4.1.4. Standard Error

Standard error values for the various rheological models were calculated using Equation (5) and are given in Table 6. A lower value of standard error represents the best-fitting of the mathematical model to the flow curve. It was observed that HB model and Modified Bingham model showed less standard error while Bingham model and Casson model showed higher values of standard error. As HB and Modified Bingham models are nonlinear, they predicted the flow behavior more accurately. Bingham model has a linear mathematical relation and hence it showed higher standard error. Casson model has a limitation in predicting the concentrated suspension [19] and hence it is believed that it showed larger values of standard error. It was noted that, for M0c sample, all of the rheological models showed large values of standard error. It may be due to some calculation and experimental error regarding the flow curve data, as, for the same high shear rate cycle range, GM3c specimen showed comparatively low standard error values. GM10 mix showed the maximum standard error values for all mathematical models. As higher dosage of nanomaterials, i.e., graphene and graphene oxide, results in the formation of flocculation structures in cement paste [7]. Therefore, it is believed that the higher values of standard error may be due to the formation of these flocculated suspensions in the cement paste.


**Table 6.** Values of standard error for various rheological models.

#### *4.2. Fresh and Hardened Properties of GNP–Cement Composite*

The results of workability test are presented in Table 7. The flow diameter of GM3 was found to be 8.5% less when compared to M0. Pan et al. [36] used 0.05% of graphene oxide in cement paste (w/c = 0.5) and observed 41.7% reduction in the slump diameter in comparison to control cement paste. Similarly, for 2% carbon nanotubes in cement paste (w/c = 0.5), 48.9% reduction in slump diameter was found in comparison to control cement paste [37]. Possible reason for reduction in flow diameter could be the large surface area of the graphene sheets, which require more amount of water for lubrication and in turn decrease the free available water. Therefore, the overall workability of the cement paste was reduced by addition of graphene in cement paste.

**Table 7.** Flow diameter, maximum compressive load and corresponding resistivity values.


Stress–strain curves for the M0 and GM3 are given in Figure 4. It can be seen that the addition of GNP greatly enhances the load carrying capacity of the cement paste (Table 7). The enhancement in compressive strength was about 30% as compared to mix M0. It was also observed that GM3 specimen showed more ductile behavior as compared to M0. The percentage strain produced in GM3 increased up to 73%, showing ductile nature of graphene–cement paste. The increase in compressive strength and strain may be attributed to the higher strength of graphene, template effect, and crack bridging by graphene sheets [38]. These results are in agreemen<sup>t</sup> with the study of Pan et al. [36] performed on graphene oxide based cement composites. For cement paste containing 0.05% graphene oxide, an improvement of about 22% in compressive strength was observed when compared to control specimen.

**Figure 4.** Stress–strain curve for cement paste (M0) and graphene–cement paste (GM3).

To study the possible reasons for the increase in compressive strength and the improvement in ductile behavior, the morphology of cement based composite was investigated. Figure 5 shows the morphology of the GNP–cement composite at 7 days and 28 days. In FESEM images of GM3 captured at seven days (Figure 5a), needle form of calcium silicate hydrate (CSH), hexagonal plates of portlandite, graphene nano particle can be observed. ESB or backscattered image was used to distinguish the carbon materials. It can be seen in Figure 5b that graphene nanoparticle are completely black. EDX image was further employed to confirm the presence of graphene and hydrated cement products. The EDX of graphene sheets (Figure 5c) for the location indicated in Figure 5a shows maximum carbon content, clearly confirming the presence of graphene. The EDX for the needle shape CSH in Figure 5d for the location marked in Figure 5b show maximum content of oxygen followed by the silicon and carbon. At 28 days, the needle form of CSH transformed into honeycomb structure of CSH as shown in Figure 5e. The backscattered image or ESB of Figure 5e shows that the hydrated products grow over graphene (Figure 5f). EDX was also employed in Figure 5g,h to confirm the presence of graphene and honeycombed CSH structure. Hence, based on the above information, it can be deduced that, due to the addition of graphene, hydrated products grow in uniform and ordered way [38], which significantly improved the ductile behavior and compressive strength.

**Figure 5.** *Cont*.

**Figure 5.** FESEM images of GNP–cement paste at seven days and 28 days: (**a**) FESEM image of GM3 at seven days; (**b**) ESB or backscattered image of (**a**); (**c**) EDX of graphene in GM3 specimen on a point indicated in (**a**); (**d**) EDX for the hydrated cement product at cross hair location in (**b**); (**e**) FESEM image of GM3 at 28 days; (**f**) ESB or backscattered image of (**e**); (**g**) EDX for the marked point in (**e**); and (**h**) EDX at cross-hair location in (**f**).

Generation and growth of cracks for M0 sample were identified using FESEM images, as shown in Figure 6a. These are the nano size cracks; later, due to externally applied forces, they become micro size cracks without any interference and play their role in failure mechanism of material. It can be seen in Figure 6b that graphene successfully interrupted these cracks at nano level and were discontinuous. This is also verified in Figure 6c,d, which shows that graphene platelets are not only holding the micro cracks but are preventing their further growth. Longitudinal growth of cracks is highlighted using green lines while the location of graphene in encircled in red color in Figure 6d. Due to these reasons, the GM3 sample showed more ductile behavior as compared to M0 mix. Pan et al. [36] made the

comparison of crack patterns for plain and graphene oxide based cement composite. The authors found that in plain cement matrix cracks passed straight through the dense hydrated product. However, in graphene oxide cement paste, cracks were fine and discontinued. Therefore, it can be deduced that the presence of graphene sheets would make the cracks fine, the crack pattern discontinuous and provide hindrance to their growth, which, in turn, would result in enhancing the ductility and compressive strength of the graphene cement composite.

