The major binding ingredient utilized in research work was Portland cement (PC), which conformed to ASTM C150/C150-16e1 (2016) requirements and had a specific gravity (SG) of 3.15. The river sand was employed as a fine aggregate, passing from a 4.75 mm mesh with a SG of around 2.65. Crushed stone, with a size of 20 mm, was employed as a coarse aggregate (CA), with an SG of about 2.60. Moreover, crumb rubber (CR), characterized by a SG of 0.92 and passed through a 1.18 mm sieve, was used as a partial substitution for sand at volumes of 10%, 20%, and 30% in this investigation. Incorporation of TiO₂ powder as a nanomaterial in rubberized concrete was a crucial aspect of the experiment. The average particle size ranged between 10 and 50 nm. To confirm uniform dispersion of nanoscale particles and prevent cohesion or agglomeration when mixed with water, a third-generation superplasticizer was employed (Zhao et al., 2017). To achieve the appropriate ability to flow, a modified polycarboxylate-based superplasticizer (SP) was utilised in the form of a water-based solution to modify the mixes. The substance is in a liquid state and has a pH of 6.2 and a SG of 1.08. Importantly, it contains no chloride ions. Water suitable for drinking and meeting standard quality criteria for concrete mixing was applied in the investigation, maintaining a water-to-cement ratio of 0.35.

The Design Expert 10 software was used to develop RSM for optimization. Employing the Central Composite Design (CCD) method, the research focused on two key factors: CR and TiO₂ as nanomaterials. Each variable underwent testing three times, with CR at 10%, 20%, and 30% volume substitution for sand, and TiO₂ at 1%, 1.5%, and 2% addition by weight of PC. This particular approach, previously utilized by researchers (Jo et al., 2015; Al-Fakih et al., 2020; Khed et al., 2020) culminated in 16 distinct mixes. These were generated by the RSM, meticulously exploring numerous Runs of the specified variables outlined in Table 1. Evaluation in the laboratory encompassed examining each mix for its flexural strength (FS), tensile strength (TS), modulus of elasticity (ME), compressive strength (CS), drying shrinkage (DS), sorptivity, and apparent porosity (AP). Moreover, the strength of each mix after a 28-day period was scrutinized. These assessments served as the basis for the RSM study and subsequent optimization of the mixtures. RSM played a pivotal role, not only in scrutinizing the results but also in conducting an analysis of variance (ANOVA) to ensure the reliability of the conducted tests. The study harnessed RSM to discern the most optimal values for the output variables (CS, TS, FS, ME, DS, sorptivity, and AP), factoring in the substantial impact of the input variables (TiO₂ and CR) on the overall outcomes.

Figure 1 shows how the CS changed at day 28 depending on how much TiO₂ nanomaterial was added to different rubberized concrete mixtures. By progressively incorporating CR, the graph organizes mixtures containing an equivalent quantity of TiO₂ in each group. The bar graph shows the average of five mixtures that came from the CCD’s central points (1.5% TiO₂ and 20% CR). This is called MCP. The composition, consisting of a replacement volume of 10% CR for sand and a weight percentage of 1.50% TiO₂, achieved an exceptionally high CS of 51.40 MPa. With a sand replacement volume of 30% CR and a weight percentage of 2% TiO₂, the minimum CS measured after a period of 28 days was 38 MPa. It is worth mentioning that as CR concentration rises, the graph illustrates a significant drop in the CS of concrete. In line with what Najim and Hall (Nazari et al., 2010). Found, this loss of strength is due to CR not sticking well enough to the cemented cement matrix. Bashar claims that the hydrophobic nature of CR particles, which hinders their ability to combine with water and thereby creates a barrier to bonding, negatively affects the strength of composites. In line with earlier studies (Kundan and Sharma, 2020; Assaggaf et al., 2021; Fauzan et al., 2021). CR makes the composite more fragile because it is weaker and has a lower elastic modulus than microaggregate particles. Prior research (Zhang Z. et al., 2015; Alaloul et al., 2020) stated that the accumulation of CR has a detrimental impact on the CS of composites. These findings corroborate these results. Figure 1 shows that TiO₂ can be used as a nanomaterial to strengthen the rubberized concrete matrix, especially when it is mixed with up to 1.5% by weight of Portland cement (PC). The augmentation in Ca(OH)₂ absorption is attributed by researchers to the participation of TiO₂ in the pozzolanic reaction. The development of C-S-H is facilitated as a result of this hydration procedure acceleration (Wang et al., 2018). Operating as a highly efficient infill, the nanoparticles hinder the unrestricted flow of water vapor within the concrete. Evidently, the CS increased when 1.50 percent TiO₂ was incorporated instead of cement but reduced as the proportion of TiO₂ increased (Figure 5). Due to two things, the performance has gone down: the nanoparticles are not evenly distributed in the concrete matrix, and there are fewer Ca(OH)₂ crystals available, which are needed to make C-S-H gels (Najim and Hall, 2013). Higher concentrations of TiO₂ with larger surface areas were also a contributing factor in these samples’ excessive water absorption. In line with the work of Rawat et al. (Rawat et al., 2023). Sorathiya et al. (Sorathiya et al., 2018) and Orakzai (Orakzai, 2021), this study found that using 1% nano-TiO₂ instead of cement increased the CS by 22.6% after 28 days.

