The incorporation of carbon nanotubes (CNTs) can enhance the mechanical properties of concrete. The stress–strain curves of CNT-reinforced concrete under uniaxial compression are investigated through an experimental program with different CNT and steel fiber proportions considered. The test results demonstrate that CNTs can increase both peak stress and peak strain, and steel fibers can further enhance the effect of CNTs. Additionally, steel fibers can effectively enhance both the strength and ductility. Theoretical models for the peak strain, initial elastic modulus, toughness index and relative absorbed energy are established. A theoretical model for the uniaxial compressive constitutive relationship of CNT-reinforced concrete considering CNT and steel fiber content is developed. Finite element (FE) modelling is developed to simulate the axial compression behavior of CNT-reinforced concrete.
Keywords: CNT; stress–strain relationship; finite element modeling; elastic modulus; toughness index; relative absorbed energy
The brittleness of ordinary concrete materials has long been a subject of concern. Studies have revealed that reinforcing fibers, such as steel, glass, and polymer fibers, can effectively mitigate crack development in concrete and enhance the mechanical properties and durability of concrete [[
As a novel nanomaterial, CNTs exhibit exceptional mechanical properties and have the characteristics of a light weight and small sizes [[
Jiang et al. investigated the strengthening effect of CNTs in cement-based material and compared it with three other nanomaterials, namely nano-SiO
Studies on the stress–strain behavior of CNT-reinforced concrete under uniaxial compression are limited, and no studies have been conducted on the theoretical model for the constitutive relationship of CNT-reinforced concrete to the authors' knowledge to date. The compressive performance of CNT-reinforced concrete is investigated in this study. The influence of the CNT and steel fiber content on uniaxial compressive stress–strain curves is analyzed through an experimental program. The theoretical models for the peak strain, initial elastic modulus, toughness index, relative absorbed energy and also the stress–strain curves are developed, and finite element modeling is performed.
In this study, the materials could be divided into three parts. The first part consisted of cement, fine sand, silica fume, quartz powder, a water-reducing agent and an accelerator, constituting the cement-based component of CNT-reinforced concrete. The second part included CNTs and GO. The third part consisted of steel fibers.
P·O 42.5 cement was used, and the average particle size of cement was 17 μm. Natural sand with an average particle size of 487 μm was used. Silica fume with a bulk density of 350 kg/m
In this study, the mass fractions of CNTs were 0, 0.08%, and 0.5%. The volume fraction of steel fibers ranged from 0 to 2%. A total of 36 specimens were prepared, with 6 specimens for each mix proportion, and 3 specimens were tested at the ages of 7 days and 28 days, respectively, as summarized in Table 5. In the group number, "C" and the number after it represent CNTs and their corresponding mass fraction (%), while "S" denotes steel fibers along with their volume fraction (%). For instance, "C0.08S2" refers to a mixture containing CNTs at a mass fraction of 0.08% and steel fibers at a volume fraction of 2%.
According to ASTM C39/C39M-17 [[
Portland cement, fine sand, silica fume, and quartz powder were mixed for 2 min. The accelerator and half of the superplasticizer and CNT solution were slowly added within 30 s, and then stirred for 1 min. The remaining superplasticizer and CNT solution were then added and stirred for 15 min. Finally, the steel fibers were gradually introduced within 30 s followed by stirring for an additional 1 min. In accordance with ASTM C31/C31M-03 [[
The MTS815.02 electro-hydraulic servo test system was employed. Following the calibration of axial load, the initial loading speed was 0.04 mm/min, and was reduced to 0.02 mm/min as the curve reached the gentle stage. The load was measured by the pressure sensor and deformation was measured using an Epsilon extensometer. All of the data were recorded. The test setup is shown in Figure 2.
The influence of CNT content on the failure modes was investigated. As shown in Figure 3a–c for 7-day specimens without steel fibers, all specimens exhibited brittle failure, and no significant difference was observed in the failure modes as CNT content varied. Upon reaching the maximum stress, vertical cracks emerged abruptly and small fragments detached from the specimen surface, leading to all-through cracks and failure. CNT content had little influence on the failure process. As internal cracks propagated, CNTs could not mitigate millimeter-scale cracks, which resulted in brittle fracture.
