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Mathematics in Engineering

2019, Volume 1, Issue 2: 359-377. doi: 10.3934/mine.2019.2.359
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Research article Special Issues

Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral

  • Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efesiou 4Athens, GR-11527, Greece
  • We investigate the long term evolution of trajectories in the Fermi-Pasta-Ulam (FPU) system, using as a probe the first non-trivial integral J in the hierarchy of integrals of the corresponding Toda lattice model. To this end we perform simulations of FPU-trajectories for various classes of initial conditions produced by the excitation of isolated modes, packets, as well as 'generic' (random) initial data. For initial conditions corresponding to localized energy excitations, J exhibits variations yielding 'sigmoid' curves similar to observables used in literature, e.g., the 'spectral entropy' or various types of 'correlation functions'. However, J(t) is free of fluctuations inherent in such observables, hence it constitutes an ideal observable for probing the timescales involved in the stages of FPU dynamics. We observe two fundamental timescales: i) the 'time of stability' (in which, roughly, FPU trajectories behave like Toda), and ii) the 'time to equilibrium' (beyond which energy equipartition is reached). Below a specific energy crossover, both times are found to scale exponentially as an inverse power of the specific energy. However, this crossover goes to zero with increasing the degrees of freedom N as εc∼N−b, with b∈[1.5,2.5]. For 'generic data' initial conditions, instead, J(t) allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori.

    Citation: Helen Christodoulidi, Christos Efthymiopoulos. Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral[J]. Mathematics in Engineering, 2019, 1(2): 359-377. doi: 10.3934/mine.2019.2.359

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  • We investigate the long term evolution of trajectories in the Fermi-Pasta-Ulam (FPU) system, using as a probe the first non-trivial integral J in the hierarchy of integrals of the corresponding Toda lattice model. To this end we perform simulations of FPU-trajectories for various classes of initial conditions produced by the excitation of isolated modes, packets, as well as 'generic' (random) initial data. For initial conditions corresponding to localized energy excitations, J exhibits variations yielding 'sigmoid' curves similar to observables used in literature, e.g., the 'spectral entropy' or various types of 'correlation functions'. However, J(t) is free of fluctuations inherent in such observables, hence it constitutes an ideal observable for probing the timescales involved in the stages of FPU dynamics. We observe two fundamental timescales: i) the 'time of stability' (in which, roughly, FPU trajectories behave like Toda), and ii) the 'time to equilibrium' (beyond which energy equipartition is reached). Below a specific energy crossover, both times are found to scale exponentially as an inverse power of the specific energy. However, this crossover goes to zero with increasing the degrees of freedom N as εc∼N−b, with b∈[1.5,2.5]. For 'generic data' initial conditions, instead, J(t) allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori.


    1.   Introduction

    In view of the importance of C–S–H gel formation in attaining strength in concrete, the addition of pozzolanic materials will help the cause as they are rich in SiO2 [1]. The SiO2 present in pozzolanic material will play a crucial role in secondary C–S–H gel reaction, which consumes the unreacted calcium in concrete [2]. Pozzolanic materials, in a broader sense, can be seen as two types for cement replacement, i.e., natural pozzolanic materials (e.g. bentonite) [3,4,5,6] and others are waste by-products from industries (e.g. Flyash, GGBS, and Silica Fume, etc.) [7]. While fly ash is a by-product of industries and has its physical and chemical characteristics more or less defined or consistent [8], natural pozzolanic materials will have their characteristics varied from region to region or even with in the same area sometimes [9]. One of the popular pozzolanic materials being utilized at a commercial level is fly ash [10].

    The significant sources of fly ash are in general carbon-emitting sources which may not be sustainable in the long run [11]. So, the utilization of fly ash as a pozzolanic replacement of cement in concrete may not be sustainable, and we shall look at other pozzolanic materials that are from non-carbon emitting sources. Bentonite is a mineral admixture (Clay) and had plenty of applications in various fields. The major applications include the utilization of bentonite are as drilling fluids, foundry bonds, pelletizing iron ore, cat litter and absorbents [12]. In India, the significant applications of bentonite are in drilling and foundry fields. A tremendous amount of bentonite resources were located in Rajasthan (424 million tonnes) and Gujarath (134 million tonnes) states in India [13].

    The researchers examined the strength of bentonite blended concrete (Bentocrete), variances in the results are were reported [14]. Bentonite is fine clay material; it shows different physical and chemical properties concerning the source of collection. Most of the research on Bentocrete was performed in Pakistan due to its abundance. Due to its variations in physicochemical properties, the performance of Bentocrete exhibits some uncertainties to the researchers. The average particle size of bentonite is nearly 4.32 µm [15]. It contains SiO2 in high (45–65) percentages, and it shows pozzolanic properties. The specific gravity (2.6–2.85) of bentonite is lower than that of other pozzolanic materials [16]. The Bentocrete may perform less thermal resistance compared to other concretes made with pozzolanic materials like fly ash. It is attributed to the higher Loss of ignition (5–14) values of bentonite[10,14,16]. Bentocrete generally exhibited longer setting times when compared with cement paste [17,18,19].

    The better performance of Bentocrete was observed when bentonite was preheated at 150 ℃ before mixing [3]. The strength activity index (SAI) of Bentocrete was found to be significantly less during early curing days (7 d), increased gradually in later curing days (28 d) and was better than cement concrete there onwards [3,17,20]. Mirza et al., in 2009, observed that the SAI was better up to 21 percentage of bentonite blending after 28 d of curing because the pozzolans will actively participate in reaction in lateral ages [21]. The workability of bentonite blended concrete was decreased, inversely proportional to the percentage of blending [5]. Utilization of little amounts (1–2%) of sodium bentonite improves the workability of concrete upon proper mixing [12]. The water absorption of bentonite blended concrete was lesser than that of conventional concrete [3,4,22].

    100% substitution of cement in Bentocrete will have feeble strength because of scarcity in binding agents like CaO (Citation). It is generally suggested that the replacement of cement in Bentocrete shall be in the range of 5–20% and should not be beyond 30% [23,24,25,26]. Bentocrete has a very little modulus of rupture values [3] and higher in the Fleuxural strength value [27,28]. Bentonite substitution (30–40%) increased the performance of concrete against Sulphate attack (Na2SO4 and MgSO4) [3,29]. There was an increment in results was observed against acid attack (H2SO4 and HCl) [4,6]. Ventura et al. conducted tests to determine the effect of bentonite on corrosion protection of steel reinforcement, reported that bentonite addition increases the corrosion protection to steel [30]. Xie et al. held experiments by utilization of bentonite slurry, evaluated the mechanical properties of foamed concrete, almost the same strength were reported for the control mix, and 10% bentonite replaced mix [31].

