# Experimental Study of the Rheology of Grease by the Example of CIATIM-221 and Identification of Its Behavior Model

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Research Objectives

- Conducting a series of full-scale experiments to determine the viscoelastic lubricants properties over a wide range of temperatures;
- Identifying a mathematical model of the viscoelastic lubricant behavior in the form of the Maxwell body based on two viscoelasticity models: the Prony series and the Anand’s model;
- Creating a unified numerical procedure to determine approaches to the parameters from item two based on the application of the multi-parameter Nelder–Mead optimization.

#### 1.2. Problem Context and Description

- Study of tribological, electromechanical, and thermal characteristics of lubricants, including those with various additives and solutions;
- Experimental research on a wide range of temperatures to identify dynamic and static characteristics;
- Identification of mathematical models of material behavior and their implementation in numerical analogues of friction nodes.

## 2. Materials and Methods

#### 2.1. Experimental Research

- -
- minimum oscillation torque 2;
- -
- minimum sustained shear torque 10;
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- maximum torque 200;
- -
- torque resolution 0.1;
- -
- minimum frequency;
- -
- maximum frequency 100;
- -
- minimum angular frequency 0;
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- maximum angular frequency 300;
- -
- displacement resolution 10, etc.

#### 2.2. Identification of a Mathematical Model of Lubricant Behavior

#### 2.2.1. Prony Series

#### 2.2.2. Anand’s Model

#### 2.3. Mathematical Model Identification Procedure

- -
- Preliminary step. It consists in the form of experimental data into the procedure, the choice of a mathematical model, and the setting of initial values of the unknowns vector $\overline{x}$;
- -
- Nelder–Mead multi-parameter optimization operations. An ANSYS file is generated with a sequence of commands to build the numerical model. The pure shift problem is solved with the generation of a results file. The function $F$ is calculated and comparing to the required error. If the condition is not met, a new vector of unknowns is generated. Then, the optimization procedure is repeated;
- -
- Obtaining a result file. If the error condition is met, the final value of the unknowns vector $\overline{x}$ is written down and the procedure is exited.

## 3. Results

#### 3.1. Results of Full-Scale Experiments

#### 3.2. Identification of a Mathematical Model of the Lubricant as a Function of Temperature

#### 3.3. Dependence of Rheological Properties of the Lubricant on the Shear Rate

## 4. Discussion

#### 4.1. Limitation Statement

- -
- The full-scale experiment is conducted in a small range of shear rates, from 0.01 to 100 Hz, which does not give a complete picture of the viscous and elastic components at small and large share rates, respectively;
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- The behavior of the lubricant was investigated using the Discovery HR2 rotational viscometer with a limited range of temperatures and share rates;
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- The lubricant is capable of operating in the temperature range of −60 to +150 °C, but the equipment allows evaluation of behavior at temperatures of −40 to +80 °C;
- -
- Lubricant is treated as a Maxwell body, in fact, the object of study has a more complex pattern of behavior;
- -
- Lubricant is considered within the problem of deformable solid mechanics; the problem of fluid and gas mechanics is not set.

- -
- Consideration of other linear viscoelastic models (Kelvin model, Voigt model, etc.);
- -
- Using a temperature–time superposition to be able to describe and analyze decreased and increased shear rates and temperatures;
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- Numerical simulation of the structure as a whole with the use of a lubricant, using the example of a spherical sliding bearing of a bridge span;
- -
- The lubricant will be examining using the DWS technology.

#### 4.2. Prediction of Structure Behavior Based on Computer Engineering

#### 4.3. On the Choice of a Mathematical Model

#### 4.4. Scope of Application Results

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- Numerical experiments to be conducted on the operation of structural elements during the life cycle;
- -
- Extension of the presented study to determine the rheological properties of polymeric materials [74];
- -
- -
- The reduction of material and time costs for field research, etc.

## 5. Conclusions

- -
- At higher frequencies, the error is minimal;
- -
- At low frequencies, there is a significant error.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\mathsf{\phi}$ | angular displacement of the rheometer; |

${G}^{*}$ | complex shear modulus; |

$\mathsf{\gamma}$ | shear strain; |

${\mathsf{\tau}}_{e}$ | tangential stress of an elastic element; |

${\mathsf{\gamma}}_{e}$ | shear strain of an elastic element; |

${\mathsf{\tau}}_{v}$ | tangential stress of a viscous element; |

$\mathsf{\eta}$ | viscosity; |

${\dot{\mathsf{\gamma}}}_{v}$ | the rate of viscous shear strain; |

$f$ | functional; |

${\mathsf{\tau}}_{\mathrm{exp}}$ | experimental tangential stress; |

${\mathsf{\tau}}_{num}$ | numerical tangential stress; |

$\overline{x}$ | vector of unknowns; |

${G}_{\infty}$ | long shear modulus; |

${G}_{0}$ | initial shear modulus; |

${\mathsf{\alpha}}_{i}$ | weight coefficients; |

${\mathsf{\beta}}_{i}^{\prime}$ | reduced time; |

$k$ | number of relaxation times; |

${A}_{WLF}\left(T\right)$ | temperature-time analogy shift function; |

$T$ | absolute temperature; |

${C}_{1},{C}_{2}$ | empirical material constants; |

${T}_{r}$ | base temperature; |

$A$ | pre-exponential multiplier; |

$U$ | activation energy; |

$R$ | universal gas constant; |

$\mathsf{\xi}$ | stress multiplier; |

$S$ | shear strain resistance; |

${h}_{0}$ | material hardening constant; |

${S}^{*}$ | saturation value of the hardening function; |

$n$ | sample saturation as a function of shear rate. |

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**Figure 1.**Experimental study on the Discovery HR2 rheometer. (