Comparing Figure 3c and Figure 3d, the incorporation of steel fibers changed the failure mode from brittle failure to ductile failure. Unlike specimens without steel fibers, those containing 2% steel fibers exhibited the initial formation of small cracks at ultimate stress and still maintained certain load-bearing capacity. Cracks propagated gradually, and eventually all-through cracks formed. The presence of steel fibers at the crack interface played a crucial role in impeding crack growth.
Regarding the failure mode of specimens at different ages, no significant difference was observed.
According to the test results, the average stress–strain curves of each mixture were plotted, as illustrated in Figure 4.
The influence of CNT content on the stress–strain curves of specimens without steel fibers is limited, as depicted in Figure 4a,b. All of the specimens without steel fibers exhibited evident brittle failure, with a small descending section observed.
As shown in Figure 4c,d, specimens with 2% steel fibers exhibited ductile failure, indicating the significant influence of steel fibers on the stress–strain curve. The presence of CNTs can affect the descending section of the curve, and the inclusion of 0.08% CNTs resulted in a more gradual decline compared to specimens without CNTs and also those with 0.5% CNTs. This suggested that 0.08% CNTs can enhance the ductility, while excessive CNT contents can have adverse effects due to agglomeration [[
Peak stress of different mixtures is compared, as shown in Figure 5a. Comparing specimens without steel fibers at 7 days of age, a 9% increase in peak stress was observed as the CNT content increased from 0 to 0.08%. With a further increase in CNT content to 0.5%, the peak stress was 5% higher than that of specimens without CNT. The addition of CNTs enhanced the peak stress, while excessive CNTs may weaken its effect on peak stress. This can be attributed to CNTs' bridging effect for nano- and micron-scale cracks within the cement matrix. However, an excessive CNT content may lead to aggregation that weakens their strengthening effect. For specimens with 2% steel fibers, an increase in the CNT content from 0 to 0.08% resulted in elevation in peak stress by 11%. And as the CNT content further increased to 0.5%, there was an enhancement of 9% compared to specimens without CNT. Steel fibers can enhance the effect of CNTs on peak stress. The effect of CNTs and steel fibers on the peak stress remains consistent for the 28-day groups.
The peak strain of different mixtures is compared, as shown in Figure 5b. Comparing specimens without steel fibers at 7 days of age, the peak strain increased by 8% as the CNT content increased from 0 to 0.08%. With a further increase in the CNT content to 0.5%, the peak strain was 1% higher than that of specimens without CNT. Similar variations were observed for other groups with different ages and steel fiber contents. Similar to the pattern for peak stress, the addition of CNTs enhanced the peak strain, while excessive CNTs may weaken its effect on peak strain. In comparison with the C0S2, the peak strain of C0.08S2 and C0.5S2 increased by 10% and 8%, respectively. Similar to peak stress, steel fibers can enhance the effect of CNTs on peak strain.
The addition of steel fibers has a significant impact on the peak stress and peak strain, as shown in Figure 5. For instance, compared with C0.08S0, the peak stress of C0.08S2 increased by 50% at 7 days. This effect of steel fibers on peak stress was consistent for different ages and CNT contents. Similarly, there was also a noticeable improvement for the peak strain ranging from 33% to 48% for various age groups and CNT content levels as 2% steel fibers was added. This can be attributed to the ability of steel fibers to control crack development.
The secant modulus at 80% of the peak stress in the stress–strain curve was calculated as the elastic modulus [[
The toughness index, denoted as
The toughness index of different mix proportions was calculated, as illustrated in Figure 8. For mixtures without steel fibers, age and CNT content had no significant influence on the toughness index. For mixtures with 2% steel fibers, those with 0.08% CNTs exhibited the highest toughness index, which was 11% and 17% higher than those without CNTs at 7 days and 28 days of age, respectively. However, as the CNT content increased to 0.5%, the toughness index decreased and was 3% and 7% higher than those without CNTs. This suggests that CNTs can enhance the toughness index when steel fibers are used, and excessive amounts may diminish its effectiveness.