    Model advancement included the utilization of Response Surface Methodology (RSM) employing theories of mathematical and statistical analysis techniques between variables and responses [32]. Moreover, RSM was utilized for the optimization of the wanted set of objectives, either independent variables or responses [33,34]. RSM has additionally was used to implement multi-objective optimization in different concrete materials [35,36]. Bashar et al. performed multi-objective optimization to accomplish a connection between variables and responses of the properties of roller-compacted concrete by keeping fly ash content as constant, considering the combined effect of both crumb rubber and nano-silica [37]. Mohammed et al. utilized RSM to attain a relationship between variables (fly ash, nano-silica and superplasticizer) and responses (flow value, setting time and compressive strength) [38]. Baris et al. developed a graphical interface by utilizing RSM to determine the optimal weight of agammaegates [20]. Geo et al. performed multi-response optimization by keeping alkali activator concentration and liquid-solid ratio as variables, 2 h and 3 h curing as responses [39]. Long et al. performed multi-objective optimization by restraining action of corrosion, fatigue and fiber content as independent variables and compressive strength, flexural strength and dynamic elastic modulus as responses [40].

    In general, the bentonite will attain its normal consistency at more water consumption than that of cement and fly ash due to its high-water absorption capacity. This will have an impact on water-cement ratios to be used during concrete mix design [21]. So, the current research on the effect of water cement-ratio on workability, compressive strength, split tensile strength, flexural strength and durability of concrete with different proportions of cement replacements with bentonite. The role of bentonite and water-cement ratio was assessed based on the RSM. Also, the study attempts to compare the costs involved in using bentonite as a replacement for cement.

    2.   Materials and methods

    2.1.   Materials

    In this research OPC-43 grade cement was used as per ASTM C150M, bentonite was collected from Unique bleaching clay (17°14ʹ27ʺN and 77°35ʹ14ʺE), Tandur (southern region of India). Table 1 shows the chemical analysis and physical properties of bentonite and cement. The quality of the water used for this study was within limits as per ASTM C1602M. In this study, the fine agammaegate and coarse agammaegate was used as per the standard procedure ASTM C33/C33M-18.

    Table 1.  Chemical analysis and physical properties of bentonite and cement.
    S. No. Component Chemical composition%
    Bentonite Cement
    1 SiO2 51.11 18.6
    2 Al2O3 16.38 8.17
    3 CaO 7.12 62.5
    4 MgO 7.57 1.43
    6 Fe2O3 7.65 3.26
    7 K2O 1.34 1.09
    8 Na2O 0.29 0.71
    9 P2O5 0.29 0.07
    10 MnO 0.14 -
    11 V2O5 0.07 -
    12 TiO2 1.29 -
    13 SO3 - 2.63
    Physical properties
    14 Blain fineness (cm2/gm) 1730 4750
    15 Specific gravity 3.12 2.79
    16 Initial setting time (min) - 90
    17 Final setting time (min) - 345
    18 Loss on Ignition 6.75 1.23
     | Show Table
    DownLoad: CSV
    2.1.1.   SEM and XRD analysis of bentonite

    The XRD analysis was performed by using X-Pert X-ray Diffractometer of PANalytical, model number: PW 3040/00. The investigation was conducted at a 2-theta range of 10–80 with a step size of 0.5 and a 5 degrees/min scan rate. The results (Figure 1) have shown different phases that can be further identified as five mineral crystalline structures. The calcite mineral has established a prominent presence in the XRD. Scanning electron microscopy (SEM) images were obtained by using Nova Nano SEM/FEI for further pursuit. The SEM images (Figure 2) have also shown the presence of crystalline structures in the bentonite sample.

    Figure 1.  XRD Analysis of bentonite.
    DownLoad: Full-Size Img PowerPoint
    Figure 2.  SEM Analysis of bentonite.
    DownLoad: Full-Size Img PowerPoint

    2.2.   Testing of specimens

    The standard consistency test was conducted on cement paste to determine the optimum water content for getting maximum strength with the Vicat apparatus as per standard procedure ASTM C187. Compressive strength of cement mortar tested for all mixes as per ASTM C109. The workability of all Bentocrete mixes were determined by slump cone test, and compaction factor test as per standard procedures ASTM C143 and IS 1199:1959. For determination of compressive strength, a total number of 396 concrete cubes (150 mm × 150 mm × 150 mm) were cast and tested to failure at different ages of concrete (7, 28 and 56 d) as per the standard procedure ASTM C39. Split tensile strength test was performed to determine the split tensile strength of Bentocrete mixes as per the standard procedure ASTM C496, a total number of 132 cylinders (150 mm × 300 mm) were cast and tested to failure after 28 d curing. Flexural strength test was performed to determine the flexural strength as per the standard procedure ASTM C78, a total number of 132 specimens (150 mm × 150mm × 700 mm) were cast and tested to failure (third-point loading) after 28 d curing. Rapid chloride ion penetration test was performed to determine chloride ion penetration into Bentocrete as per standard procedure ASTM C 1202-97. A total number of 528 specimens, 44 sets (a set include four samples) were tested against chloride ion permeability after 7, 28 and 56 curing; respectively, Figure 3 represents the rapid chloride penetration test setup. The mix design of each mix was prepared as per ACI committee 211; the details of each blend were shown in Table 2.

