5 (Chinese Standard of GB 175–2007) and river sand with a fineness modulus of 2.5 (Chinese Standard of GB 14684–2022) were used to prepare steel fiber-reinforced cement mortar. A straight steel fiber with a length of 30 mm and 0.5 mm diameter was used, and the properties of steel fiber were listed in Table 1.

A superplasticizer was used to control the rheology of the cement mortar with a 0.36 water-cement ratio. Two reference mortar mixes were designed with a water-cement ratio of 0.36 and 0.42, respectively (Table 2). The rheology properties of these mortars are also tested by using the rheometer ViskomatTM NT, following the method proposed by (Banfill, 1994). The rheology properties of cement mortar with different W/C were also listed in Table 2.

Each reference mortar mix was used to prepare four types of steel fiber-reinforced specimens by incorporating steel fibers at volume fractions of 0.4%, 0.8%, 1.2%, and 1.6%. For each steel fiber-reinforced cement mortar mix, three types of fiber orientation of specimens were prepared from the same batch: SFRC with random fiber orientation (RSFRC), ASFRC prepared using an internal uniform magnetic field offered by solenoid coil (I-ASFRC), and ASFRC prepared using an external magnetic field offered by assembled magnets (E-ASFRC).

The I-ASFRC was prepared following the method proposed by Mu (Mu et al., 2017). During the preparation of the I-ASFRC, an aligning setup with an electric magnetic coil and adjustable DC power was used to provide an approximately uniform magnetic field with adjustable magnetic field strength, as shown in Figure 6. After mixing, fresh composites were poured into a plastic mold. The plastic mold was placed inside the electric magnetic coil device. Then, the mold, together with the device of the electric magnetic coil, was placed on the compacting table. The aligning setup and compacting table were turned on for approximately 40 s, and then the compacting table was first switched off. When the compacting table completely stopped, the alignment setup was switched off. The plastic mold was pulled out from the coil and placed in a curing room.

To prepare the E-ASFRC, the assembled magnets shown in Figure 7 were used to offer a stable external magnetic field. The device consists of rubidium iron boron magnet pieces. The size of the rubidium iron boron magnet was 20 mm × 20 mm×5mm, and the surface magnetic density of the magnet was 280 mT. Several rubidium iron boron magnets are assembled into bar magnets, and several bar magnets are combined to form a square device that offers an external magnetic field. The device’s dimensions can be adjusted by controlling the number of iron boron magnet pieces and iron boron magnet pieces to meet the preparation requirements of E-ASFRC with different sizes. The advantage of the assembled magnets is that the size of the assembled magnets (or the number of rubidium iron boron magnets) can be flexibly adjusted according to the size of the specimen to be prepared. The larger the size of the specimen, the greater the size of the assembled magnet and the number of magnetic sheets required. The size of the specimens used in this study was 100 mm × 100 mm × 400 mm. Based on the size of this specimen, assembled magnets with dimensions of 130 mm × 150 mm × 26 mm, 130 mm × 450 mm × 26 mm, and 390 mm × 150 mm × 26 mm were prepared to investigate the influence of the device dimensions on the magnetic field distribution of the external magnetic field and the influence of magnetic field on the fiber orientation of E-ASFRC preparation.

The E-ASFRC was prepared using the magnet assembled above. In the preparation of E-ASFRC, the fully mixed fresh composites were poured into a plastic mold, and the mold was placed on the compacting table. A partition board was placed on the plastic mold to prevent the steel fiber in the mixture from being taken up by the assembled magnets. After the assembled magnets and partition board were placed appropriately on the mold, the compacting table was turned on to vibrate. After vibration, the assembled magnets and partition board were removed and the specimen was demolded after curing for 24 h. To investigate the influence distance between the assembled magnets and specimen surface, E-ASFRCs were prepared using assembled magnets with heights of 0, 10, 20, and 30 mm, respectively, as shown in Figure 8. The specimen was divided into four distinct regions with a height of 25 mm in each region to investigate the variation in fiber orientation with different height positions (as shown in Figure 8). The fiber orientation was separately measured for each region. The fractured cross-section and fiber distribution of the prepared E-ASFRC and I-ASFRC are shown in Figure 9. As shown in Figure 9, the steel fibers in both E-ASFRC and I-ASFRC are aligning well and homogeneously distributed in the matrix. The fiber orientation factor is calculated according to the angle between the fiber and the normal vector of the fractured cross-section, and the angle of each fiber was measured using a protractor.