The toughness index of mixtures without steel fibers was less than 0.26, and it increased and ranged between 0.60 and 0.72 as 2% steel fibers were added. Steel fibers can significantly improve the toughness index by limiting crack propagation. Furthermore, the toughness index increased by 2% to 15% with the increase in age.
The energy absorption rate is defined as the area enclosed by the stress–strain curve and the transverse axis, spanning from the origin to the peak stress point [[
The relative absorbed energy of different mix proportions was calculated, as illustrated in Figure 10. CNTs had negligible influence on the relative absorbed energy. Steel fibers can increase the relative absorbed energy, and the maximum increase was found at a CNT content of 0.08% and 28 days. No pattern can be found in the influence of age on the relative absorbed energy.
The peak strain is a critical parameter that influences the characteristics of both the ascending and descending sections of the stress–strain curve. Table 6 presents the theoretical equations for peak strain proposed in various literatures.
Based on the test data in this study, a new theoretical model for peak strain is developed, as shown in Equation (
(
The test results of this study and the predicted results for the peak strain produced by different theoretical models are compared, as shown in Figure 11, and the model established in this study shows the best fit. The coefficient of determination (R
The theoretical equations for the initial elastic modulus from previous studies are presented in Table 7.
Based on the test data in this study, a new theoretical model for initial elastic modulus is proposed, as shown in Equation (
(
The test results of this study and the predicted results for the initial elastic modulus produced by different theoretical models are compared, as shown in Figure 12. The R
Based on the test data in this study, a new theoretical model for toughness index is developed, as shown in Equation (
(
The test results and predicted results for the toughness index are compared, as shown in Figure 13. The R
The theoretical equations for the relative absorbed energy from previous studies are presented in Table 8.
Based on the experimental data in this study, a new theoretical model for relative absorbed energy is proposed, as shown in Equation (
(
The test results and predicted results for the relative absorbed energy produced by different models are compared, as shown in Figure 14. The R
The theoretical models for stress–strain curves from the literature are presented in Table 9.
Based on the test data in this study, a new theoretical model is proposed.
A fourth-order polynomial is employed for the ascending section to consider the effect of CNTs and steel fibers, as shown in Equation (
(
A fractional model is employed for the descending section, as shown in Equation (
(
To prevent overfitting, the coefficients
(
(
By initializing the values of the linear coefficients
The test and predicted stress–strain curves produced by different theoretical models are compared, as shown in Figure 15 and Figure 16. The R
As shown in Figure 15 and Figure 16, the model established in this study shows the best fit for most mixtures. The prediction provided by the proposed model has the highest R
Finite element analysis was conducted using the representative volume element (RVE) method. The software Ansys, version R18.0 was utilized. Steel fibers are distributed in a homogeneous RVE cylinder of Φ50 mm × 100 mm. The specimen consists of two components: steel fibers and RVE. The RVE is composed of a homogeneous cement-based material and distributed CNTs. Firstly, a 100 μm × 100 μm cube RVE was modeled as a composite material and its constitutive relationship was analyzed. Then, the cylinder specimen was modeled and the simulation results were compared with experimental data.
In the FE model, the CNTs and steel fibers are assumed to be distributed uniformly and randomly in the matrix, which was realized by the rand() function in Ansys. The total number of fibers is determined by dividing the overall fiber volume in the matrix by the volume of one individual fiber, as shown in Equation (
(
where
To enhance model convergence and computational efficiency, simplifications were implemented in this study. GO was solely employed as a surfactant and thus was not considered in modeling. The cement matrix in RVE was treated as a homogeneous entity with mechanical properties derived from control specimens without CNTs and steel fibers.