    Figure 3.  Setup for rapid chloride penetration test.
    DownLoad: Full-Size Img PowerPoint
    Table 2.  The mix design details of all mixes.
    Si No. Mix name Cement (kg) Bentonite (kg) Fine aggregate (kg) Coarse aggregate (kg) Water (L) Ratio
    1 CM60 320 0 819.299 1077.072 192 1:2.56:3.37
    2 CM61 320 0 819.265 1068.34 195.2 1:2.56:3.34
    3 CM62 320 0 819.197 1059.643 198.4 1:2.56:3.31
    4 CM63 320 0 819.096 1050.981 201.6 1:2.56:3.28
    5 CM64 320 0 818.961 1042.355 204.8 1:2.56:3.26
    6 CM65 320 0 818.793 1033.763 208 1:2.56:3.23
    7 CM66 320 0 818.592 1025.207 211.2 1:2.56:3.20
    8 CM67 320 0 818.357 1016.685 214.4 1:2.56:3.18
    9 CM68 320 0 818.088 1008.088 217.6 1:2.56:3.15
    10 CM69 320 0 817.787 999.748 220.8 1:2.56:3.12
    11 CM70 320 0 817.452 991.332 224 1:2.55:3.10
    12 10BC60 288 32 818.145 1075.548 192 1:2.56:3.36
    13 10BC61 288 32 815.54 1063.482 195.2 1:2.55:3.32
    14 10BC62 288 32 815.68 1055.8 198.4 1:2.55:3.30
    15 10BC63 288 32 816.27 1047.365 201.6 1:2.55:3.27
    16 10BC64 288 32 815.18 1037.55 204.8 1:2.55:3.24
    17 10BC65 288 32 815.23 1029.27 208 1:2.55:3.22
    18 10BC66 288 32 814.98 1019.54 211.2 1:2.55:3.19
    19 10BC67 288 32 814.77 1012.23 214.4 1:2.55:3.16
    20 10BC68 288 32 814.48 1003.74 217.6 1:2.55:3.14
    21 10BC69 288 32 814.169 995.325 220.8 1:2.54:3.11
    22 10BC70 288 32 816.29 994.29 224 1:2.55:3.11
    23 20BC60 256 64 812.346 1067.931 192 1:2.54:3.34
    24 20BC61 256 64 812.215 1058.625 195.2 1:2.54:3.31
    25 20BC62 256 64 812.18 1050.568 198.4 1:2.54:3.28
    26 20BC63 256 64 812.04 1041.939 201.6 1:2.54:3.26
    27 20BC64 256 64 811.827 1034.547 204.8 1:2.54:3.23
    28 20BC65 256 64 811.783 1024.787 208 1:2.54:3.20
    29 20BC66 256 64 811.689 1016.561 211.2 1:2.54:3.18
    30 20BC67 256 64 811.263 1008.368 214.4 1:2.54:3.15
    31 20BC68 256 64 810.885 988.38 217.6 1:2.53:3.09
    32 20BC69 256 64 810.552 990.903 220.8 1:2.53:3.10
    33 20BC70 256 64 810.185 982.52 224 1:2.53:3.07
    34 30BC60 224 96 812.513 1059.536 192 1:2.54:3.31
    35 30BC61 224 96 812.415 1050.87 195.2 1:2.54:3.28
    36 30BC62 224 96 812.285 1042.241 198.4 1:2.54:3.26
    37 30BC63 224 96 811.921 1039.418 201.6 1:2.54:3.25
    38 30BC64 224 96 811.544 1029.841 204.8 1:2.54:3.22
    39 30BC65 224 96 811.229 1020.299 208 1:2.54:3.19
    40 30BC66 224 96 811.088 1011.792 211.2 1:2.53:3.16
    41 30BC67 224 96 810.87 1003.914 214.4 1:2.53:3.14
    42 30BC68 224 96 810.53 995.77 217.6 1:2.53:3.11
    43 30BC69 224 96 810.282 987.66 220.8 1:2.53:3.09
    44 30BC70 224 96 810.007 985.457 224 1:2.53:3.08
     | Show Table
    DownLoad: CSV

    2.3.   Specimen designation

    Two types of nomenclatures were framed for simple identification of the mixes. Each specimen was represented by the percentage of partial replacement followed by bentonite cement (BC) followed by water/cement ratio multiplied with 100. For example, 20BC68 represents 20% of cement was replaced by bentonite with 0.68 as water/cement ratio, 10BC represents 10% of cement was replaced by calcium bentonite, control mix (CM) represents 0BC hence no blending of bentonite in that mix.

    3.   Results and discussion

    3.1.   Normal consistency of cement paste

    The determination of normal consistency for any cement is essential, the water content of paste which will produce the desired consistency. The consistency of cement was directly proportional to the percentage of bentonite blended, as shown in Figure 4. There is a strong linear correlation between percentage replacement of bentonite and normal consistency with R2 values reaching up to 0.99. According to the analysis, there is an approximately 2.5% increase in normal consistency for every 1% increase in cement replacement with bentonite; this attributes the bentonite's high water absorption capacity [12].

    Figure 4.  Normal consistency of bentonite mixtures.
    DownLoad: Full-Size Img PowerPoint

    3.2.   Compressive strength of cement mortar

    The compressive strength of hardened cement is the property of the material that is required for structural use. Figure 5 shows the compressive strengths of hardened cement mixes; maximum compressive strength was demonstrated by 10% bentonite blended mix. The strength of hardened cement mixes was examined as per standard procedure IS:650. In cement mortar mixes, 10BC and 30BC exhibits 6.41% higher and 17.00% less compressive strengths than CM.

    Figure 5.  Compressive strength (MPa) of mixes at different ages.
    DownLoad: Full-Size Img PowerPoint

    3.3.   Workability of concrete

    The workability was delineated as determination of the ease of placement and resistance to the segregation of concrete. The workability was determined by the help of a slump cone test, and compaction factor test, slump values and compaction factor were measured for each mix. 45 mm slump value was fixed by the help of trail mixes for CM61, remaining values measured while casting of each mix. The results were drawn after testing of all specimens (section testing of specimens), Figure 6 shows the slump values of all mixes.

    Figure 6.  Slump values of all bentonite mixes.
    DownLoad: Full-Size Img PowerPoint

    The degree of compaction is known as compaction factor, the ratio of achieved density of the concrete in the same at compacted state. The results were taken after performing the test for all mixes; Figure 7 shows the compaction factor values. The workability of blended mixes was reduced as the percentage of bentonite blending increased. This can be attributed to the high-water absorption capacity of bentonite. The lowest workability was observed with a 30BC60 mix. The causative is the combination of higher replacement (30%) of cement with bentonite and low water-cement ratio (0.60). Simultaneously, the highest workability was observed with CM70. The workability of blended mixes was reduced as the percentage of bentonite blending increased.

    Figure 7.  Compaction factor values of all bentonite mixes.
    DownLoad: Full-Size Img PowerPoint

    3.4.   Compressive strength of concrete

    The compressive strength of concrete is always a crucial aspect of structural design and is specified for compliance purposes. The compressive strength of all mixes is inversely proportional to the water-cement ratio as per the basic principle of concrete [21].

    Figure 8 shows the compressive strength of a set of all specimens (Table 2) after seven days of curing. CM mixes shown higher compressive strength among was all, because of 43-grade cement's property to gain early age strength [21]. The pozzolanic materials like bentonite will tend not to have early age compressive strength. Thus, the mixes with high bentonite show less compressive strength.

    Figure 8.  Compressive strength of bentonite mixes at the age of 7 d.
    DownLoad: Full-Size Img PowerPoint

    Figure 9 shows the compressive strength of a set of all specimens (Table 2) after 28 d of curing. There is an improvement in the compressive strength of bentonite mixes as the curing days were increased. The compressive strength of 10BC mixtures increases and nearly equals with CM. The occurrence of pozzolanic reactions may result in the improvement of the compressive strength for all bentonite mixes. The increase in the percentage of strength for 20BC and 30BC was observed, but those mixes showed lower compressive strength than that of CM.

    Figure 9.  Compressive strength of bentonite mixes at the age of 28 d.
    DownLoad: Full-Size Img PowerPoint

    Figure 10 shows the compressive strength of a set of all specimens (Table 2) after 56 d of curing. The compressive strength of 10BC and 20BC shown higher than CM. 10BC and 30BC shown highest and lowest compressive strength among all mixes, whereas 20BC shown greater than CM. A little amount of strength gain percentage was observed for CM. It can be assessed that bentonite fills the voids among cement particles in concrete (pore filling effect) and a pozzolanic reaction is taking place between inactive calcium in OPC and excess of SiO2 present in bentonite.

