# Study on Temperature-Dependent Uniaxial Tensile Tests and Constitutive Relationship of Modified Polyurethane Concrete

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Material and Proportioning

_{3}, contains both inorganic and organic functional groups and is compatible with inorganic and organic materials [20,21]. After mixing MPU with an aggregate, SCA can form a strong covalent bond between PU and inorganic aggregate, which enhanced the interfacial strength between PU and the aggregate [22,23]. The reaction mechanism of SCA is shown in Figure 3.

#### 2.2. Tensile Specimen and Fixture Design

#### 2.3. Temperature Dependent Tensile Test

#### 2.3.1. Specimen Production

#### 2.3.2. Test Procedure

^{3}, with an average density of about 2200 kg/m

^{3}. Secondly, the tensile strains of the specimens were measured by both a non-contact video extensometer (DIC) and resistive strain gauges. Preparations for DIC technology: 1. Spray white paint on the specimen surface in advance; 2. Randomly spray black spots of certain deformation capacities; 3. Set up a high-frequency camera. The black dots on the specimen surface were identified and the relative displacements between the black dots were calculated by digital image processing techniques during the tensile process. After further calculation, the displacement field, deformation gradient, and strain field of the specimen were obtained in a specific time interval. The non-contact measurement system, VIC-3D manufactured by Correlated Solutions, Inc., was used to identify the black dots and calculate them. Three foil strain gauges were pasted on the back of the specimen. The size of the strain gauge sensitive gate was 50 mm × 4 mm, the resistance was 120 Ω, and the sensitivity was 2.0 ± 1%. The strain data, measured by the strain gauge, were used to check the data of the DIC. The arrangement position of the strain gauge and DIC scatter point is shown in Figure 8.

#### 2.3.3. Test Phenomenon

#### 2.3.4. Test Data Results

## 3. Analysis of Test Results

#### 3.1. Stress-Strain Curve

#### 3.2. Tensile Peak Stress

_{0}are the test temperature and 15 °C, respectively, and T takes a value range of −10 °C ≤ T ≤ 60 °C. The room temperature during the test was 15 °C. f

_{t,T}and ${f}_{{\mathrm{t},\mathrm{T}}_{0}}$ are the tensile peak stresses of MPUC at temperatures T and T

_{0}, respectively. The correlation coefficient is R

^{2}= 0.999 and the residual sum of squares is v

^{2}= 2.145 × 10

^{−5}, between the fitted curve and the test data. The peak stress fitting agrees well with the test results.

#### 3.3. Tensile Peak Strain

_{t,T}and ${\epsilon}_{{\mathrm{t},\mathrm{T}}_{0}}$ are the tensile peak strains of MPUC at temperatures T and T

_{0}, respectively. The correlation coefficient is R

^{2}= 0.998 and the residual sum of squares is v

^{2}= 2.370 × 10

^{−4}between the fitted curve and the test data. The peak strain fitting agrees well with the test results.

#### 3.4. Elastic Modulus

_{t,T}, E

_{t,T0}are the elastic modulus of MPUC at temperatures T and T

_{0}, respectively. The correlation coefficient is R

^{2}= 0.992, and the sum of residual squares is v

^{2}= 0.010. The elastic modulus fitting is in good agreement with the test results.

## 4. Constitutive Relationship

#### 4.1. Uniaxial Tensile Constitutive Model

**M**independent fiber elements. The elements are randomly damaged, and the stress-strain curve of the damaged elements is linear. When

**m**elements are damaged, the external load will be uniformly distributed over the remaining elements (

**M**−

**m**). The damage variable (

**D**) is usually defined as the ratio of the number of damaged elements (

**m**) to the total number of elements (

**M**). The stress-strain relationship under arbitrary loading conditions is shown in Equation (4).

_{t}and ε

_{t}are the tensile peak stress and peak strain of concrete, respectively; E is the modulus of elasticity of concrete.