A 100 μm cubic RVE matrix was simulated, and Solid65 unit was used. For the failure criterion, C1 was set to 0.5, C2 to 1.0, and C3 to −1, semi-brittle cracking behavior was considered for C9, and default values were used for other parameters. The multi-linear kinematic hardening model (KINH) was employed for the cement matrix, with the material properties determined by the test stress–strain curve of C0S0 specimens. CNTs within the RVE were simulated using Link10 elements and restricted to tension only. CNTs were considered as ideal elastic–plastic materials, and a bilinear kinematic model (BKIN) material model with an elastic modulus of 750 GPa and tensile strength of 50 GPa was adopted.
The cement matrix had a partition density of 12 units along each side. Each CNT was divided into four segments. The meshing result of the RVE is illustrated in Figure 22.
The CEINTF command was utilized to couple the cell nodes of CNTs and the cement matrix, with CNTs as the dense grid region and the cement matrix as the sparse grid region. A tolerance value of 25% (TOLER = 25%) was selected. All nodes on the bottom of the RVE cube were subjected to constraints in the X, Y, and Z directions. A displacement load was applied to the top surface. Force convergence criteria were employed with a convergence accuracy of 5%. The number of load steps was 1000.
The test specimen of Φ50 mm × 100 mm cylinder was modeled. The Link10 unit was employed for steel fibers, with only tension considered. The real constant of the Link10 unit was the cross-sectional area of steel fibers, 0.038 mm
The steel fibers were meshed into two segments. The grid for the RVE matrix was a hexahedron, with a partition density of 20 units along the cylinder's height and 24 units along the top and bottom surface circumferences. The meshing result is illustrated in Figure 23.
The coupling between steel fibers and the RVE matrix was simulated through the CEINTF command, with the former considered as the dense grid region and the latter as the sparse one. A tolerance of 25% (TOLER = 25%) was selected. All element nodes on the bottom surface of the cylinder were subjected to constraints in the directions of X, Y, and Z, while a fixed vertical displacement load was applied to the top surface. Automatic load step adjustment was employed.
The stress–strain curves of C0.08S2 and C0.5S2 from the FE simulation are compared with the test results, as shown in Figure 24.
The stress–strain curves from the FE simulation exhibit good agreement with the measured curves in the ascending section, except that the FE curves show higher peak stress and peak strain values. For the descending section, enhanced ductility is demonstrated in the FE curves. These differences can be attributed to defects present in the test specimens compared with the ideal condition in the simulation. The actual effect of steel fibers on cracks is also limited compared with the ideal condition in FE modelling. Also, a simplification was made in the modelling, and refined models considering properties of CNTs at the nanoscale need be further studied.
The crack development of the C0.08S2 specimen was analyzed using the FE modelling. The crack propagation under various load steps is shown Figure 25. The cracks were first observed after the 31st loading step near the top and bottom of the specimen, where displacement loads and constraint were applied. As the load increased, axial cracking on the side occurred, and previously observed cracks near the top and bottom developed. After the 41st loading step, cracks widely distributed on the surface of the specimen, which is consistent with the experiment.
The compressive properties of CNT-reinforced concrete were studied through an experimental investigation. The theoretical models of peak strain, elastic modulus, toughness index, relative absorbed energy and also compressive stress–strain curves of CNT-reinforced concrete were proposed. The following conclusions can be drawn:
- (
1 ) CNT content had little influence on the failure process and elastic modulus. The incorporation of CNTs can enhance the peak stress and peak strain due to CNTs' bridging effect for nano- and micron-scale cracks, while an excessive CNT content may weaken this effect due to CNTs' aggregation. Steel fibers can enhance the effect of CNTs on peak stress and peak strain. - (
2 ) In the absence of steel fibers, CNTs had negligible influence on toughness index, while steel fibers can enhance the effect of CNTs on toughness index. Nevertheless, excessive CNTs diminished this enhancement effect. - (
3 ) Steel fibers can increase the peak stress, peak strain, elastic modulus, toughness index and relative absorbed energy of concrete, and can result in ductile failure. - (
4 ) Theoretical models for peak strain, initial elastic modulus, toughness index and relative absorbed energy were established. The models were in good agreement with experiment results. - (
5 ) A theoretical model for the stress–strain curve of CNT-reinforced concrete was developed considering the content of CNTs and steel fibers. The proposed model demonstrated the best fit among the different theoretical models. - (
6 ) Considering the random distribution of CNTs and steel fibers, a simplified FE model was developed using the RVE method. The specimen was divided into two components: steel fibers and RVE with CNTs and cement-based material. There is agreement between the experimental and simulation results. The differences between the FE modelling and the experimental results can be attributed to defects in the test specimens compared with the ideal condition in the simulation. Refined models considering the properties of CNTs at the nanoscale need be further studied for better simulation accuracy. - (
7 ) Due to the high cost of the materials, the mixture proportions considered and tests performed in this study were limited. More tests are needed to study the influence of CNT on concrete properties. Additionally, a simplified FE model was developed, and refined FE models considering properties of CNTs at the nanoscale need be further studied for better simulation accuracy.