    Figure 10.  Compressive strength of bentonite mixes at the age of 56 d.
    DownLoad: Full-Size Img PowerPoint

    The compressive strength needs to increase as workability decreases, as per the fundamentals of concrete. In this study, this fundamental was applied up to 20% replacement of bentonite. It can be attributed to pore filling effect and pozzolanic reaction with bentonite. The introduction of bentonite in 30 percentage does not follow the fundamentals. Those mixes show a decrease in strength as well as workability. The causative is less CaO content presence in bentonite than cement (Table 1) and the assimilating bentonite behaviour.

    3.5.   Split tensile strength of concrete

    In the split tensile strength test, a cylinder was placed in compression testing, and it was tested up to failure. Figure 11 shows the split tensile strength of all bentonite mixes. CM has demonstrated better performance than other mixes; equal split tensile strength was observed in a few 10BC mixes compared with CM. 30BC shown lower performance among all mixes. It was observed that split tensile strength was decreasing in the case of more than 20% of bentonite blending. It can be assessed that the poor bond formation between bentonite and cement particles.

    Figure 11.  Split tensile strength of bentonite mixes.
    DownLoad: Full-Size Img PowerPoint

    3.6.   Flexural strength of concrete

    In the flexural strength test, a plain concrete beam is subjected to flexure using third-point loading test. Figure 12 shows the flexural strength of all bentonite mixes. CM exhibits better flexural strength than other mixes comparatively. Few 10BC mixes shown equal flexural strength. The flexural strength of concrete is inversely proportional to bentonite blending (after 20%) and W/C ratio. It can be attributed to the week bond formation between bentonite and cement particles.

    Figure 12.  Flexural strength of bentonite mixes.
    DownLoad: Full-Size Img PowerPoint

    3.7.   Rapid chloride penetration test (RCPT) of concrete

    Figure 13 shows the average charge (Columbus) passed through 4 cells after seven days of curing. CM exhibits less Charge Passed (CP) (1200–1250) through it compared to others. All bentonite substituted mixes shown poor performance against chloride ion penetration at the age of seven days. This result may be because the pozzolanic reaction will not happen within seven days. The chloride ion penetration was decreased by increasing the water-cement ratio.

    Figure 13.  Charge passed in concrete after 7 d curing.
    DownLoad: Full-Size Img PowerPoint

    Figure 14 shows the average charge passed through 4 cells after 28 d curing. 10 BC exhibits less CP (360–410) through it comparatively with other mixes. 30 BC has shown abysmal performance against chloride ion penetration. This may attribute the occurrence of pozzolanic reaction at lateral (28 d) ages.

    Figure 14.  Charge passed in concrete after 28 d curing.
    DownLoad: Full-Size Img PowerPoint

    Figure 15 shows the average charge passed through 4 cells after 56 d curing. 10 BC exhibits less CP (290–320) through it comparatively with other mixes. CM and 20BC performed 30 BC shown abysmal performance against chloride ion penetration. This may attribute the occurrence of pozzolanic reaction at lateral (56 d) ages.

    Figure 15.  Charge passed in concrete after 56 d curing.
    DownLoad: Full-Size Img PowerPoint

    3.8.   Cost analysis

    Lesser cost is one of the reasons to blend bentonite with cement. The bentonite was brought from the company named "Unique Bleaching Clay" located at Tandur in Telangana state in India. The cost comparison was made according to "Standard Schedules of Rates" (SSR), Andhra Pradesh-2018-19, India, the points of interest of the cost of all blends have appeared in Table 3.

    Table 3.  Cost comparison of all mixes per cubic meter.
    Description of item CM 10BC 20BC 30BC
    Quantity Price Quantity Price Quantity Price Quantity Price
    OPC (kg) 320 2240 288 2016 256 1026 224 896
    Calcium bentonite (kg) 0 0 32 32 64 64 96 96
    Fine aggregate (m3) 819.2 148.7 818.1 148.7 812.3 148.7 812.5 148.7
    Coarse aggregate (m3) 1077.0 487.6 1075.5 487.6 1067.9 487.6 1059.5 487.6
    Total price in Indian rupees 2876.3 2684.3 2492.4 2312.9
     | Show Table
    DownLoad: CSV

    4.   RSM modeling

    4.1.   Development of the model

    In RSM, various types of models are available like Central Composite, Box Behnken and optimal (custom) in a randomized design. Choosing a model type depends on the nature of accessible information and levels for each factor. For the most part, for an experiment matrix that has been as of now built up, the historical data or user-defined model or is applied for the development and investigation of a model [41]. The linear model is signified by the first order as shown in Eq 1. Similarly, polynomial models were signified by the second order as shown in Eq 2.

    y=β0+β1X1+β2X2+β3X3+⋯⋯+βnXn+ε (1)

    where y represents the response modeled, β0 is the y-intercept at X1 = X2 = 0, whereas β1 and β2 are the coefficients of first and second independent variables, respectively, while X1 and X2 are the first and second independent variables, and ε is the error.

    y=β0+∑ki=1βiXi+∑ki=1βiiX2j+∑kj=2∑j=1i=1βijXiXj+ε (2)

    where y represents the response's model, the coded values of independent variables are represented by Xi and Xj, i and j are the coefficients for the linear and quadratic equations, respectively, βo stands for the y-intercept at X1 = X2 = 0, whereas k represents the number of independent variables used in the analyses, and ε is the error [41].

    4.2.   Mix matrix design

    Design Expert 11.0 version was used for modelling in this investigation. The design of experiments was generated using central composite design method based on two variables (bentonite replacement and W/C ratio). Four levels of % of bentonite replacement (0%, 10%, 20% and 30%) and ten levels of W/C ratio (0.60, 0.61, 0.62, 0.63, 0.64, 0.65, 0.66, 0.67, 0.68, 0.69 and 0.70) were used. A total number of 44 combinations of mixtures was developed in RSM. Table 4 represents the details of all mixtures and their combination of variables. The responses of bentonite blended concrete (slump value, compaction factor, CS 28 d, split tensile strength, flexural strength and 28 d CP) were determined for all mixtures, considered for RSM analysis and optimization.