The TS is a critical measure of concrete’s tensile capacity, impacted by both TiO₂ as nanoparticle and CR content as a sand replacement by volume. Figure 2 illustrates the experimental results for rubberized concrete’s TS at 28 days, analysed via ANOVA in the RSM framework. The impact of these factors on the TS is statistically evaluated through RSM analysis. The highest recorded tensile strength, at 4.47 MPa, was achieved with 1.50% TiO2 and 10% CR, while the lowest, at 3.30 MPa, occurred with 2% TiO₂ and 30% CR replacing sand after 28 days Figure 2 highlights a reducing trend in TS with growing CR content in concrete. Partially substituting sand with CR has a detrimental effect on TS due to water’s repulsion on CR, trapping air on its surface, increasing air content in the concrete, and thickening the ITZ between aggregate and PC paste (Khed et al., 2020). This process weakens the bond between PC paste and aggregate. When a load is put on the material, the strain compatibility difference between the hardened state of PC and CR at ITZ causes stress to build up and cracks to form around the CR parts. These cracks have been recognised as weak spots by many studies (Posi et al., 2019; Shaji et al., 2019; Al-Fakih et al., 2020; Shahrul et al., 2021). However, the accumulation of TiO₂ as nanoparticle in rubberized concrete showcases a noticeable enhancement in TS. Figure 2 indicates that optimal strength is achieved when TiO₂ is up to 1.5%, after which strength begins to decline. This observed enhancement in TS might be accredited to the great surface area of TiO₂ nanoparticles, promoting pozzolanic reactions that foster C-S-H gel formation and overall strength increase (Chen et al., 2012). However, further accumulation of TiO₂ leads to decreased strength in rubberized concrete, linked to non-uniform nanoparticle dispersion, hindering Ca(OH)₂ crystal formation crucial for C-S-H gel development (Khed et al., 2020). Comparable remarks were completed by researchers (Rawat et al., 2023).

It is the measure used to quantify the capability of composite to withstand bending. The adhesion between aggregate and PC paste affects the FS (Khed et al., 2020). At the 28-day, the experimental outcomes for rubberized concrete comprising several contents of TiO₂ as nanoparticle are illustrated in Figure 3. The material exhibiting the greatest FS noted by 5.91 MPa when composed of 1.5% TiO₂ and 10% CR by 28 days. Conversely, the least FS was attained by 4.37 MPa at 2% TiO₂ as nanoparticle and 30% CR in place of sand together after 28 days. The trend identified suggests that as CR substitution increases, the FS of concrete declines. The decrease in toughness of CR particles relative to FA which cause of the reduction in FS. In line with this, Sharul et al. (Shahrul et al., 2021) detected that the FS of CR concrete is diminished in comparison to sand particles due to its lesser SG, strength, and load-bearing capacity. Additionally, when sand is replaced with CR, the FS is dropped as a consequence of water repulsion, air entrapment on CR surfaces, and a rise in air content in CR concrete. By increasing the thickness of ITZ that separates aggregate and cement material, their bond is weakened (Khed et al., 2020). The incorporation of TiO₂ significantly contributes to the enhancement of FS. As exposed in Figure 3, the FS of PC rises with the TiO₂ content reaching 1.5% by weight. The observed improvement can be ascribed to the filling of micropores in rubberized concrete by nanoscale particles of TiO₂, which consequently enhance its strength. Nevertheless, as TiO₂ continues to accumulate, its strength diminishes as a result of the uneven distribution of particles; this impedes the formation of Ca(OH)₂ crystals, which are vital for the development of C-S-H gels (Khed et al., 2020). Scholars have encountered comparable observations (Rawat et al., 2023).