Graph: Figure 1 SWCNT and MWCNT [[
Graph: Figure 2 Test setup.
Graph: Figure 3 Failure modes of specimens at 7 days of age: (a) C0S0; (b) C0.08S0; (c) C0.5S0; (d) C0.5S2.
Graph: Figure 4 Stress–strain curves of different mixture proportions: (a) no steel fibers at 7 d; (b) no steel fibers at 28 d; (c) 2% steel fibers at 7 d; (d) 2% steel fibers at 28 d.
Graph: buildings-14-00418-g004b.tif
Graph: Figure 5 Peak stress and peak strain of different mixture proportions: (a) peak stress; (b) peak strain.
Graph: Figure 6 Elastic modulus of different mixture proportions.
Graph: Figure 7 Calculation of toughness index.
Graph: Figure 8 Toughness index of different mixture proportions.
Graph: Figure 9 Calculation of relative absorbed energy: (a) perfectly linear elastic material; (b) perfectly plastic material.
Graph: Figure 10 Relative absorbed energy of different mixture proportions.
Graph: Figure 11 Test results and predicted results for the peak strain produced by different theoretical models [[
Graph: Figure 12 Test results and predicted results for the initial elastic modulus produced by different theoretical models [[
Graph: Figure 13 Test results and predicted results for the toughness index.
Graph: Figure 14 Test results and predicted results for the relative absorbed energy produced by different theoretical models [[
Graph: Figure 15 Test and predicted stress–strain curves at 7 days produced by different theoretical models [ 41 , 44 , 46 , 53 , 54 ]: (a) C0S0; (b) C0S2; (c) C0.08S0; (d) C0.08S2; (e) C0.5S0; (f) C0.5S2.
Graph: buildings-14-00418-g015b.tif
Graph: Figure 16 Test and predicted stress–strain curves at 28 days produced by different theoretical models [ 41 , 44 , 46 , 53 , 54 ]: (a) C0S0; (b) C0S2; (c) C0.08S0; (d) C0.08S2; (e) C0.5S0; (f) C0.5S2.
Graph: buildings-14-00418-g016b.tif
Graph: Figure 17 The R 2 and RMSE of prediction according to different theoretical models for stress–strain curves [ 41 , 44 , 46 , 53 , 54 ]: (a) R 2 ; (b) RMSE.
Graph: buildings-14-00418-g017b.tif
Graph: Figure 18 The R2 and RMSE of prediction according to different theoretical models for the ascending section of stress–strain curves [[
Graph: Figure 19 The R2 and RMSE of prediction according to different theoretical models for the descending section of stress–strain curves [[
Graph: Figure 20 Distribution of CNTs (0.5%) in RVE.
Graph: Figure 21 Distribution of steel fibers (2%) in the specimen.
Graph: Figure 22 Meshing of RVE: (a) cement matrix; (b) CNTs (0.5%).
Graph: Figure 23 Meshing of the specimen: (a) RVE matrix; (b) steel fibers (2%).