    Table 4.  Design matrix of experiments and responses.
    Run F:1 F:2 R1 R2 R3 R4 R5 R6
    Bentonite substitution (%) W/C Slump value (mm) Compactor factor CS 28 d (MPa) Split tensile strength (MPa) Flexural strength (MPa) CP 28 d (Columbus)
    1 0 0.6 30 0.82 34.65 3.54 4.63 488.3
    2 0 0.61 45 0.85 32.04 3.40 4.43 494.3
    3 0 0.62 65 0.88 30.09 3.26 4.25 495.0
    4 0 0.63 75 0.91 28.54 3.26 4.06 503.8
    5 0 0.64 90 0.92 26.22 3.11 3.89 505.5
    6 0 0.65 115 0.93 24.44 2.97 3.69 506.5
    7 0 0.66 125 0.94 23.07 2.83 3.54 507.0
    8 0 0.67 140 0.95 22.22 2.83 3.47 508.0
    9 0 0.68 165 0.97 21.33 2.69 3.38 511.5
    10 0 0.69 210 0.98 20.00 2.69 3.24 519.5
    11 0 0.7 235 0.99 19.03 2.55 3.13 532.0
    12 10 0.6 5 0.74 32.23 3.40 4.45 367.5
    13 10 0.61 10 0.76 31.62 3.40 4.38 369.5
    14 10 0.62 30 0.81 30.12 3.26 4.23 379.0
    15 10 0.63 50 0.84 28.63 3.26 4.12 384.0
    16 10 0.64 75 0.88 26.02 3.11 3.83 388.3
    17 10 0.65 95 0.9 24.09 2.83 3.66 390.8
    18 10 0.66 110 0.92 22.96 2.83 3.54 391.0
    19 10 0.67 120 0.93 21.85 2.83 3.43 396.0
    20 10 0.68 135 0.94 20.34 2.69 3.28 403.3
    21 10 0.69 160 0.95 19.32 2.55 3.17 407.0
    22 10 0.7 195 0.97 17.99 2.41 3.03 411.8
    23 20 0.6 0 0.72 28.33 3.26 4.08 521.5
    24 20 0.61 0 0.74 27.44 3.11 3.96 529.5
    25 20 0.62 10 0.76 26.11 3.11 3.84 532.5
    26 20 0.63 15 0.78 25.22 2.97 3.74 533.0
    27 20 0.64 45 0.82 24.56 2.97 3.68 535.5
    28 20 0.65 60 0.85 22.05 2.83 3.46 540.8
    29 20 0.66 70 0.88 21.12 2.69 3.33 541.8
    30 20 0.67 85 0.9 19.96 2.55 3.21 543.8
    31 20 0.68 95 0.92 18.34 2.41 3.07 545.8
    32 20 0.69 110 0.93 17 2.41 2.91 555.8
    33 20 0.7 135 0.94 15.37 2.26 2.72 563.5
    34 30 0.6 0 0.71 25.03 2.97 3.78 655.8
    35 30 0.61 0 0.72 23.96 2.83 3.63 657.0
    36 30 0.62 5 0.74 22.45 2.83 3.5 659.3
    37 30 0.63 10 0.76 21.23 2.69 3.35 673.3
    38 30 0.64 20 0.79 19.43 2.60 3.17 676.3
    39 30 0.65 30 0.82 18.96 2.55 3.1 677.5
    40 30 0.66 45 0.84 17.2 2.41 2.91 683.0
    41 30 0.67 50 0.86 16.99 2.41 2.87 683.5
    42 30 0.68 70 0.89 15.62 2.26 2.74 684.3
    43 30 0.69 85 0.91 14.6 2.12 2.58 689.5
    44 30 0.7 95 0.93 13.33 2.12 2.47 691.8
     | Show Table
    DownLoad: CSV

    4.3.   Analysis of variance

    The summaries of ANOVA for the responses of the Bentocrete properties (slump value, compaction factor, CS 28 d, split tensile strength, flexural strength and 28 d CP) were shown in Tables 5–11. The models' F-values are 553.10,363.29,550.10,381.30,609.69 and 5635.79 for slump value, compaction factor, CS 28 d, split tensile strength, flexural strength and 28 d CP respectively, pointing all models to be significant, with 0.01% chances for all models. The Confidence Interval (CI) of the data presented in the model is 95%. This could be used to find out the significance of all the models and their terms. For slump value and split tensile strength, a quadratic model was used, and A, B, AB, A2, B2 were the terms, as per the p-value of 0.05. The significant models of compaction factor, CS 28 d, flexural strength, and CP 28 d, and its terms were cubic and, A, B, AB, A2, B2, A2B, AB2, A3 and, B3, respectively, based on the p-values were less than 0.05.

    Table 5.  ANOVA for Quadratic model, R1: slump value.
    Source Sum of squares df Mean square F-value p-value
    Model 1.550E + 05 5 31007.90 553.10 < 0.0001 significant
    A-bentonite replacement 41319.20 1 41319.20 737.02 < 0.0001
    B-water/cement ratio 1.063E + 05 1 1.063E + 05 1896.65 < 0.0001
    AB 5923.68 1 5923.68 105.66 < 0.0001
    A2 205.11 1 205.11 3.66 0.0633
    B2 1260.61 1 1260.61 22.49 < 0.0001
    Residual 2130.37 38 56.06
    Cor total 1.572E + 05 43
     | Show Table
    DownLoad: CSV
    Table 6.  ANOVA for Cubic model, R2: compaction factor.
    Source Sum of squares df Mean square F-value p-value
    Model 0.2826 9 0.0314 363.29 < 0.0001 significant
    A-bentonite replacement 0.0068 1 0.0068 78.30 < 0.0001
    B-water/cement ratio 0.0332 1 0.0332 384.19 < 0.0001
    AB 0.0030 1 0.0030 35.00 < 0.0001
    A2 0.0012 1 0.0012 13.91 0.0007
    B2 0.0027 1 0.0027 31.56 < 0.0001
    A2B 0.0017 1 0.0017 19.44 < 0.0001
    AB2 0.0011 1 0.0011 12.23 0.0013
    A3 4.091E-06 1 4.091E-06 0.0473 0.8291
    B3 0.0001 1 0.0001 0.8009 0.3771
    Residual 0.0029 34 0.0001
    Cor total 0.2856 43
     | Show Table
    DownLoad: CSV
    Table 7.  ANOVA for Cubic model, R3: CS 28 d.
    Source Sum of squares df Mean square F-value p-value
    Model 1150.93 9 127.88 550.10 < 0.0001 significant
    A-bentonite replacement 24.93 1 24.93 107.24 < 0.0001
    B-water/cement ratio 113.81 1 113.81 489.57 < 0.0001
    AB 9.70 1 9.70 41.74 < 0.0001
    A2 20.78 1 20.78 89.40 < 0.0001
    B2 2.49 1 2.49 10.71 0.0025
    A2B 0.4925 1 0.4925 2.12 0.1547
    AB2 2.54 1 2.54 10.94 0.0022
    A3 1.19 1 1.19 5.12 0.0302
    B3 0.1151 1 0.1151 0.4951 0.4865
    Residual 7.90 34 0.2325
    Cor total 1158.84 43
     | Show Table
    DownLoad: CSV
    Table 8.  ANOVA for Quadratic model, R4: split tensile strength.
    Source Sum of squares df Mean square F-value p-value
    Model 5.66 5 1.13 381.30 < 0.0001 significant
    A-bentonite replacement 1.46 1 1.46 492.99 < 0.0001
    B-water/cement ratio 4.08 1 4.08 1374.26 < 0.0001
    AB 0.0052 1 0.0052 1.74 0.1954
    A2 0.1108 1 0.1108 37.31 < 0.0001
    B2 0.0007 1 0.0007 0.2208 0.6411
    Residual 0.1129 38 0.0030
    Cor total 5.78 43
     | Show Table
    DownLoad: CSV
    Table 9.  ANOVA for Cubic model, R5: flexural strength.
    Source Sum of squares df Mean square F-value p-value
    Model 12.07 9 1.34 609.69 < 0.0001 significant
    A-bentonite replacement 0.2878 1 0.2878 130.83 < 0.0001
    B-water/cement ratio 1.12 1 1.12 510.87 < 0.0001
    AB 0.0305 1 0.0305 13.86 0.0007
    A2 0.2490 1 0.2490 113.18 < 0.0001
    B2 0.0114 1 0.0114 5.17 0.0294
    A2B 0.0006 1 0.0006 0.2902 0.5936
    AB2 0.0172 1 0.0172 7.81 0.0085
    A3 0.0139 1 0.0139 6.33 0.0168
    B3 0.0006 1 0.0006 0.2662 0.6093
    Residual 0.0748 34 0.0022
    Cor total 12.15 43
     | Show Table
    DownLoad: CSV
    Table 10.  ANOVA for Cubic model, R6: CP 28 d.
    Source Sum of squares df Mean square F-value p-value
    Model 4.626E + 05 9 51397.11 5635.79 < 0.0001 significant
    A-bentonite replacement 1.050E + 05 1 1.050E+05 11509.10 < 0.0001
    B-water/cement ratio 388.24 1 388.24 42.57 < 0.0001
    AB 0.0525 1 0.0525 0.0058 0.9399
    A2 1.745E+05 1 1.745E + 05 19135.34 < 0.0001
    B2 0.4838 1 0.4838 0.0531 0.8192
    A2B 8.67 1 8.67 0.9502 0.3365
    AB2 38.92 1 38.92 4.27 0.0465
    A3 43851.07 1 43851.07 4808.35 < 0.0001
    B3 116.64 1 116.64 12.79 0.0011
    Residual 310.07 34 9.12
    Cor total 4.629E + 05 43
     | Show Table
    DownLoad: CSV
    Table 11.  Validation of models.
    Response R2 Adj. R2 Pred. R2 Adj.R2-Pred.R2 SD Adeq. Prec. Mean
    Slump value 0.9864 0.9847 0.9821 0.0026 7.49 85.9629 75.34
    Compaction factor 0.9897 0.9870 0.9802 0.0068 0.0093 62.4137 0.8634
    28 d CS 0.9932 0.9914 0.9873 0.0041 0.4821 91.3261 22.98
    Split tensile strength 0.9805 0.9779 0.9743 0.0036 0.0545 72.18 2.82
    Flexural strength 0.9938 0.9922 0.9889 0.0033 0.0469 96.7725 3.52
    CP 28 d 0.9993 0.9992 0.9988 0.0004 3.02 225.9156 528.03
     | Show Table
    DownLoad: CSV