The ME, defining concrete’s resistance to deformation, is crucial and influenced by multiple features such as concrete’s compatibility, aggregate properties, and the ITZ (Silva et al., 2016). The ME test, following ASTM C 469 (ASTM C469, 2002), is strongly correlated with CS and often expressed by empirical equations derived from experimental studies for various concrete types (Alsalman et al., 2017). For rubberized concrete, equations estimating the modulus of elasticity based on various w/c ratios have been developed Figure 4 illustrates the modulus of elasticity results for sixteen experimental runs conducted with rubberized concrete blended with various TiO₂ concentrations at 28 days. The highest ME was observed by 40.15 GPa with addition of 1.50% TiO₂ and 10% CR, while the lowest ME was noted by 34.52 GPa at 2% TiO₂ and 30% CR replacing sand by volume after 28 days respectively. The reduction in ME with increased CR replacement is accredited to the lower stiffness of rubber elements (Khed et al., 2020). However, a noteworthy trend reversal occurred with TiO₂ addition. Across all CR replacement levels, mixes containing 1.5% TiO₂ exhibited higher ME. This improvement is associated to the nanoscale fineness of TiO₂ particles, filling micro pores within rubberized concrete and enhancing its stiffness. Yet, further TiO₂ addition led to decreased ME in rubberized concrete. This decline is associated with reduced or unavailable Ca(OH)₂ crystals necessary for C-S-H gel formation and non-uniform nanoparticle distribution in matrix (Nazari et al., 2010). These outcomes are dependable with the studies done by Rawat et al. (Rawat et al., 2023).

It is a significant concern in concrete in which the volume change associated to water loss within the capillary pores of matrix. Figure 5 illustrates a noteworthy decrease in rubberized concrete shrinkage with increasing TiO₂ as a nanomaterial addition. The shrinkage values for rubberized concrete blended with TiO₂ ranged between 0.05% and 0.066%, higher than the control sample’s shrinkage value of 0.04% and the typical composite shrinkage values reported (ranging from 1,200 × 10⁻⁶ to 1800 × 10⁻⁶) (Zhang et al., 2009). Observations show an increase in concrete shrinkage as the CR content in the mixture rises. This rise is accredited to the lower stiffness of CR constituent part compared to sand, causing increased deformation under drying shrinkage stress (Wang et al., 2019). Also, because CR is flexible, there is less internal restraint within the matrix compared to substituted sand particles. This makes the volume less stable and causes it to shrink more when it dries (Zhang Z. et al., 2015). Moreover, higher CR content leads to improved porosity in the composite owing to the hydrophobic character of CR. This prompts capillary water rise and subsequent volume change upon evaporation, further increasing DS (Huang X. et al., 2013). Adding TiO₂ as a nanomaterial to rubberized concrete dramatically decreased its drying shrinkage at all levels of CR replacement, which is a big surprise. This decrease is ascribed to the nanoscale fineness of TiO₂ particles, which effectively fill micropores within the mixture. During the concrete drying process, excess water evaporates from the surface, creating an air/water interface within capillary pores. The phenomenon of surface tension creates pressure on the innermost layers of such tiny pores, resulting in an inward push that causes the pore structures to contract and decreases the volume of capillaries. The higher the capillary pores, the greater the potential for concrete to shrink. Nanoparticles are very important for lowering the number of capillary voids, which in turn lowers surface tension and, as a result, lessens the shrinkage that happens when water evaporates from the concrete matrix. Joshaghani (Joshaghani, 2018) noted similar observations.