Graph: Figure 24 Stress–strain curves from experiment and FE modelling: (a) C0.08S2; (b) C0.5S2.
Graph: Figure 25 Crack development of the C0.08S2 specimen at different load steps in the FE model: (a) the 31st step; (b) the 35th step; (c) the 41st step; (d) the 56th step.
Table 1 Relevant research on mechanical properties of CNT-reinforced concrete.
Literature Amount of CNT (% of Cement) Dispersion Technique Properties Increase (%) Jiang et al. [ 0.1, 0.5, 1.0 US 1 CS 2 16.8, 10.0, −3.5 Kumar et al. [ 0.5, 0.75, 1.0 US CS 13.7, 1.5, −25.7 STS 3 16.7, 4.4, −21.8 Chaipanich et al. [ 0.5, 1.0 US CS 6.4, 4.1 FS 4 2.9, 2.3 Gillani et al. [ 0.05, 0.1 ST 5 and US CS 19.1, 24.7 STS 26.0, 18.0 Xu et al. [ 0.025, 0.05, 0.1 ST and US CS 6.3, 12.7, 14.6 0.025, 0.05, 0.1, 0.2 STS 7.5, 15.0, 30.0, 40.0 Wang et al. [ 0.05, 0.08, 0.10, 0.12, 0.15 ST and US FTI 6 31.0, 57.5, 47.1, 31.6, 10.3 Nochaiya et al. [ 1.0 US TP 7 16.0 Li et al. [ 0.5 HMF 8 CS 18.9 FS 25.1 TP 39.2 Musso et al. [ 0.5 AS 9 and US CS 10.6 Al-Rub et al. [ 0.2 SP 10 and US FS 269.0 D 11 81.0 Konsta-Gdoutos et al. [ 0.08 ST and US YM 12 45.0 FS 25.0 Luo et al. [ 0.2 MS 13, ST and US CS 29.5 FS 35.4 Collins et al. [ 0.5 MS, PCA 14 and US CS 25.0 Cwirzen et al. [ 0.045 CF 15, PAP 16 and US CS 50.0
Table 2 Physical properties of CNTs.
Average Diameter of Outer Layer (nm) Average Diameter of Inner Layer (nm) Average Length (μm) Surface Area (m2/g) Loose Bulk Density (g/cm3) Tapped Bulk Density (g/cm3) >50 5~15 15 250~300 0.18 2.1
Table 3 Physical and chemical properties of GO.
Particle Size (μm) Solvent Concentration (mg/mL) pH Proportion of Single-Layer GO Sheets (%) Diameter of Single-Layer GO (μm) Thickness (nm) <10 μm water 4 2.2~2.5 >95 0.5~5 0.8~12
Table 4 Physical and mechanical properties of steel fibers.
Length Diameter Length/ Tensile Strength (MPa) Number of Fibers (/kg) 13 0.22 60 2850 224,862
Table 5 Mix proportions.
Group Cement Silica Fume Fine Sand Quartz Powder Water Superplasticizer Accelerator CNT GO Steel Fiber * C0S0 100 32.5 145 30 24 4.3 4.2 0 0 0 C0S2 100 32.5 145 30 24 4.3 4.2 0 0 2 C0.08S0 100 32.5 145 30 24 4.3 4.2 0.08 0.04 0 C0.08S2 100 32.5 145 30 24 4.3 4.2 0.08 0.04 2 C0.5S0 100 32.5 145 30 24 4.3 4.2 0.5 0.04 0 C0.5S2 100 32.5 145 30 24 4.3 4.2 0.5 0.04 2
Table 6 Theoretical equations for peak strain.
Literature Theoretical Model CEB/FIP [ Tadros [ Tomaszewicz [ Wee [ Lee [
Table 7 Theoretical equations for initial elastic modulus.
Literature Theoretical Model GB50010–2010 [ ACI 318-11 [ Eurocode 2-04 [ JCI-08 [ CSA A23.3-04 [ Kollmorgen [
Table 8 Theoretical equations for relative absorbed energy.
