Viscoelastic Properties of Cementitious Paste without Magnetic FieldFig. 2 presents the typical oscillatory strain-sweep testing results of the studied cementitious pastes without a magnetic field. It can be seen that the storage modulus of all the cementitious pastes shows linear behavior at relatively low shear strain. As soon as the shear strain reaches a critical value, the storage modulus starts to decline dramatically. This indicates that all cementitious pastes exhibit viscoelastic behavior regardless of the solid volume fractions, which is in agreement with (Schultz and Struble 1993; Conte and Chaouche 2016). As the solid volume fraction increases, the storage modulus at LVER exhibits an increase, indicating an increase in the stiffness. This is due to the improvement of the interparticle contacts and the network strength with increasing solid volume fraction (Roussel et al. 2010; Mahaut et al. 2008b; Jiao et al. 2017).With regard to the critical strain, it decreases with increasing the solid volume fraction from 0.101 to 0.326. When the solid volume fraction is higher than 0.326, however, the critical strain almost remains unchanged. Similar phenomena were also observed in Yuan et al. (2017), Ukrainczyk et al. (2020), and Mostafa and Yahia (2016), but no detailed explanation was provided. The critical strain from the oscillatory strain-sweep test corresponds to the breakage of early hydration products between cement particles (Roussel et al. 2012; Huang et al. 2019; Jiao et al. 2021a). Cementitious paste with lower solid volume fraction and higher free water generally has a higher dissolution rate of cement particles and thus a higher concentration of Ca2+ ions (Hošková et al. 2009; Liu et al. 2017), resulting in more bridges of initial C–S–H and ettringite. Note that the formation of more hydration products does not necessarily mean a higher stiffness or more solid-like properties of the suspension, as a higher w/c system requires more products to develop a percolating structure due to the larger solid-to-solid spacing (Zhang et al. 2010). Furthermore, the C–S–H gel in a cement paste with a higher particle volume fraction possibly becomes more fragile (Ukrainczyk et al. 2020). Consequently, the deformation capacity of the bridges of early hydration products increases with the decrease of solid volume fraction, and thus a slightly higher critical strain is observed for the cementitious pastes with lower solid volume fraction.Magnetorheological ResponseThe rheological response of cementitious paste to an external magnetic field is characterized by the early structural build-up. Fig. 3 shows the evolutions of the structural build-up of cementitious pastes at various solid volume fractions. The cementitious pastes with ϕT of 0.101, 0.417, and 0.469 were selected to represent low, moderate, and high solid volume fractions, respectively. It can be seen that the storage modulus, loss modulus, and phase angle of the representative cementitious pastes show distinctly different evolutions with and without an external magnetic field. Specifically, the storage modulus was higher than the loss modulus at the beginning of applying the external magnetic field due to the microagitation effect of the magnetic nano-Fe3O4 particles (Jiao et al. 2021c, d). With elapsed time of magnetization, the storage modulus gradually increased, while the loss modulus increases to a peak and then decreases to a steady state. This results in a gradual reduction in the phase angle. After a sufficiently long period of magnetization, e.g., 100 s, the storage modulus under the magnetic field was significantly higher than that without a magnetic field, while the phase angle showed an opposite behavior. This indicates a higher stiffness of the cementitious paste under an external magnetic field, which can be attributed to the formation of magnetic chains or clusters (Jiao et al. 2019, 2021b; Nair and Ferron 2014).For the cementitious paste with ϕT of 0.101 under an external magnetic field of 0.5 T, despite the significant increase in the storage modulus [e.g., the storage modulus at 300 s (denoted as Gt=300 s’) increasing from 4 to 36 kPa by approx. 800% in Fig. 3(a)], the peak of the loss modulus was less pronounced compared to the cementitious paste with moderate solid volume fraction (i.e., ϕT=0.417) in Fig. 3(c). Furthermore, it seems that the phase angle of the cementitious paste with ϕT of 0.101 required a slightly shorter time to reach a stable state compared to that of paste with ϕT of 0.417, as can be observed from Figs. 3(b and d). This indicates that after exposing a cementitious paste to an external magnetic field, the moving nanoparticles and disturbed cement particles (by the microagitation effect) arrive at their equilibrium state more quickly for the cementitious paste with low solid volume fractions possibly due to the relatively low movement resistance. On the other hand, a limited MR response was observed in the cementitious paste with a high solid volume fraction (ϕT=0.469). More specifically, the increase of the loss modulus only happened at the beginning of applying the external magnetic field, as shown in Fig. 3(e), and it reached its peak very fast compared to the cementitious paste with ϕT of 0.417. Meanwhile, the phase angle needs a shorter time to arrive at the steady state in Fig. 3(f). The results indicate that the movement of the nano-Fe3O4 particles is possibly limited by the dense packing of the solid particles. Nevertheless, some magnetic clusters of nano-Fe3O4 particles still might be formed in the gaps between cement particles, as reflected by the increase in the storage modulus and the slight reduction in the phase angle.The storage modulus at the end of the time-sweep test (i.e., Gt=300 s’) and the magnetorheological effect of the cementitious pastes are shown in Fig. 4. Two parameters, one is the difference of Gt=300 s’ between 0 and 0.5 T (absolute MR effect, kPa) and the other is the relative change of Gt=300 s’ (relative MR effect, %), calculated by Eqs. (1) and (2), respectively, are used to describe the MR response of the cementitious pastes (1) Absolute MR effect=Gt=300s′(0.5T)−Gt=300s′(0T)(2) Relative MR effect=Gt=300s′(0.5T)−Gt=300s′(0T)Gt=300s′(0T)×100where Gt=300 s’(0 T) and Gt=300 s’(0.5T) = storage modulus at 300 s under 0 and 0.5 T, respectively. It can be seen that the solid volume fraction can be divided into three regions according to the magnitude of the absolute MR effect. At low solid volume fractions (e.g., ϕT<0.3), the values of Gt=300 s’ obtained without a magnetic field are in a low magnitude, almost independent of the solid volume fraction. After applying an external magnetic field of 0.5 T, Gt=300 s’ showed a lower increase, and thus only a small difference of Gt=300 s’ was observed. It should be mentioned that the MR response can still be regarded as obvious if described by the relative MR effect [as shown in Fig. 4(b)] because of the extremely low value of the Gt=300 s’ at 0 T. At moderate solid volume fractions (e.g., 0.3<ϕT<0.45), Gt=300 s’ increases exponentially in the absence of the magnetic field. While under the magnetic field of 0.5 T, the stiffness or solid-like behavior of the cementitious pastes is significantly enhanced. Thereby, the absolute MR effect increases dramatically with increasing solid volume fraction, while the relative MR effect keeps a relatively low value of 0.7 due to high Gt=300 s’ at 0 T. By contrast, at relatively high solid volume fractions (e.g., ϕT>0.45), the cementitious pastes possess very high stiffness and solid-like properties even without magnetic field, and thus Gt=300 s’ shows less change after applying an external magnetic field of 0.5 T. Consequently, the absolute MR effect decreases and the relative MR effect is almost close to zero, indicating a very limited MR response. In a word, at the same concentration of magnetic nanoparticles, only cementitious pastes with appropriate viscoelasticity can exhibit an obvious rheological response to an external magnetic field.