Figure 12 shows the variation of the component of magnetic induction intensity in the Z-direction offered by assembled magnets with sizes of 130 mm × 150 mm × 26 mm, as calculated theoretically and measured experimentally. The theory calculation results are in excellent agreement with the experimental results, with an error of less than 5%. A possible cause of the low errors in the simulation may be that the magnetic leakage between two adjacent magnets disturbs the magnetic field distribution in a real situation. As shown in Figure 12, the magnetic induction intensity component in the Z-direction exceeded 10 mT at a position near the assembled magnets (Y = 26 mm), and the maximum value can reach 65 mT. As the distance between the measured height and the assembled magnets increases, the magnetic induction intensity component along the Z-direction decreases, and the minimum magnetic induction intensity component along the Z-direction decreases to 3 mT at the plane of Y = 126 mm.

The variations of the magnetic induction intensity component in the X-direction, B _xt, generated by the assembled magnets with sizes of 130 mm × 150 mm × 26 mm at the plane of Y = 26 mm and Y = 126 mm are shown in Figure 13. The results indicate that the magnetic induction intensity component in the X-direction is very small (less than 10 mT), except for the four corner areas of the plane. Therefore, the magnetic induction intensity component along the X-direction has little effect on aligning steel fibers for ASFRC, consistent with the analysis and assumptions in Section 2. The external magnetic field generated by the device of the assembled magnets can realize the alignment of steel fibers in a certain direction (the Z direction) to realize the preparation of E-ASFRC.

The variations in the magnetic induction intensity component along the Z-direction with the height for different assembled magnet sizes are shown in Table 3. The results indicate that at a position near the assembled magnets, the magnetic induction intensity component along the Z-direction at both ends of the plane is higher than that in the middle zones. As the height (Y) increased, the decay rate of the induction intensity component along the Z-direction at both ends was faster than that in the middle zones. The difference in magnetic induction intensity between both ends and the middle zone gradually decreases, or even the magnetic induction intensity component along the Z-direction in the middle zone exceeds that at both ends when the height (Y) is greater than 76 mm (except for the assembled magnets of dimensions 130 mm × 450 mm × 26 mm). The results also indicate that increasing the length of the magnetic pieces (along the Z-direction) increases the magnetic field intensity of each area. Referring to the study by Mu (Mu et al., 2021), assembled magnets with dimensions of 130 mm × 150 mm × 26 mm can be used to prepare the E-ASFRC specimen with dimensions of 100 mm × 100 mm × 400 mm. However, increasing the width of the magnetic pieces (along the X-direction) does not increase the magnet induction intensity. Instead, it increased the decay rate of the magnetic induction intensity in the height (Y) direction. Therefore, it is unnecessary to increase the width of the magnetic devices. In applications, the width only needs to meet the size of the prepared E-ASFRC.

The variation of fiber orientation with the assembled magnets device (130 mm × 150 mm×26 mm) for E-ASFRC with different water-cement ratios and fiber contents is predicted by Eq. 34, and the experiment. The parameters used in Eq. 34 are listed in Table 4, and the calculated and experimental results are listed in Table 5.