Literature Theoretical Model Tasdemir [ Nematzadeh [
Table 9 Theoretical models for stress–strain curves.
Literature Theoretical Model Smith and Young [ Desayi and Krishnan [ CEB/FIP [ Wee [ GB50010-2010 [
Table 10 Coefficients
Age (Days) Group 7 C0S0 1.162 0.256 C0S2 1.402 0.384 C0.08S0 1.239 0.235 C0.08S2 1.415 0.335 C0.5S0 1.076 0.281 C0.5S2 1.320 0.371 28 C0S0 1.225 0.238 C0S2 1.218 0.412 C0.08S0 1.130 0.265 C0.08S2 1.458 0.297 C0.5S0 1.153 0.258 C0.5S2 1.410 0.446
Table 11 Coefficients
Coefficient value 1.18 −7.17 10.32 0.48 13.28 11.87
Table 12 R
Model 7 Days Old 28 Days Old Average C0S0 C0S2 C0.08 S0 C0.08 S2 C0.5S0 C0.5S2 C0S0 C0S2 C0.08 S0 C0.08 S2 C0.5S0 C0.5S2 Smith and Young [ 0.748 0.052 0.764 0.440 0.701 0.352 0.767 0.079 0.717 0.484 0.744 0.101 0.496 Desayi and Krishnan [ 0.956 0.550 0.962 0.822 0.933 0.785 0.961 0.598 0.936 0.808 0.951 0.568 0.819 CEB/FIP [ 0.999 0.675 0.999 0.517 0.995 0.487 0.999 0.567 0.988 0.285 0.995 0.785 0.774 Wee [ 0.997 0.903 0.996 0.862 0.985 0.897 0.991 0.841 0.970 0.799 0.982 0.799 0.919 GB50010-2010 [ 0.997 0.991 0.996 0.892 0.985 0.949 0.991 0.939 0.970 0.947 0.982 0.966 0.967 Proposed model 0.999 0.912 0.999 0.995 0.995 0.991 0.999 0.945 0.995 0.987 0.999 0.935 0.979
Table 13 RMSE of prediction according to different theoretical models for stress–strain curves.
Model 7 Days Old 28 Days Old Average C0S0 C0S2 C0.08 S0 C0.08 S2 C0.5S0 C0.5S2 C0S0 C0S2 C0.08 S0 C0.08 S2 C0.5S0 C0.5S2 Smith and Young [ 9.54 24.06 9.85 20.81 10.68 22.41 10.90 29.02 12.97 27.06 12.14 33.64 18.59 Desayi and Krishnan [ 4.01 16.58 3.96 11.72 5.05 12.91 4.44 19.17 6.18 16.49 5.34 23.33 10.77 CEB/FIP [ 0.36 7.89 0.41 20.69 1.35 21.42 0.83 19.85 2.64 36.57 1.77 16.38 10.85 Wee [ 1.08 7.71 1.23 10.35 2.42 8.92 2.08 12.07 4.25 16.89 3.21 15.92 7.18 GB50010-2010 [ 1.08 2.30 1.23 9.12 2.42 6.26 2.08 7.49 4.25 8.67 3.21 6.59 4.56 Proposed model 0.65 7.34 0.66 1.98 1.42 2.36 0.66 7.07 1.75 4.22 0.78 9.06 3.16
Table 14 R
Model 7 Days Old 28 Days Old Average C0S0 C0S2 C0.08 S0 C0.08 S2 C0.5S0 C0.5S2 C0S0 C0S2 C0.08 S0 C0.08 S2 C0.5S0 C0.5S2 Smith and Young [ 0.748 0.898 0.764 0.845 0.701 0.819 0.767 0.779 0.717 0.927 0.744 0.874 0.799 Desayi and Krishnan [ 0.956 0.997 0.962 0.985 0.933 0.973 0.961 0.954 0.936 0.999 0.951 0.991 0.967 CEB/FIP [ 0.999 0.997 0.999 0.998 0.995 0.995 0.999 0.976 0.988 0.994 0.995 0.998 0.994 Wee [ 0.997 0.999 0.996 0.987 0.985 0.981 0.991 0.945 0.970 0.999 0.982 0.986 0.985 GB50010-2010 [ 0.997 0.999 0.996 0.987 0.985 0.981 0.991 0.945 0.970 0.999 0.982 0.986 0.985 Proposed model 0.999 0.997 0.999 0.999 0.995 0.995 0.999 0.981 0.995 0.993 0.999 0.998 0.996
Table 15 RMSE of prediction according to different theoretical models for ascending section of stress–strain curves.