**a**) General view of the setup; (

**b**) application of liquid nitrogen.

**Figure 4.**Linear viscoelasticity limit analysis: (

**a**) 50 °C, (

**b**) 80 °C; measuring systems: black line—plane–plane, gray line—cone–plane.

**Figure 6.**Comparison of measuring systems: (

**a**) 50 °C, (

**b**) 80 °C; measuring systems: black line—plane–plane, gray line—cone–plane.

**Figure 7.**Dependence of tangential stress on temperature in a series of experiments: solid line—experiment 1; dashed line—experiment 2; dots—experiment 3.

**Figure 8.**Dependence of physical and mechanical characteristics of a lubricant on temperature: solid line—complex shear modulus; dashed line—accumulation modulus; dashed line—loss modulus.

**Figure 10.**Tangential stress distribution as a function of temperature: red line—approximation of experimental data; solid black line—Anand’s model; dashed black line—Prony series.

**Figure 12.**Tangential stress distribution as a function of shear rate over a wide range of temperatures: (

**a**) −40 °C, (

**b**) −20 °C, (

**c**) 0 °C, (

**d**) 20 °C, (

**e**) 50 °C, and (

**f**) 80 °C; red markers—experimental data; solid black line—Anand’s model; dashed black line—Prony series.

Values | ${\mathit{S}}_{0}$, MPa | $\mathit{A}$$,1/\mathit{s}$ | $\mathit{U}/\mathit{R}$, K | ${\mathit{S}}_{1}$, MPa | $\mathit{n}$ | $\mathit{a}$ |
---|---|---|---|---|---|---|

Initial | 1 | ${10}^{10}$ | 1000 | 1 | 1 | 1 |

Final | 15.6218 | 3.518 × 10^{9} | 985.0634 | 7.023 × 10^{−7} | 4.498 × 10^{−4} | 2.8112 |

Matehematical Model | Scheme of the Model | Equation |
---|---|---|

Kelvin–Voigt model | $\mathsf{\sigma}\left(t\right)=E\mathsf{\epsilon}\left(t\right)+\mathsf{\eta}\frac{d\mathsf{\epsilon}\left(t\right)}{dt}$ | |

Burgers Material (Maxwell representation) | $\mathsf{\sigma}+\left(\frac{{\mathsf{\eta}}_{1}}{{E}_{1}}+\frac{{\mathsf{\eta}}_{2}}{{E}_{2}}\right)\dot{\mathsf{\sigma}}+\frac{{\mathsf{\eta}}_{1}{\mathsf{\eta}}_{2}}{{E}_{1}{E}_{2}}\ddot{\mathsf{\sigma}}=\left({\mathsf{\eta}}_{1}+{\mathsf{\eta}}_{2}\right)\dot{\mathsf{\epsilon}}+\frac{{\mathsf{\eta}}_{1}{\mathsf{\eta}}_{2}\left({E}_{1}+{E}_{2}\right)}{{E}_{1}{E}_{2}}\ddot{\mathsf{\epsilon}}$ | |

Burgers Material (Kelvin representation) | $\mathsf{\sigma}+\left(\frac{{\mathsf{\eta}}_{1}}{{E}_{1}}+\frac{{\mathsf{\eta}}_{2}}{{E}_{1}}+\frac{{\mathsf{\eta}}_{2}}{{E}_{2}}\right)\dot{\mathsf{\sigma}}+\frac{{\mathsf{\eta}}_{1}{\mathsf{\eta}}_{2}}{{E}_{1}{E}_{2}}\ddot{\mathsf{\sigma}}={\mathsf{\eta}}_{2}\dot{\mathsf{\epsilon}}+\frac{{\mathsf{\eta}}_{1}{\mathsf{\eta}}_{2}}{{E}_{1}}\ddot{\mathsf{\epsilon}}$ |

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**MDPI and ACS Style**

Nosov, Y.O.; Kamenskikh, A.A.
Experimental Study of the Rheology of Grease by the Example of CIATIM-221 and Identification of Its Behavior Model. *Lubricants* **2023**, *11*, 295.
https://doi.org/10.3390/lubricants11070295

**AMA Style**

Nosov YO, Kamenskikh AA.
Experimental Study of the Rheology of Grease by the Example of CIATIM-221 and Identification of Its Behavior Model. *Lubricants*. 2023; 11(7):295.
https://doi.org/10.3390/lubricants11070295

**Chicago/Turabian Style**

Nosov, Yuriy O., and Anna A. Kamenskikh.
2023. "Experimental Study of the Rheology of Grease by the Example of CIATIM-221 and Identification of Its Behavior Model" *Lubricants* 11, no. 7: 295.
https://doi.org/10.3390/lubricants11070295