# Bridging Law Application to Fracture of Fiber Concrete Containing Oil Shale Ash

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Work

#### 2.1. Mix Design with Composite Basalt Fibers for Three-Point Bending Test Experiments

^{−3}, min value of modulus of elasticity 42 GPa, and min tensile strength of 1000 MPa [42], see Figure 2.

^{3}of concrete. These doses correspond to volume 0.00321 m

^{3}(0.32 vol%), equivalent to 25 kg of steel fibers, respectively, and volume 0.00513 m

^{3}(0.51 vol%), equal to 40 kg of steel fibers. Mixes were prepared with natural sand fractions 0–8 mm and crushed coarse aggregates in fractions 8–22.4 mm. The cement used in this study was blended cement, CEM II/B-S 52.5 N, with an apparent particle density of 3080 kg · m

^{−3}. Pigment BAYFERROX 920 was used to distinguish better basalt fibers from cement matrix and, also due to additional research, focused on recycled concrete aggregate testing that is not part of this study. Water and superplasticizer Dynamon SX-23 from Mapei were used for consistency adjustments. For all cases, the W/C ratio was 0.4. The designs of analyzed cases are summarized in Table 1.

^{3}) does not significantly affect the compressive strength. In contrast, when the BF content is increased to 10.8 kg per 1 m

^{3}, the compressive strength decreases by 4.1 MPa (6.4%). This observation is consistent with the results available in the literature [43,44].

#### 2.2. Fabrication of Basalt Fiber Concrete with OSA for Four-Point Bending Test Experiments

^{3}of concrete and Batch C with 23.7 kg per 1 m

^{3}of concrete. Mixtures with 0%, 10%, 15%, and 30% OSA were used for further analysis from the samples of Batch B. The tested compressive strength of the corresponding sample cubes with 100 mm edges was 11.7 MPa, 13.2 MPa, 11.5 MPa, and 10.5 MPa. The corresponding tensile strengths ranged from 1.13 to 1.27 MPa. Samples with 10%, 20%, and 30% OSA were used in modeling from Batch C. The compressive strength was 7.1 MPa, 7.2 MPa, and 6.6 MPa, respectively. The tensile strength ranged from 0.93 to 1.3 MPa. These results do not coincide with common views about increasing the concrete strength if you add fly ash or OSA in small amounts. It is necessary to point out that we are investigating quite high OSA concentrations.

^{3}of concrete and 10% OSA. However, considering the insufficiency of the statistical amount of the analyzed experiments, it is not yet possible to determine the optimal composition of BFs and OSA in concrete at this research stage.

## 3. Modeling Methodology of Crack Propagation and Related Parameters

#### 3.1. Finite Element Model

_{xx}< 0) on the crack path near the opposite edge (see Figure 4). While the crack grows, the compressive region’s length shrinks, and the compressible stress component’s absolute value increases. It is known [48] that the material may fail under compression at certain bridged crack sizes in polymeric composites. Still, preliminary analysis shows that this does not happen in the concrete of the type analyzed here due to large differences in tensile and compressive strength. In the model, the need for the force balance mentioned above is considered by fixing a single node (point B) with no crack opening at the final stages when the rest of the crack path is already abridged.

_{i}can be either a displacement or an angle. The corresponding matrix K constants are found using the FE model, which solves the following six sub-models in each iteration:

- The constant displacement du
^{s}= 0 and the rotation angle θ^{s}= 0 is defined for the support (lower) roller. The roller of the test machine (upper) is loaded with the displacement du^{t}, but the angle θ^{t}= 0 (see Figure 4). - The constant displacement du
^{s}= 0 is defined for the support roller; it is loaded with the rotation angle dθ^{s}. For the roller of the test machine, the displacement is constant du^{t}= 0, and the angle is zero. - The support roller has the constant displacement du
^{s}= 0 and the rotation angle θ^{s}= 0, while the roller of the test machine is loaded with the rotation angle dθ^{t}, and the displacement is constant du^{t}= 0. - The support roller has the constant displacement du
^{s}= 0, but it is loaded with the rotation angle dθ^{s}, while the roller of the test machine is loaded with the displacement du^{t}, but θ^{t}= 0. - The support roller has a constant displacement du
^{s}= 0, and the rotation angle is zero. The roller of the test machine is loaded with both the displacement du^{t}and the angle of rotation dθ^{t}. - Both rollers are loaded with the angle of rotation, but the displacements of both rollers are constant.