    For slump value, A2 was the insignificant term and interaction in the quadratic model. The insignificant terms in compaction factor model were A3 and B3, while A2B and B3 were insignificant terms in CS 28 d cubic model, whereas the insignificant terms in split tensile strength quadratic model were AB and B2. The insignificant terms in flexural strength cubic model were A2B and B3, whereas AB, B2 and A2B were insignificant terms in CP 28 d. A few interactions were not significant in all bentonite mixes influencing factors because of their higher (>0.05) p-values. The positive and negative signs were assigned in the models and their interactions to show the effects of the variables on all bentonite mixes factors.

    The final models for slump value, compaction factor, CS 28 d, split tensile strength, flexural strength and 28 d CP of all mixes comprising all the terms are presented in Eqs 3–8, respectively.

     Slump value =+1342.72727+17.94318 A−5831.96970 B−32.81818AB+0.021591 A2+6060.60606 B2 (3)
     Compaction factor =+7.45830+0.151008 A−44.12313 B−0.563364AB+0.001302 A2+86.85606 B2−0.001955 A2 B+0.496503AB2+4.54545E−07 A3−52.93318 B3 (4)
    CS28 d=−196.85548−10.37951 A+1752.40855 B+31.98212 A B−0.039650 A2−3573.76457 B2+0.033455 A2 B+0.000245 A3+2158.11966 B3 (5)
     Spilt tensile strength =+7.73014−0.021167 A−4.40756 B+0.030625AB−0.000502 A2−4.37166 B2 (6)
     Flexural strength =−8.19684−0.856959 A+117.10348 B+2.63983AB−0.002728 A2−251.96096 B2+0.001205 A2 B−2.00117AB2+0.000027 A3+153.94328 B3 (7)
    CP28 d=−17862.46208−77.38314 A+85306.42113 B+128.13371AB+2.83872 A2−1.32425E+05 B2−0.140341 A2 B−95.25058AB2−0.047061 A3+68703.86558 B3 (8)

    The properties of formulated models of for slump value, compaction factor, CS 28 d, split tensile strength, flexural strength and 28 d CP of all mixes were presented in Table 11. The adequacy of the formulated models was validated by their degree of correlation (R2). A high degree of correlation (R2) was observed for all models based on their values were close to unity (R2 > 0.97). This shows that the experimental values slump value, compaction factor, CS 28 d, split tensile strength, flexural strength and 28 d CP cannot be represented by the models only about 1.36%, 1.03%, 0.68%, 1.95%, 0.62%, and 0.07%, respectively.

    3-dimensional (3D) response surface plot was used to illustrate the relationship between responses and independent variables. Figures 16–21 show the 3D response surface plots illustrating the relationship between responses (slump value, compaction factor, CS 28 d, split tensile strength, flexural strength and 28 d CP) and independent variables (W/C ratio and bentonite) for all mixes. The high-water absorption capacity of bentonite may have caused a rise in slump values with increasing w/c ratio, which decreases upon the addition of bentonite [12]. Bentocrete mixes exhibit Lower CS at early ages (7 d) curing, higher compressive strength at later curing period (28 d and 56 d). The reason behind this was the formation of secondary C–S–H gel since the bentonite obeys pozzolanic properties [42]. The split tensile strength and flexural strength of CM exhibits equal for 10BC for a few mixes, less strength was observed for 20BC and 30BC mixes. Bentocrete mixes display better durability upon addition of bentonite up to some extent (20%). This attributes the pore filling effect and happening of pozzolanic reaction caused by the addition of bentonite.

    Figure 16.  Response surface model for slump value.
    DownLoad: Full-Size Img PowerPoint
    Figure 17.  Response surface model for compaction factor.
    DownLoad: Full-Size Img PowerPoint
    Figure 18.  Response surface model for CS 28 d.
    DownLoad: Full-Size Img PowerPoint
    Figure 19.  Response surface model for split tensile strength.
    DownLoad: Full-Size Img PowerPoint
    Figure 20.  Response surface model for flexuleral strength.
    DownLoad: Full-Size Img PowerPoint
    Figure 21.  Response surface model for CP 28 d.
    DownLoad: Full-Size Img PowerPoint

    4.4.   Optimization

    Optimization was used with the aim of finding an optimized value among all mixes, performed by using Design Expert Software. The upper limit, lower limit and goal were set to perform the optimization, displayed in Table 12. The optimized solution was found with a desirability of 0.881, Table 13 shows the optimized Bentocrete mixes. The optimized mix was achieved at 3.92% of bentonite substitution and 0.62 W/C ratio. The optimum values of responses were 49.65 mm slump value, 0.85 compaction factor, 30.04 MPa CS 28 d, 3.28 MPa split tensile strength, 4.23 MPa flexural strength and 401.87 28 d CP.