**Figure 6.** Effect of graphene on propagation of cracks (**a**) Crack propagation in control specimen; (**b**) blockage of the cracks by graphene; (**c**) crack bridging phenomena by graphene; and (**d**) backscattered electron image to identify the graphene in paste.

#### *4.3. Electrical Resistivity Values of the GNP–Cement Composite*

Piezo-resistive properties of the M0 and GM3 samples were investigated using the four-probe method. Electrical resistivity for unequal spacing between the probes was calculated using Equation (6).

$$\rho = \frac{\text{V}}{\text{I}} \times 2\pi \times \frac{1}{\left(\frac{1}{\text{S}\text{I}} + \frac{1}{\text{S}\text{S}} - \frac{1}{\text{S}\text{I} + \text{S}\text{I}} - \frac{1}{\text{S}\text{I} + \text{S}\text{S}}\right)}\tag{6}$$

where V is the floating potential difference between inner two probes, I is current measured by outer two probes, ρ is the resistivity in ohm-cm and S is the spacing in cm and was calculated from current carrying probe to voltage measuring probe. S1 = S3 = 40 cm and S2 = 60 cm.

The results of electrical resistivity are presented in Table 7. The electrical resistivity value at maximum compressive load for GM3 sample was found to be 42% less as compared to M0. To observe the sensing ability of the specimens, normalized compressive loading (NCL) values were calculated. It is the ratio between the applied loads to the maximum compressive load before specimen failure. For electrical resistance values, the fractional change in resistance (FCR) was used. Equations (7) and (8) present the calculation procedure for the NCL and FCR values.

$$\text{NCL} = \frac{\text{P}}{\text{P}\_{\text{max}}} \tag{7}$$

$$\text{FCR} = \frac{\rho\_{\text{t}} - \rho\_{\text{o}}}{\rho\_{\text{o}}} \times 100\% \tag{8}$$

where ρt is the electrical resistivity at the given time during the test; ρo is the electrical resistivity at the start of the test; P is the compressive loading at the given time during the test; and Pmax is the maximum compressive loading for the specimen.

Figure 7 shows the fractional change in resistance (in percentage) for M0 and GM3 against the normalized compressive load. It can be seen that in M0 sample the fractional change in resistance is very less as compared to GM3. The results are in line with Li et al. [39]. As graphene is expensive, therefore, it would be very costly to use GNP–cement composite material in a mega project. However, because of superior self-sensing properties, it can be used as a substitute to do health monitoring of the structure. In order to strengthen this view point and show its practical application, full length RC beam was tested in the laboratory by placing GM3 specimen at the time of casting as shown in Figure 2. This beam was subjected to flexural loading and, due to applied loading, cracks were generated in the region of maximum bending moment. Figure 8 shows the fractional change in resistance of GM3 specimen in RC beam. It can be seen that FCR values were varied with the increase in the applied loading on the beam and a sharp response was noted at the time of beam failure. As the beam was subjected to the flexural loading, resistance values were positive. The in-set figure (enlarged view) shows that with the increase in flexural loading, the fractional change in resistance values are also increasing. It is important to mention here that this response in not linear, however, due to the occurrence of damage and propagation of cracks, the FCR values varied as shown in in-set figure (enlarged view of Figure 8). A sudden drop in electrical resistivity value was observed at 22 kN load. The possible reason may be the occurrence of tensile cracks. Initially, tensile cracks started to occur and electrical resistivity values increased linearly. Thereafter, the steel reinforcement started to carry stresses and the significant variation in resistance was noted in GM3 specimen. This effect was more significant as the specimen (GNP cement based composite) was placed in the tensile region, i.e., just above the reinforcement bars. Finally, at the failure stage abrupt increase in FCR was observed, which shows that widening and irreversible crack opening has occurred inside the beam. Hence, structural member is not capable of carrying additional load.

**Figure 7.** Fractional change in resistance against normalized compression load.

**Figure 8.** Fractional change in resistance of GM3 specimen against applied compressive loading on the RC beam.

A comparison of strain produced strain against the applied load in reinforcement bars and graphene–cement composite sample is shown in Figure 9. In reinforcement bars, the strain increased linearly and slight variation was observed from 60 kN to 100 kN load as shown in Figure 9a. This variation is very minute and can be neglected for the reinforcement as steel bar was in the linear and elastic region. The GM3 sample also showed variation in strain from 60 kN to 100 kN load as marked in Figure 9a. However, in comparison to reinforcement bar, the variation was large and the strain values for the GM3 sample decreased significantly. This may be related to the redistribution of stresses in the RC concrete beam. Similarly, when the applied loading exceeded 200 kN, the reinforcement bar showed significant variance in strain values. At the same point, an increase in strain values in graphene cement composite sample was noted. It may be related to crushing of concrete in RC concrete beam It is pertinent to mention here that the cost of 5 g of GNPs as per Graphene Laboratories, Inc. USA is 50 USD and 25 cement based composite samples can be casted with GM3 specimen. Hence, GM3 specimen can be used in an efficient and economical way to predict the damages in concrete structures.

**Figure 9.** *Cont*.

**Figure 9.** Strain produced by applied compressive load in: (**a**) reinforcement; and (**b**) GNP–cement composite specimen.