#### 3.2. Analyzed Bridging Law Functions

_{t}and the fracture energy G

_{F}.

_{F}is defined as the area average of the fracture work calculated by Equation (4).

_{lig}is the ligament area formed at the front of the initial crack. The fracture work absorbed by the material throughout the fracture process in the bending test can be calculated by Equation (5).

#### 3.2.1. Bilinear Bridging Law Function

_{1}= 1.2 (G

_{F}/f

_{t}) is the horizontal intercept of the steeper segment, and σ

_{k}, w

_{k}are the coordinates of the kink point, given by Equations (7) and (8), respectively.

_{c}is obtained from the quadratic equation:

#### 3.2.2. Nonlinear Bridging Law Function

_{1}, c

_{2}, and w

_{c}are material constants. For analyzed concrete, the three parameters are considered as c

_{1}= 2.5, c

_{2}= 6.93 and the critical opening is obtained by Equation (12).

#### 3.3. Surrogate Modeling

_{i}.

_{1},x

_{2},…,x

_{3}),

_{1}, x

_{2}, … x

_{n}are coordinates from the factor space. The response function F in the factor space represents the response surface.

^{2}and Pearson’s χ

^{2}test were used to characterize the fit of the surrogate model. In addition, a leave-one-out cross-validation approach was used for surrogate model verification. Leave-one-out cross-validation is a cross-validation method that involves using one observation as validation data, and the remaining data are used to train the model. For example, point i = 1 is selected for validation in the first iteration for a data set with N points, and the remaining (N-1) points are used for model building. For the second iteration, point i = 2 is selected for validation, and the remaining ones are for building the model. The process is repeated N times. A smaller cross-validation value means a higher accuracy of the surrogate model. The cross-validation percentage error is defined in Equation (14) to avoid overestimating the prediction error.

## 4. Numerical Examples

_{max}as a function of the tensile strength f

_{t}and the fracture work W was created using the 4PBT modeling results for a concrete specimen with BFs and OSA.

#### 4.1. Three-Point Bending Tests

^{3}(see Figure 9b, Batch A1) and 10.8 kg per 1 m

^{3}(see Figure 9c, Batch A2) doses of composite BFs in the mixture are analyzed. In the figures, the black dotted line represents the mean values of three experiments for Batch A0 and the mean values of six samples for the other groups. Error bars show the standard deviation for 3PBT. The blue line represents the P-CMOD curve obtained by the FE modeling using the bridging law concept.

^{3}of BFs (Batch A1) is 15.8 kN when CMOD = 0.036 mm. The comparison of the experimental and modeling results of Batch A1 shows that the bilinear bridging law function better describes the trends of the linear part of the P-CMOD curve and cracking phase in the concrete matrix but incorrectly predicts the fiber failure phase. The nonlinear function overestimates the peak load (P

_{max}= 17.76 kN at 0.054 mm). However, it best describes the fiber failure phase.

^{3}to the mixture (Batch A2), tested on six samples, reduces the peak load value to 14.1 kN. CMOD value is 0.033 mm. It can be seen in Figure 9c that the FE model predicts a significantly higher load peak value (P

_{max}= 17.32 kN at CMOD = 0.074 mm) and a slower sample rupture than was obtained in the experiment. The addition of longer composite BFs causes considerable damage to the specimen volume in the bending test. Therefore, the assumption used in the FE modeling about damage formation only around the fiber surface is not true in this case. For this reason, a faster decrease in the P-CMOD curve is observed in the experiment after the maximum loading than predicted by the FE model. Modeling such cases (with longer fiber pull-outs) should also consider micromechanics models.