The results indicate that the calculation results agree well with the experimental results. This model can predict the fiber orientation coefficient of an E-ASFRC prepared using the assembled magnets. As the distance between assembled magnets and specimen surface (i.e., the height of assembled magnets) increases, the fiber orientation coefficient of the four measured regions in the E-ASFRC decreases. For 0.36 of water-cement ratio, the largest decay of the fiber orientation coefficient is observed in region D, which decreases from 0.893 to 0.649 when the height of assembled magnets and specimen surface increases from 0 to 30 mm. For the E-ASFRC prepared using the 0 mm height of the assembled magnets device, the orientation coefficient of region D reaches 0.953. However, some steel fibers on the substrate surface of the matrix are attracted by the assembled magnets device, which is almost perpendicular to the substrate surface of the matrix, as shown in Figure 14. The phenomenon that the steel fibers stick out from the surface is caused by the magnetic leakage between the two adjacent rubidium iron boron magnets. Although the leakage between adjacent magnetic sheets is small, when the assembled magnet is very close to the surface of the specimen, this leakage is sufficient to attract steel fibers to expose the surface of the specimen. Therefore, the orientation coefficient of region A is only 0.642 when the distance between the assembled magnets and the specimen surface is 0 mm. For the E-ASFRC prepared using the 10 mm height of the assembled magnets device, the orientation coefficients of the four regions are higher than 0.9, and the magnets do not attract the steel fiber on the substrate surface of the matrix. Therefore, a height of 10 mm is optimum for preparing E-ASFRC with dimensions of 100 mm × 100 mm × 400 mm. This height can be used when preparing test pieces of other sizes, and the required magnetic field strength can be calculated using Eq. 34. In addition, the orientation coefficient also decreases as the water-cement ratio decreases. Therefore, a stronger magnetic field is needed to prepare the E-ASFRC with a lower water-cement ratio.

The flexural load-deflection curves of E-ASFRC prepared using a magnet with a height of 10 mm and size of 130 mm × 150 mm×26 mm, I-ASFRC, and RSFRC are shown in Figures 15, 16. The flexural strength changes with fiber content and water-cement ratio are shown in Figure 17. Because the fiber orientation coefficient of E-ASFRC is closer to that of I-ASFRC, the flexural strength and deflection curve of E-ASFRC are closer to those of I-ASFRCs and are much higher than those of RSFRC. These results indicate that using the assembled magnets to prepare ASFRCs can replace the solenoid coil.

For a water-cement ratio of 0.42, compared to that of the RSFRC with the same fiber content, the flexural strength of E-ASFRC with 0.4, 0.8, 1.2 and 1.6 vol% fiber content increases to 30.8%, 40.7%, 68.3%, and 91.3%, respectively (as shown in Figure 16). For a water-cement ratio of 0.36, compared to that of the RSFRC with the same fiber content, the flexural strength of E-ASFRC with 0.4, 0.8, 1.2, and 1.6 vol% fiber content increases to 16.5%, 14.6%, 67.3%, and 60.3%, respectively. As the water-cement ratio decreases, the bonding strength between the steel fiber and matrix increases (Li et al., 2022), increasing the bridging effect of the steel fiber and resulting in a more pronounced increase in the flexural strength of E-ASFRC with lower water-cement ratios, as shown in Figure 17.

Table 6 shows the toughness indices of RSFRC, E-ASFRC, and I-ASFRC. For a water-cement ratio of 0.42, compared to that of the RSFRCs with the same fiber content, the flexural toughness of E-ASFRCs with 0.4, 0.8, 1.2, and 1.6 vol% fiber content increases to 135.7%, 129.8%, 148.2%, and 87.3%, respectively. For a water-cement ratio of 0.36, compared to that of the RSFRCs with the same fiber content, the flexural toughness of E-ASFRCs with 0.4, 0.8, 1.2, and 1.6 vol% fiber content increases to 69.8%, 92.9%, 54.8%, and 95.2%, respectively. In conclusion, the flexural toughness of E-ASFRC and I-ASFRC are similar and significantly improved compared to RSFRC.