    Table 12.  Criteria setting for optimization.
    Variables/responses Goal Lower limit Upper limit
    Bentonite replacement In range 0 30
    W/C ratio In range 0.60 0.70
    Slump value Target-50 0 235
    Compaction factor Target-0.85 0.71 173
    CS 28 d Maximize 13.33 34.65
    Split tensile strength Maximize 2.12 3.53
    Flexural strength Maximize 2.47 4.63
    28 d CP Minimize 367.5 691.75
     | Show Table
    DownLoad: CSV
    Table 13.  Optimized Bentocrete mixes.
    Bentonite replacement W-C ratio Slump value Compaction factor CS 28 d Split tensile strength Flexural strength 28 d CP
    3.92 0.62 49.65 0.85 30.04 3.28 4.23 401.87
     | Show Table
    DownLoad: CSV

    5.   Conclusions

    The following conclusions were drawn based on the experiments and analyses performed: The standard consistency of cement paste is directly proportional to the bentonite substitution. 10 BC showed maximum compressive strength among all cement mortar mixes due to pozzolanic nature of bentonite. Workability is directly proportional to W/C ratio, because of the high-water absorption capacity of bentonite. The compression strength of Bentocrete shown lesser at 7 d equals at 28 d, and higher at 56 d than cement concrete. However, Bentocrete with more than 20 percentage of bentonite exhibits lesser compressive strength than concrete at all curing periods. In split tensile strength and flexural strength, the lower performance was observed for more than 10% of bentonite substitution. The durability against chloride ion was improved for Bentocrete than cement concrete up to 20% upon curing for enough time; this attribute the pore filling effect since bentonite's particle size is lesser than cement. Cost analysis was performed, 9.91% of the cost can be cut down for Bentocrete with 20 percentage bentonite. The RSM model has fitted the experimental data with the full agreement with R2 values not less than 0.985 in all the cases. 3.92 % of bentonite substitution and 0.62 W/C ratio provided optimum solution for the intended goals with desirability of 0.881.

    Acknowledgments

    The authors are thankful to the authorities of the Koneru Lakshmaiah Education Foundation for funding this research as a part of the internal funding project scheme.

    Conflict of interest

    All authors declare no conflicts of interest in this paper.



    [1] Bambusi D, Ponno A (2006) On metastability in FPU. Commun Math Phys 264: 539–561. doi: 10.1007/s00220-005-1488-1
    [2] Berman GP, Izrailev FM (2005) The Fermi-Pasta-Ulam problem: 50 years of progress. Chaos 15: 015104. doi: 10.1063/1.1855036
    [3] Benettin G, Livi R, Ponno A (2009) The Fermi-Pasta-Ulam problem: Scaling laws vs. initial conditions. J Stat Phys 135: 873–893.
    [4] Benettin G, Ponno A (2011) Time-scales to equipartition in the Fermi-Pasta-Ulam problem: Finite-size effects and thermodynamic limit. J Stat Phys 144: 793–812. doi: 10.1007/s10955-011-0277-9
    [5] Benettin G, Christodoulidi H, Ponno A (2013) The Fermi-Pasta-Ulam problem and its underlying integrable dynamics. J Stat Phys 152: 195–212. doi: 10.1007/s10955-013-0760-6
    [6] Benettin G, Pasquali S, Ponno A (2018) The Fermi-Pasta-Ulam problem and its underlying integrable dynamics. J Stat Phys 171: 521–542. doi: 10.1007/s10955-018-2017-x
    [7] Carati A, Galgani L, Giorgilli A, et al. (2007) Fermi-Pasta-Ulam phenomenon for generic initial data. Phys Rev E 76: 022104.
    [8] Carati A, Galgani L (1999) On the specific heat of Fermi-Pasta-Ulam Systems and their glassy behavior. J Stat Phys 94: 859–869. doi: 10.1023/A:1004531032623
    [9] Carati A, Maiocchi A, Galgani L, et al. (2015) The Fermi-Pasta-Ulam system as a model for glasses. Math Phys Anal Geom 18: 31. doi: 10.1007/s11040-015-9201-x
    [10] Carati A, Ponno A (2018) Chopping time of the FPU α-model. J Stat Phys 170: 883–894. doi: 10.1007/s10955-018-1962-8
    [11] Casetti L, Cerruti-Sola M, Pettini M, et al. (1997) The Fermi-Pasta-Ulam problem revisited: Stochasticity thresholds in nonlinear Hamiltonian systems. Phys Rev E 55: 6566–6574. doi: 10.1103/PhysRevE.55.6566
    [12] Christodoulidi H, Efthymiopoulos C, Bountis T (2010) Energy localization on q-tori, long-term stability, and the interpretation of Fermi-Pasta-Ulam recurrences. Phys Rev E 81: 016210/1–16.
    [13] Christodoulidi H, Efthymiopoulos C (2013) Low-dimensional q-Tori in FPU lattices: Dynamics and localization properties. Phys D 261: 92–113. doi: 10.1016/j.physd.2013.07.007
    [14] Christodoulidi H (2017) Extensive packet excitations in FPU and Toda lattices. EPL 119: 40005. doi: 10.1209/0295-5075/119/40005
    [15] Chirikov BV (1960) Resonance processes in magnetic traps. Soviet J At Energy 6: 464–470. doi: 10.1007/BF01483352
    [16] Chirikov BV (1979) A universal instability of many-dimensional oscillator systems. Phys Rep 52: 263–379. doi: 10.1016/0370-1573(79)90023-1
    [17] Danieli C, Campbell DK, Flach S (2017) Intermittent many-body dynamics at equilibrium. Phys Rev E 95: 060202(R).
    [18] Danieli C, Mithun T, Kati Y, et al. (2018) Dynamical glass in weakly non-integrable many-body systems. [arxiv:1811.10832].
    [19] Dauxois T (2008) Fermi, Pasta, Ulam, and a mysterious lady. Phys Today 61: 55–57.
    [20] Ferguson WE Jr, Flaschka H, McLaughlin DW (1982) Nonlinear normal modes for the Toda Chain. J Comput Phys 45: 157–209. doi: 10.1016/0021-9991(82)90116-4
    [21] Fermi E, Pasta J, Ulam S (1995) Studies of non linear problems. Los Alamos report No LA-1940.
    [22] Flach S, Ivanchenko MV, Kanakov OI (2005) q-Breathers and the Fermi-Pasta-Ulam problem. Phys Rev Lett 95: 064102. doi: 10.1103/PhysRevLett.95.064102
    [23] Flach S, IvanchenkoMV, Kanakov OI (2006) q-Breathers in Fermi-Pasta-Ulam chains: Existence, localization, and stability. Phys Rev E 73: 036618. doi: 10.1103/PhysRevE.73.036618
    [24] Flach S, Ponno A (2008) The Fermi-Pasta-Ulam problem Periodic orbits, normal forms and resonance overlap criteria. Phys D 237: 908–917. doi: 10.1016/j.physd.2007.11.017
    [25] Flaschka H (1974) The Toda lattice. II. Existence of integrals. Phys Rev B 9: 1924–1925.
    [26] Goldfriend T, Kurchan T (2019) Equilibration of Quasi-Integrable Systems. Phys Rev E 99: 022146. doi: 10.1103/PhysRevE.99.022146
    [27] Izrailev FM, Chirikov BV (1966) Statistical properties of a nonlinear string. Dokl Akad Nauk SSSR 166: 57–59.
    [28] Gallavotti G (2008) The Fermi-Pasta-Ulam Problem: A Status Report. Springer, Berlin- Heidelberg, vol. 728.
    [29] Genta T, Giorgilli A, Paleari S, et al. (2012) Packets of resonant modes in the Fermi-Pasta-Ulam system. Phys Lett A 376: 2038–2044. doi: 10.1016/j.physleta.2012.05.006
    [30] Hénon M (1974) Integrals of the Toda lattice. Phys Rev B 9: 1921–1923. doi: 10.1103/PhysRevB.9.1921
    [31] Kantz H, Livi R, Ruffo S (1994) Equipartition thresholds in chains of anharmonic oscillators. J Stat Phys 76: 627–643. doi: 10.1007/BF02188678
    [32] Kantz H (1989) Vanishing stability thresholds in the thermodynamic limit of nonintegrable conservative systems. Phys D 39: 322–335. doi: 10.1016/0167-2789(89)90014-6
    [33] Kruskal MD, Zabusky NJ (1965) Interaction of "solitons" in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15: 240–243.
    [34] Parisi G (1997) On the approach to equilibrium of a Hamiltonian chain of anharmonic oscillators. EPL 40: 357–362. doi: 10.1209/epl/i1997-00471-9
    [35] Penati T, Flach S (2007) Tail resonances of Fermi-Pasta-Ulam q-breathers and their impact on the pathway to equipartition. Chaos 17: 023102/1-16.
    [36] Paleari S, Penati T (2005) Equipartition times in a Fermi-Pasta-Ulam system. Discrete Contin Dyn S 2005: 710–719.
    [37] Ponno A, Bambusi D (2005) Korteweg-de Vries equation and energy sharing in Fermi-Pasta- Ulam. Chaos 15: 015107. doi: 10.1063/1.1832772
    [38] Ponno A, Christodoulidi H, Skokos Ch, et al. (2011) The two-stage dynamics in the Fermi-Pasta- Ulam problem: From regular to diffusive behavior. Chaos 21: 043127. doi: 10.1063/1.3658620
    [39] Livi R, Pettini M, Ruffo S, et al. (1985) Equipartition threshold in nonlinear large Hamiltonian systems The Fermi-Pasta-Ulam model. Phys Rev A 31: 1039–1045.
    [40] Lvov YV, Onorato M (2018) Double scaling in the relaxation time in the β-Fermi-Pasta-Ulam- Tsingou model. Phys Rev Lett 120: 144301. doi: 10.1103/PhysRevLett.120.144301
    [41] Shepelyansky DL (1997) Low-energy chaos in the Fermi-Pasta-Ulam problem. Nonlinearity 10: 1331–1338. doi: 10.1088/0951-7715/10/5/017
    [42] Toda M (1970) Waves in Nonlinear Lattice. Prog Theor Phys Suppl 45: 174–200. doi: 10.1143/PTPS.45.174