#### 4.2. Four-Point Bending Tests

^{3}). It follows from Table 4 that the maximum load obtained with the FE model is close to the experimentally measured values. The standard deviation stdev of four experiments is shown in brackets. The midspan deflection δ

_{Pmax}, at which these maximum values are reached, is 0.05 mm in the experiments, regardless of the amount of added OSA, but in the FE model, the deflection is larger, i.e., 0.09–0.1 mm. However, comparing the P-δ curve obtained in the experiments with the results of the FE model, it can be concluded that the model predicts the general trends of crack propagation.

^{3}), the experimental P-δ curve is most accurately represented by the FE model with a nonlinear bridging law function. The obtained results are summarized in Table 5. It can be seen that the FE model can predict the maximum loads and the corresponding midspan deflection values of the samples of this group according to the experimentally measured values under the 4PBT.

^{3}, a more accurate P-δ curve can be obtained with a nonlinear bridging law function. At higher BF doses, the model fits well with the experimental data for the peak loading P

_{max}and the corresponding midspan deflection δ

_{Pmax}value. Although preliminary results show that the nonlinear function gives a more accurate bending test result in mixtures with higher fiber content, more research is needed to confirm this conclusion.

#### 4.3. Surrogate Model to Predict Maximum Load in 4PBT

_{t}and fracture work W) for the 4PBT. In the design of experiments, orthogonality is essential because it leads to the best models and reduces the variance of the regression coefficients. The building set uses BF concrete supplemented with OSA, as in Batch C. The design of the experiments is shown in Figure 10. Based on the experiments, the following factor space was chosen: f

_{t}= [0.98; 1.31] (MPa) and W = [6.5; 12.7] (Nm). The validation set consists of five experimental data points of Batch C.

^{2}for all cases shows that the surrogate model agrees with the FE model. On the other hand, from the calculated cross-validation error, it can be concluded that the first-order polynomial approximation provides the worst fitting to the FE model results, and the third-order polynomial approximation provides the best. The cross-validation error of the surrogate model using a third-order polynomial for approximation is less than 0.5%. In all cases, Pearson’s χ

^{2}criterion is met.

_{max}as a function of f

_{t}and W is shown in Equation (15).

_{max}(t

_{f}, W) = −1.9534401 + 14.166294f

_{t}+ 0.060235196W − 7.5494805f

_{t}

^{2}+ 0.25515142f

_{t}W − 0.012196241W

^{2}+ 1.6287772f

_{t}

^{3}− 0.037482758f

_{t}

^{2}W − 0.0042167212f

_{t}W

^{2}+ 0.00041631893W

^{3}.

_{max}values predicted by the surrogate model are lower than the experimentally measured ones. When calculated, the coefficient of determination is 0.764. To increase the accuracy of the surrogate model, the methodology for finding the tensile strength f

_{t}should be improved.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

- (1)
- Data input/output and memory management;
- (2)
- Two-dimensional mesh generator and 3D mesh automatic generator by mesh extrude method;
- (3)
- Common sparse FEM matrix generator by energy minimum and Galerkin methods;
- (4)
- Solver of linear real/complex equation system, direct, and Krylov space methods;
- (5)
- Plastic flow modulus, dynamic, Prager, and Armstrong–Frederich types of plasticity;
- (6)
- Linear-elastic, thermal-elastic and plastic/triangular, tetrahedral, and beam elements;
- (7)
- The procedural script interpretation modulus gives access to all numerical modules mentioned, used for boundary conditions and extra physical feature coding.

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**Figure 4.**Sketch of FE model with (

**a**) midspan deflection δ and crack mouth opening displacement CMOD and (

**b**) introduced generalized coordinates.

**Figure 6.**An example of a computational domain mesh. On the x and y axis is the length in millimeters.

**Figure 8.**An example of the stress field distribution in the computational domain at different failure phases: (

**a**) linear phase; (

**b**) matrix failure phase. Four-point bending test. On the axes are the sample dimensions in mm. The color scale corresponds to stress σ in MPa.