_{t,T}is the damage variable of the temperature correction. After substituting Equation (7) into Equation (6), Equation (8) is obtained. In Equation (8), f

_{t,T}, and ε

_{t,T}are calculated by Equations (1) and (2), although α and β are unknown. When ε = 0, dσ/dε = E

_{t,T}. Therefore, α could be obtained by using Equation (9).

_{t,T}, and E

_{tc,T}are the initial elastic modulus and peak secant modulus of MPUC at different temperatures, respectively, which are calculated by Equations (1)–(3). By bringing Equation (9) into Equation (8), the coefficient β could be obtained by Equation (8). The fitting results are shown in Table 5.

_{1}are the test temperature and 1 °C, respectively.

#### 4.2. Comparison of Models

## 5. Conclusions

- A novel tensile test fixture (SJ-4) is developed. The fixture with the dumbbell-shaped specimens (arc transition) can effectively avoid stress concentration and ensure the specimen breaks within measurement length, which is suitable for the stretching of high-strength brittle materials.
- The bonding effect between MPU and aggregate is less affected by temperature. The tensile strength and elastic modulus of MPUC decrease with increasing temperature, while the fracture strain and fracture energy are the opposite. The variation in tensile strength, fracture strain, and elastic modulus of MPUC with temperature is well reflected by the proposed temperature-dependent equations, and the calculation results show good agreement with the experimental ones.
- The shapes of the tensile stress-strain curve of MPUC at low temperatures (−10 °C and 0 °C) and ambient temperature (15 °C) are similar to an elastomeric brittle material. As the temperature increases, the plasticity of MPUC increases. The relative error in measuring deformation with the DIC technique and strain gauges is related to the material properties. The stronger the plasticity of the test material is, the smaller the relative error is.
- The temperature-dependent uniaxial tension constitutive relation of the MPUC ascending segment is established. The prediction of MPUC is improved by introducing temperature-related parameters a and b, which are significantly better than the constitutive model for the Chinese code. The results provide a reference for the engineering application and numerical analysis of MPUC.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Symbol | Description | Unit |
---|---|---|

w | Thickness of SJ-2 and SJ-3 | mm |

D | Diameter of SJ-1 | mm |

T | Test temperature | °C |

T_{0} | 15 °C | °C |

T_{1} | 1 °C | °C |

f_{t,T} | Tensile peak stress of MPUC at temperature T | MPa |

f_{t,T0} | Tensile peak stress of MPUC at temperature T_{0} | MPa |

ε_{t,T} | Tensile peak strain of MPUC at temperature T | / |

ε_{t,T0} | Tensile peak strain of MPUC at temperature T_{0} | / |

E_{t,T} | Elastic modulus of MPUC at temperature T | GPa |

E_{t,T0} | Elastic modulus of MPUC at temperature T_{0} | GPa |

E_{tc,T} | Peak secant modulus of MPUC at temperature T | GPa |

E | Elastic modulus of concrete | GPa |

D | Damage variable of concrete | / |

D_{t,T} | Temperature-dependent damage variables of MPUC | / |

σ | Tensile stress of concrete | MPa |

ε | Tensile strain of concrete | / |

α | Temperature parameter | / |

β | Temperature parameter | / |

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**Figure 1.**Aggregate and specimen slices, (

**a**) coarse aggregate, (

**b**) fine aggregate, (

**c**) specimen section.

**Figure 7.**Schematic diagram of strain concentration of specimens, (

**a**) transition zone fracture, (

**b**) loading end fracture.

**Figure 12.**Comparison of specimen fracture and DIC digital imaging. (

**a**) 40 °C—0.1 s before crack, (

**b**) 40 °C—crack, (

**c**) 60 °C—0.1 s before crack, (

**d**) 60 °C—crack.

**Figure 18.**Comparison of algorithms for predicting stress-strain curves after different temperatures.

**Figure 19.**Comparison of tensile constitutive models. (

**a**) −10 °C, (

**b**) 0 °C, (

**c**) 15 °C, (

**d**) 40 °C, (

**e**) 60 °C.

Sieve Pore Diameter (mm) | 9.5 | 4.75 | 2.36 | 0.6 | 0.3 | 0.15 | 0.075 | |
---|---|---|---|---|---|---|---|---|