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    4. M. Achyutha Kumar Reddy, V. Ranga Rao, Veerendrakumar C. Khed, Kavuri Naga Chaitanya, Optimization of reinforced bentocrete column parameters under eccentric compression, 2022, 41, 23520124, 1027, 10.1016/j.istruc.2022.05.050
    5. Jawad Ahmad, Karolos J. Kontoleon, Mohammed Zuhear Al-Mulali, Saboor Shaik, Mohamed Hechmi El Ouni, Mohammed A. El-Shorbagy, Partial Substitution of Binding Material by Bentonite Clay (BC) in Concrete: A Review, 2022, 12, 2075-5309, 634, 10.3390/buildings12050634
    6. Hany A. Dahish, Ahmed D. Almutairi, Effect of Elevated Temperatures on the Compressive Strength of Nano-silica and Nano-clay Modified Concretes Using Response Surface Methodology, 2023, 22145095, e02032, 10.1016/j.cscm.2023.e02032
    7. Raman Saini, Gyanendra Kumar Chaturvedy, Umesh Kumar Pandey, Examining the effects of nano iron oxide on physical and mechanical characteristics of rubberized concrete, 2024, 9, 2364-4176, 10.1007/s41062-024-01494-6
    8. Dhrub Kumar Das, Aditya Kumar Tiwary, Influence of nano bentonite clay and nano fly ash on the mechanical and durability properties of concrete, 2024, 26, 1618-954X, 3881, 10.1007/s10098-023-02610-3
    9. Mohammed Salman Khizar, M. Achyutha Kumar Reddy, Naga Chaitanya Kavuri, Veerendrakumar C. Khed, 2023, 2759, 0094-243X, 060005, 10.1063/5.0144106
    10. Kamrul Hasan, Ramadhansyah Putra Jaya, Fadzil Mat Yahaya, Application of bentonite in cement-based composites: A review of current status, challenges and future prospects, 2024, 98, 23527102, 111171, 10.1016/j.jobe.2024.111171
    11. Muhammad Saeed Zafar, Adnan Shahid, Reza Sedghi, Maryam Hojati, Optimization of biopolymer additives for 3D printable cementitious systems: A design of experiment approach, 2025, 22, 22145095, e04515, 10.1016/j.cscm.2025.e04515
    12. Muhammad Saqib Khan, Muhammad Sarfaraz Khan, Muhammad Imran Khan, Rania Al-Nawasir, Nelson Maureira-Carsalade, Siva Avudaiappan, Rafiq M. Choudhry, Enhancing rigid pavement performance: Experimental study and design optimization of bentonite clay-blended concrete with a focus on durability, 2025, 22, 22145095, e04641, 10.1016/j.cscm.2025.e04641
    13. Ajith S., Karthik R., Manoj Kanti Debnath, Laxmanarayanan M., Deepranjan Sarkar, Rakesh S., Rahul Datta, Sachidanand Singh, 2025, Chapter 19, 978-981-96-1336-6, 469, 10.1007/978-981-96-1337-3_19
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Mathematics in Engineering
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Article outline
Abstract
   Introduction
   Materials and methods
   Results and discussion
   RSM modeling
   Conclusions
Acknowledgments
Conflict of interest
References

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