**Figure 9.**Comparison of FE model and experiments for the 3PBT: (

**a**) Batch A0 (without BFs); (

**b**) Batch A1; (

**c**) Batch A2.

**Figure 11.**The response surface using the third-order polynomial approximation and experimental points.

Mix | BF (kg) | CEMII/B-S 52.5 N | Pigment | Aggregates 0–8 mm | Aggregates 8–22.4 mm | Dynamon SX-23 (kg) | Dynamon SX-23 (% of Binder) |
---|---|---|---|---|---|---|---|

A0 | 0 | 350 | 7 | 1279 | 656 | 3.68 | 1.03 |

A1 | 6.7 | 350 | 7 | 1271 | 652 | 4.36 | 1.22 |

A2 | 10.8 | 350 | 7 | 1270 | 652 | 4.36 | 1.22 |

Mix | Fresh Density (kg · m^{−3}) | Air Content (%) | Slump (mm) | Hardened Density (kg · m^{−3}) | Compressive Strength (MPa) | E-Modulus (GPa) |
---|---|---|---|---|---|---|

A0 | 2466 | 2.0 | 210 | 2460 | 64.4 | 28.7 |

A1 | 2429 | 2.4 | 200 | 2440 | 64.1 | 29.5 |

A2 | 2460 | 2.0 | 210 | 2420 | 60.3 | 29.8 |

Case | The Number of Elements in the FE Model | The Number of Nodes | P_{max} (kN) |
---|---|---|---|

M1 | 8600 | 3891 | 9.79 |

M2 | 2456 | 1025 | 9.83 |

M3 | 749 | 274 | 9.98 |

**Table 4.**Comparison of experimental and FE model (with bilinear bridging law) results for Batch B samples.

OSA (%) | FE Modeling | Experiments | ||
---|---|---|---|---|

P_{max} (kN) | δ_{Pmax} (mm) | P_{max} (kN) (stdev) | δ_{Pmax} (mm) (stdev) | |

10 | 10.72 | 0.094 | 9.93 (0.16) | 0.05 (0.004) |

15 | 10.23 | 0.104 | 10.45 (0.33) | 0.05 (0.006) |

30 | 9.75 | 0.10 | 9.97 (0.42) | 0.05 (0.007) |

**Table 5.**Comparison of experimental and FE model (with nonlinear bridging law) results for Batch C samples.

OSA (%) | FE Modeling | Experiments | ||
---|---|---|---|---|

P_{max} (kN) | δ_{Pmax} (mm) | P_{max} (kN) (stdev) | δ_{Pmax} (mm) (stdev) | |

10 | 7.82 | 0.104 | 7.44 (0.64) | 0.14 (0.04) |

20 | 7.73 | 0.112 | 7.90 (0.67) | 0.16 (0.05) |

30 | 9.91 | 0.104 | 9.64 (0.59) | 0.15 (0.05) |

Polynomial | Cross-Validation Error | R^{2} |
---|---|---|

1st order | 10.65% | 0.993 |

2nd order | 1.17% | 0.999 |

3rd order | 0.47% | 0.999 |

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Upnere, S.; Novakova, I.; Jekabsons, N.; Krasnikovs, A.; Macanovskis, A.
Bridging Law Application to Fracture of Fiber Concrete Containing Oil Shale Ash. *Buildings* **2023**, *13*, 1868.
https://doi.org/10.3390/buildings13071868

**AMA Style**

Upnere S, Novakova I, Jekabsons N, Krasnikovs A, Macanovskis A.
Bridging Law Application to Fracture of Fiber Concrete Containing Oil Shale Ash. *Buildings*. 2023; 13(7):1868.
https://doi.org/10.3390/buildings13071868

**Chicago/Turabian Style**

Upnere, Sabine, Iveta Novakova, Normunds Jekabsons, Andrejs Krasnikovs, and Arturs Macanovskis.
2023. "Bridging Law Application to Fracture of Fiber Concrete Containing Oil Shale Ash" *Buildings* 13, no. 7: 1868.
https://doi.org/10.3390/buildings13071868