# Development of Damage Type Viscoelastic Ontological Model for Soft and Hard Materials under High-Strain-Rate Conditions

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{−4}to 10

^{3}s

^{−1}. Initially, Tang first proposed a nonlinear viscoelastic instantonal equation that can describe the kinetic behavior of epoxy resin, and the instantonal model can describe eight magnitudes of strain rates, which is called the ZWT instantonal model. Many scholars have improved the ZWT intrinsic model and established the instrinsic model for certain materials. The ZWT intrinsic model has been widely used in the fields of polymer materials, concrete materials, and soils. Luo et al. [17] studied the mechanical properties of TDE86 epoxy resin at different temperatures, established the ZWT intrinsic structure model of epoxy resin containing nine parameters, and verified the applicability of the ZWT intrinsic structure model at different temperatures by developing a subroutine. Liu et al. [18], according to the principle of building the constitutive model of polymer material, added the one-dimensional structure of the ZWT material constitutive model to the dashpot element in parallel, in which strain rate and coefficient of viscosity were introduced and the nonlinear viscoelastic constitutive model of Polycarbonate material was achieved; this intrinsic structure model contains 14 parameters. Xu et al. [19] developed a ZWT nonlinear viscoelastic model, which contains 10 parameters. The standard ZWT nonlinear viscoelastic model was selected to predict the elastic behavior of LNBR/epoxy composites under a wide range of strain rates in this study.

## 2. Establishment of Constitutive Model

^{0}and 10

^{2}s. Therefore, the loading time in this experiment is not one order of magnitude as that of the low-frequency Maxwell body, and the former is much smaller than the latter. Therefore, when the dynamic load is completed, the low-frequency Maxwell body is not able to relax. Therefore, the second medium-low-frequency Maxwell body in the ZWT constitutive model can be replaced by a simple spring. The simplified physical model is shown in Figure 2, and the formula of the ZWT constitutive model can be further simplified as:

_{0}and E

_{1}are parallel, and a simple spring ${E}_{a}$ can replace the two parallel springs. The equivalent physical model is shown in Figure 3. The ZWT constitutive model is further simplified to:

_{a}represents effective stress; σ

_{r}represents the original stress; and D represents the damage variable.

## 3. Experimental Research

#### 3.1. Preparation of Test Pieces

#### 3.1.1. Preparation of Coal Rock

#### 3.1.2. Preparation of Foam Concrete

#### 3.2. Experimental Method

#### 3.3. Verification of Stress Balance

#### 3.4. Data Analysis

_{0}, and A are the elastic modulus, elastic wave velocity, and sectional area of the rod, respectively. ${\epsilon}_{i}$, ${\epsilon}_{r}$, and ${\epsilon}_{t}$ are incident wave strain, reflected wave strain, and transmitted wave strain, respectively. A

_{0}and L

_{0}are the cross-sectional area and length of the specimen, and t is the stress wave continuous specimen.

#### 3.5. Experimental Results

## 4. Validation of the Viscoelastic Constitutive Model with Damage

^{2}was greater than 0.9, indicating that the constitutive equation established in this paper has wide applicability. 1stOpt is a mathematical optimization analysis synthesis tool developed by 7D-Soft High Technology Inc. in China, which is widely used in the fields of nonlinear regression, curve fitting, and nonlinear complex engineering mode parameter evaluation solutions, etc. The data in the references in this paper were extracted by GetData.

_{a}and E

_{2}. The parameter alpha and the parameter beta jointly determine the trend of the whole curve. The parameter alpha has a greater influence on the peak strain and peak stress of the curve. The parameter beta has a greater influence on the direction of the opening of the curve. The parameter E

_{a}is determined by the rising section of the curve, and E

_{2}is determined by the falling section of the curve. The intrinsic constitutive model established in this paper also has certain defects, for example, the model was only able to describe the large trend of stress–strain, and it could not accurately describe the curve wavy fluctuation. The description of material strength and weakness by the intrinsic constitutive model is described by E

_{a}and E

_{2}. Due to the limited experimental data, the model could only prove to be valid for a certain range of material strengths at present.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 17.**Curve fitting. (

**a**) Steel fiber reinforced concrete; (

**b**) Granite; (

**c**) Lightweight foramed concrete; (

**d**) EPS concrete.

Cement | PO42.5 ordinary Portland cement (PC) with a density of 3150 kg/m^{3}, an initial setting time of 50 min, and a final setting time of 3 h and 30 min |

Fly ash | Fly ash with an average grain diameter of 1–15 μm, class I parameters, and a density of 2400 kg/m^{3} |