Following a curing period of 28 days, each specimen underwent a sorptivity test in order to evaluate capillary suction. It was possible to determine the slope of the line that links absorption with the square root of time (Nazari, 2011; Saleem et al., 2021). The absorption curve and square root of the absorption coefficient for concrete mixtures are illustrated in Figure 6 (Chen et al., 2012). The provided data illustrates the sorptive outcomes of rubberized concrete mixtures supplemented with varying quantities of TiO₂ nanomaterials. After a period of 28 days, the mixture containing 2% TiO₂ nanomaterial and 10% CR sand replacement by volume exhibited the lowest sorptivity. The water-only solution exhibited the greatest sorptivity. A higher CR content in concrete is positively correlated with increased sorptivity, according to these findings. This structure forms because rubber is hydrophobic, which raises the capillary water pressure, causes more volume change during evaporation, and then contraction. As a result, a higher CR content leads to more porosity (Huang X. et al., 2013). Notably, adding TiO₂ nanoparticles to rubberized concrete greatly decreased its ability to soak up water, no matter how much CR was replaced. The fluctuations in sorptivity levels exhibit a strong correlation with the properties of the pores. TiO₂ has a lower sorptivity because its particles are smaller and more C-S-H gel forms, which makes the pores smaller and, in turn, the sorptivity lower (Jayapalan et al., 2009; Zhang R. et al., 2015). When samples with more TiO₂ are put into the C-S-H gel, the sorptive levels go down. As a consequence, the interconnection of pores diminishes, resulting in a reduction in absorption. These observations resonate with finding from earlier research (Rawat et al., 2023).

Figure 7 illustrates the apparent porosity of rubberized concrete blended with various concentrations of TiO₂ as nanomaterial after a 28-day period. Interestingly, the porosity of all concrete including TiO₂ was lower than control concrete. However, the porosity of the concrete increased with rising CR content, attributed to the hydrophobic character of CR. This characteristic led to capillary water rise and increased drying shrinkage, thereby augmenting porosity with higher CR content (Huang X. et al., 2013). Conversely, across all CR replacement levels, a significant decrease in porosity was detected upon the introduction of TiO₂ as a nanomaterial in rubberized concrete. This decline in porosity can be linked to TiO₂ nanoparticles acting as effective fillers. As the nanoparticle conglomerates expanded, the surrounding void spaces were gradually filled. These “nuclei” significantly accelerated hydration rates, leading to reduced porosity as hydration accumulated rapidly within water-filled pores (Mohammadi et al., 2014). Studies by Mohammadi et al. (Mohammadi et al., 2014) showcased reduced total porosity with the substitution of titanium dioxide to calcium phosphate cement. These observations resonate with finding from earlier research by Riahi and Nazari (Nazari and Riahi, 2011b; 2011a), who observed decreased porosity when PC was partially substituted with TiO₂ at different concentrations. Their results demonstrated that the decrease in porosity was achieved with a TiO₂ content of 3%, showing reductions of 1.64%, 4.3%, 5.67%, and 5.07% in porosity with additions of 1%, 2%, 3%, and 4% TiO₂, respectively. Comparable observations were made by other researchers (Zhang and Li, 2011; Jalal et al., 2013; Sorathiya et al., 2018; Rawat et al., 2022).

At a 5% significance level (Bheel et al., 2023c; 2023e; 2023f; 2023d), ANOVA was used to look at how the factors interacted with each other and how each parameter affected the responses. Subsequently, any model or term with a p-value below 0.05 is deemed significant. The ANOVA findings are shown in Table 2, indicates that all generated models are significant at a significance level of <0.01%. Looking at the specific models, certain terms stand out as significant contributors. For instance, in the CS model, terms A, B, A², and B² demonstrate significance. In the meantime, only terms A, B, and A² are important for the TS, FS, ME, and Sorptivity models. On the other hand, the DS and AP models show significance for terms A, B, and B². The significance of terms A and B across all models highlights the direct influence of both TiO₂ and CR on the responses. Additionally, the significance of A² and B² suggests that the quadratic effects of these independent factors also impact the responses. It is important to note that for the interaction between factors to affect the response, the AB term must be significant.

Table 3 showcases the model validation parameters, revealing remarkably high R² values across all models, fluctuating from 97% to 99.08%. These values signify the strong alignment between the models and the actual data. Additionally, the models exhibit a narrow difference (<0.2) between the adjusted R² and predicted R², meeting the criteria for a good fit. Additionally, the signal-to-noise ratio, which measures adequacy precision, exceeds the threshold of 0.4, confirming the accuracy of the models’ predictions.

Figures 8–14 show portray the interactions between input parameters and outputs (responses) through contour plots and 3D surface plots. They specifically highlight the relationships between different input variables. For instance, Figure 8A,B demonstrate the 2D diagram and 3D plots for CS, showcasing that a significant concentration of CS was observed at a 1.5% TiO₂ level in the nanomaterial category among all TiO₂ concentrations. Additionally, it is evident that the value of strength notably enhanced when using up to 1.50% TiO₂ in the CR concrete. This enhancement can be owing to TiO₂’s ability, as a nanomaterial, to fill pores and enhance densification in the CR combination. Similar patterns are observed in the remaining response surface graphs for TS, FS, ME, DS, sorptivity, and AP, indicating the impactful relationships between input parameters and outputs.