Passing rate (%) | Gradation upper limit | 100 | 88 | 75 | 50 | 22 | 10 | 6 |

Gradation lower limit | 95 | 60 | 41 | 15 | 6 | 3 | 0 | |

Designed gradation | 100 | 65 | 49 | 28 | 12 | 6 | 3 |

Component | Coarse Aggregate | Fine Aggregate | Modified Polyurethane Binder | Curing Agent |
---|---|---|---|---|

Mass fraction (%) | 30.2 | 54.4 | 15.2 | 0.2 |

Fineness modulus | 3.4 | 2.5 | / | / |

Apparent density (kg/m^{3}) | 2600 | 2580 | 1005 | / |

Number | Test Method | Specimen Diagram (mm) | Thickness w/Diameter D (mm) | Specimen Failure | Damage Feature |
---|---|---|---|---|---|

SJ-1 | S1 + Type I | D = 150 | Undamaged specimen, adhesive layer debonding | ||

SJ-2 | S2 + Type II | w = 75 | Cracks mostly occur at the loading end, and the stress concentration is obvious. | ||

SJ-3 | S3 + Type II | w = 75 | Main crack is in the gauge section, yet an obvious crack appears at the load end. | ||

SJ-4 | S3 + Type III | w = 75 | Ideal tensile fracture effect |

Temperature (°C) | Force (kN) | Peak Stress (MPa) | Peak Strain-Strain Gauge (10^{−3}) | Peak Strain-DIC (10^{−3}) | Relative Error (%) | Elastic Modulus (GPa) | Fracture Energy Density (N·mm^{−2}) |
---|---|---|---|---|---|---|---|

−10 °C | 46.26 | 10.28 | 0.681 | 0.618 | 9.25 | 16.86 | 3.304 |

0 °C | 48.60 | 10.80 | 0.690 | 0.623 | 9.71 | 16.16 | 3.929 |

15 °C | 44.06 | 9.79 | 0.958 | 0.899 | 6.16 | 11.93 | 4.247 |

40 °C | 26.42 | 5.87 | 1.432 | 1.524 | 6.42 | 6.23 | 5.561 |

60 °C | 17.78 | 3.95 | 4.967 | 4.873 | 1.89 | 1.21 | 12.843 |

Temperature (°C) | −10 | 0 | 15 | 40 | 60 |
---|---|---|---|---|---|

β | 0.93 | 1.05 | 0.94 | 1.58 | 2.39 |

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**MDPI and ACS Style**

Han, Y.; Meng, X.; Feng, F.; Song, X.; Huang, F.; Wen, W.
Study on Temperature-Dependent Uniaxial Tensile Tests and Constitutive Relationship of Modified Polyurethane Concrete. *Materials* **2023**, *16*, 2653.
https://doi.org/10.3390/ma16072653

**AMA Style**

Han Y, Meng X, Feng F, Song X, Huang F, Wen W.
Study on Temperature-Dependent Uniaxial Tensile Tests and Constitutive Relationship of Modified Polyurethane Concrete. *Materials*. 2023; 16(7):2653.
https://doi.org/10.3390/ma16072653

**Chicago/Turabian Style**

Han, Yanqun, Xiandong Meng, Fan Feng, Xuming Song, Fanglin Huang, and Weibin Wen.
2023. "Study on Temperature-Dependent Uniaxial Tensile Tests and Constitutive Relationship of Modified Polyurethane Concrete" *Materials* 16, no. 7: 2653.
https://doi.org/10.3390/ma16072653