Water | Tap water |

Polypropylene fibers | Density of 0.91 g/m^{3}, Elastic Modulus of 5000 MPa |

Admixture | Htq-1 compound foaming agent |

Foam Agent | Calcium stearate (C_{36}H_{70}CaO_{4}) |

Design Plastic Density (kg/m^{3}) | Fly Ash (kg) | Water (kg) | Cement (kg) | Cement Type | Foam Agent (kg) |
---|---|---|---|---|---|

658 | 164 | 329 | 658 | PC | 5.57 |

$\stackrel{\xb7}{\mathit{\epsilon}}$ | A | α | β | E_{a} | E_{2} | θ_{2} | δ | R^{2} | |
---|---|---|---|---|---|---|---|---|---|

Coal rock | 120 | 3 | −0.6 | −1 | −1.77042 × 10^{10} | 1.77042 × 10^{10} | 3 | −0.0002 | 0.9648 |

145 | 3 | −0.59 | −1 | −2.76368 × 10^{10} | 2.76368 × 10^{10} | 3 | −0.0002 | 0.9401 | |

186 | 3 | −0.545 | −1 | −2.1608 × 10^{10} | 2.1608 × 10^{10} | 3 | −0.0002 | 0.9584 | |

212 | 3 | −0.51 | −1 | −1.5297 × 10^{10} | 1.5297 × 10^{10} | 3 | −0.0002 | 0.9739 | |

Foamed concrete | 156.47 | 3 | 0.80 | 0.30 | 3005.86 | −3005.93 | 3 | 0.0002 | 0.9953 |

235.29 | 3 | 0.68 | 0.20 | 12,479.65 | −12,479.82 | 3 | 0.0002 | 0.9945 | |

254.12 | 3 | 0.64 | 0.30 | 14,062.01 | −14,062.27 | 3 | −0.0002 | 0.9946 | |

298.82 | 3 | 0.50 | 0.25 | 88,387.15 | −88,388.46 | 3 | 0.0002 | 0.9946 | |

342.94 | 3 | 0.70 | 0.35 | 21,752.35 | −21,752.57 | 3 | −0.0002 | 0.9881 | |

Steel fiber-reinforced concrete | 38.24 | 3 | −0.485 | −1 | −2.58732 × 10^{8} | 2.58731 × 10^{8} | 3 | −0.0002 | 0.9625 |

120.13 | 3 | −0.14 | −0.2 | 3.84645 × 10^{8} | −3.84677 × 10^{8} | 3 | −0.0002 | 0.9712 | |

Granite | 70.99 | 3 | −0.27 | −0.47 | −2.77818 × 10^{9} | 2.77818 × 10^{9} | 3 | −0.0002 | 0.9583 |

246.93 | 3 | 0.02 | 0.33 | 2.18859 × 10^{10} | −2.18859 × 10^{10} | 3 | −0.0002 | 0.9943 | |

Lightweight foamed concrete | 64 | 3 | −0.0026 | −0.7 | 155,087.9264 | −155,093.636 | 3 | −0.0002 | 0.9078 |

266 | 3 | −0.34 | −0.7 | 5.30233 × 10^{7} | −5.30245 × 10^{7} | 3 | −0.0002 | 0.9105 | |

EPS concrete | 51.12 | 3 | −0.077 | 0.06 | −2.97949 × 10^{9} | 2.9796 × 10^{9} | 3 | −0.0002 | 0.9776 |

74.29 | 3 | 0.1 | 0.5 | −1.00786 × 10^{9} | 1.00788 × 10^{9} | 3 | −0.0002 | 0.9838 |

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**MDPI and ACS Style**

Liu, W.; Xu, X.; Mu, C. Development of Damage Type Viscoelastic Ontological Model for Soft and Hard Materials under High-Strain-Rate Conditions. *Appl. Sci.* **2022**, *12*, 8407.
https://doi.org/10.3390/app12178407

**AMA Style**

Liu W, Xu X, Mu C. Development of Damage Type Viscoelastic Ontological Model for Soft and Hard Materials under High-Strain-Rate Conditions. *Applied Sciences*. 2022; 12(17):8407.
https://doi.org/10.3390/app12178407

**Chicago/Turabian Style**

Liu, Wei, Xiangyun Xu, and Chaomin Mu. 2022. "Development of Damage Type Viscoelastic Ontological Model for Soft and Hard Materials under High-Strain-Rate Conditions" *Applied Sciences* 12, no. 17: 8407.
https://doi.org/10.3390/app12178407