The RSM evaluation comprises essential diagrams, including crucial model diagnostic graphs like the normal diagram of residuals (Figures 15–21). These graphs, split into (a) run versus predicted and (b) real versus predicted, serve to validate the significance of the quadratic models used for rubberized concrete with TiO₂ as nanomaterials. The plots depict residuals plotted against the run number (Figures 15–21A), showcasing a consistent distribution without any observable model drift. The data points, encapsulated within red boundary lines, maintain a sinusoidal pattern, affirming the stability of the model throughout the process (Memon et al., 2020). Additionally, the predicted versus actual plot (Figures 15–21B) demonstrates strong agreement between experimental and predicted 28-day data for CS, TS, FS, ME, DS, sorptivity, and AP models. The close alignment of distribution points and the minimal variance between experimental and predicted values confirm this agreement. Notably, the points align closely along a straight line, indicating a normal distribution pattern, further validating the model’s accuracy.

It referred to as multi-response optimization is a technique aimed at identifying the most beneficial variables to maximize various responses simultaneously. Real-world optimization problems often involve multiple conflicting objectives, necessitating the search for several optimal solutions (Bheel et al., 2023b; 2023a). Hence, this approach becomes the preferable strategy (Liu et al., 2017; Habibi et al., 2021). Optimization focuses on the model’s independent variables, enhancing their efficacy in the process. The objective function guides this optimization, setting the goals for the variables and potentially encompassing significance levels (Liu et al., 2017; MIR et al., 2017). In this process, both independent and dependent variables have objectives established using diverse criteria (Burke and Kendall, 2005; Waqar et al., 2023a). The ultimate aim is to attain objective functions without compromising any of the responses. The desired values, which are between 0 and 1, are what matter in the optimization process for each response, which ranges from 0 to 1 (Waqar et al., 2023b; 2023a; Khan et al., 2023). Elevating the DJ value enhances the favourability of the outcome, often represented as a percentage. In multi-objective optimization, the geometric mean of how desirable each response is found. This lets you find the desirability value of the composite response, which is shown in Eq. 13 (Achara et al., 2019). D = d 1 r 1 × d 2 r 2 × d 3 r 3 × ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ × d n r n 1 n (13)

In this context, “n” represents the total number of responses considered during the optimization process, while “ri” denotes the assigned weight of significance to each function, ranging from 1 (least important) to 5 (most crucial). In this specific model, the desirability limit was set between 0% and 1%, where a value closer to 1% indicates a more favourable outcome, strengthening the optimized results impact on determining the dependent variable. These parameters outline the optimization goals of the variables. Notably, the optimization aims to maximize the responses for CS, TS, FS, and ME, whereas it aims to minimize responses for DS, sorptivity, and AP. The optimized results indicate maximum values for CS (49.47 MPa), TS (4.31 MPa), FS (5.69 MPa), ME (39.41 GPa), DS (0.051%), sorptivity (0.035 mm/s^0.5), and AP (16.48%), detailed in Table 4. The variability analysis of the model reflects a desirability of 66.10%, indicating the models ability to produce relevant and promising outcomes. These desirability results stem from the optimization method as shown in Table 4.

Experimental validation aimed to understand the disparities between the predicted response values derived from the optimization process and the actual experimental values. The optimal parameters, concrete samples were created, incorporating various proportions of CR as a sand substitute alongside TiO₂ nanomaterials as cementitious components. The comparison between predicted and experimental values of CS, TS, FS, ME, DS, Sorptivity and AP is expressed as a percentage error, i.e 2.45%, 2.80%, 1.93%, 1.50%, 5.88%, 5.71%, and 4.92% respectively and all responses exhibit an error percentage of less than 6%. This low error percentage across all responses signifies a high level of precision in approach. The discrepancy ( δ ) between empirical and anticipated readings for individual responses was calculated using Eq. 14, demonstrating the model’s precision in predicting outcomes accurately. In essence, this signifies a high degree of consistency between the anticipated and observed values, affirming the model’s accuracy. δ = ϑ E − ϑ P ϑ P × 100 % (14)