Model 7 Days Old 28 Days Old Average C0S0 C0S2 C0.08 S0 C0.08 S2 C0.5S0 C0.5S2 C0S0 C0S2 C0.08 S0 C0.08 S2 C0.5S0 C0.5S2 Smith and Young [ 9.54 9.44 9.85 12.54 10.68 13.62 10.90 15.96 12.97 10.74 12.14 13.24 11.8 Desayi and Krishnan [ 4.01 1.54 3.96 3.86 5.05 5.26 4.44 7.30 6.18 1.23 5.34 3.46 4.3 CEB/FIP [ 0.36 1.72 0.41 1.38 1.35 2.33 0.83 5.29 2.64 3.05 1.77 1.61 1.89 Wee [ 1.08 1.08 1.23 3.66 2.42 4.43 2.08 7.94 4.25 1.36 3.21 4.39 3.09 GB50010-2010 [ 1.08 1.08 1.23 3.66 2.42 4.43 2.08 7.94 4.25 1.36 3.21 4.39 3.09 Proposed model 0.65 1.66 0.66 1.17 1.42 2.26 0.66 4.65 1.75 3.43 0.78 1.79 1.74
Table 16 R
Model 7 Days Old 28 Days Old Average C0S2 C0.08 S2 C0.5S2 C0S2 C0.08 S2 C0.5S2 Smith and Young [ −0.981 0.062 0.097 −0.458 0.271 −0.371 −0.230 Desayi and Krishnan [ −0.005 0.663 0.675 0.313 0.715 0.303 0.444 CEB/FIP [ 0.497 0.296 0.510 −2.486 −17.185 −0.599 −3.161 Wee [ 0.783 0.739 0.850 0.765 0.701 0.682 0.753 GB50010-2010 [ 0.982 0.801 0.933 0.943 0.922 0.954 0.923 Proposed model 0.807 0.991 0.992 0.919 0.985 0.896 0.932
Table 17 RMSE of prediction according to different theoretical models for descending section of stress–strain curves.
Model 7 Days Old 28 Days Old Average C0S2 C0.08 S2 C0.5S2 C0S2 C0.08 S2 C0.5S2 Smith and Young [ 29.99 24.63 25.31 34.89 31.07 40.72 31.1 Desayi and Krishnan [ 21.36 14.77 15.18 23.95 19.42 29.04 20.62 CEB/FIP [ 23.89 33.74 34.40 30.21 51.61 25.82 33.28 Wee [ 9.92 12.99 10.32 14.02 19.89 19.60 14.45 GB50010-2010 [ 2.82 11.35 6.88 6.88 10.17 7.47 7.59 Proposed model 9.37 2.35 2.34 8.21 4.41 11.24 6.32
Conceptualization, P.Z.; Resources, P.Z.; Formal analysis, Z.L.; Writing—original draft preparation, Q.J.; Writing—review and editing, P.Z., Y.W. and Z.J.M.; Validation, Y.W.; Funding acquisition, P.Z.; Supervision, Z.J.M. All authors have read and agreed to the published version of the manuscript.
The raw data supporting the conclusions of this article will be made available by the authors on request.
The authors declare no conflicts of interest.
CNTs and GO used in this study were provided by Nanocyl SA and Graphenea SA, respectively. The writers gratefully acknowledge their kind support.
By Peng Zhu; Qihao Jia; Zhuoxuan Li; Yuching Wu and Zhongguo John Ma
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