# The Study Influence Analysis of the Mathematical Model Choice for Describing Polymer Behavior

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## Abstract

**:**

## 1. Introduction

#### 1.1. Research Objectives

- The behavior description of polymeric materials in viscoelastic terms using the generalized Maxwell model;
- Comparison of experimental data and numerical solutions of three different mathematical models of material behavior: elastic, plastic, and viscoelastic;
- Carrying out verification of the numerical Hertz problem;
- Constructing numerical models of the specimen behavior (elastic–plastic, viscoelastic);
- The influence study of mathematical models of polymers’ behavior in a dynamic setting (loading and subsequent exposure for 1 h at room temperature).

#### 1.2. Problem Context

#### 1.3. Problem Description

## 2. Materials and Methods

#### 2.1. Materials

- -
- The specimen was tested at a strain rate of 0.006 mm/min up to 10%;
- -
- Constant strain was held for 15 min;
- -
- The load was removed from the material until normal stresses of 0.1 MPa was reached at a rate of 0.006 mm/min.

#### 2.2. Description of Polymer Behavior

- -
- Elastic body:

- -
- Elastoplastic body:

- -
- Viscoelastic body:

- -
- First (preliminary) stage: input of experimental data in the form of a text file with data, model selection, and generation of the initial vector of unknowns from Equation (5);
- -
- The second stage is based on the Nelder–Mead optimization algorithm: creation of a script-file describing the numerical experiment in ANSYS Mechanical APDL, conducting the numerical experiment, obtaining the results file, comparison of numerical and experimental data, generation of the vector of unknowns from Equation (5) at step i, and transition to the next iteration;
- -
- The third (and final) step is performed when the error between the experimental and numerical data reaches 5%: formation of the final vector of unknowns from Equation (5) and exit from the procedure.

#### 2.3. The Hertz Formulation

- -
- Sliding friction: ${u}_{n}^{1}={u}_{n}^{2}$, ${u}_{{\mathsf{\tau}}_{1}}^{1}\ne {u}_{{\mathsf{\tau}}_{1}}^{2}$, ${u}_{{\mathsf{\tau}}_{2}}^{1}\ne {u}_{{\mathsf{\tau}}_{2}}^{2}$, ${\mathsf{\sigma}}_{n}^{1}={\mathsf{\sigma}}_{n}^{2}$, ${\mathsf{\sigma}}_{n{\mathsf{\tau}}_{1}}^{1}={\mathsf{\sigma}}_{n{\mathsf{\tau}}_{1}}^{2}$, ${\mathsf{\sigma}}_{n{\mathsf{\tau}}_{2}}^{1}={\mathsf{\sigma}}_{n{\mathsf{\tau}}_{2}}^{2}$, when $\left|{\mathsf{\sigma}}_{n{\mathsf{\tau}}_{1}}\right|=\mathsf{\mu}\left({\mathsf{\sigma}}_{n}\right)\left|{\mathsf{\sigma}}_{n}\right|$;
- -
- No contact: $\left|{u}_{n}^{1}-{u}_{n}^{2}\right|\ge 0$, ${\mathsf{\sigma}}_{n{\mathsf{\tau}}_{1}}={\mathsf{\sigma}}_{n{\mathsf{\tau}}_{2}}={\mathsf{\sigma}}_{n}=0$;
- -
- Adhesion: ${\overline{u}}^{1}={\overline{u}}^{2}$, ${\mathsf{\sigma}}_{n}^{1}={\mathsf{\sigma}}_{n}^{2}$, ${\mathsf{\sigma}}_{n{\mathsf{\tau}}_{1}}^{1}={\mathsf{\sigma}}_{n{\mathsf{\tau}}_{1}}^{2}$, ${\mathsf{\sigma}}_{n{\mathsf{\tau}}_{2}}^{1}={\mathsf{\sigma}}_{n{\mathsf{\tau}}_{2}}^{2}$,

## 3. Results

#### 3.1. Invastigation of Mathematical Models

#### 3.2. Hertz Contact Calculation Model

## 4. Discussion

#### 4.1. Limitation Statement

- The material behavior is considered at a constant temperature of 20 °C;
- The model problem of spherical indenter penetration into a half-space is considered;
- For each material, it is necessary to carry out a separate description of the mathematical model;
- Long time ranges are not considered, while the material works for a long time.

- Investigation of the material on a large range of operating temperatures;
- Study of the material on the dependence on the load impact rate on the polymer material;
- Study of temperature characteristics of the material;
- Realization of the problem on the example of a bridge bearing structure under cyclic loading.

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

#### 4.3. Applicability of the Research

## 5. Conclusions

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- The use of an elastic–plastic body to describe the material behavior can be used only in static problems to determine the strength properties of the structure;
- -
- The use of a viscoelastic body to describe the mathematical model of material behavior allows for evaluation of the performance of a structure at the entire stage of its life cycle.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Experimental studies: (

**a**) experiment scheme; (

**b**) full-scale sample. Sections 1 and 2 are upper and lower steel sections, respectively; 3—polymer.

**Figure 3.**Dynamic characteristics determination of the material: (

**a**) strain–time dependence, (

**b**) stress–time dependence; (

**c**) stress–strain dependence.

**Figure 6.**Dependence of weight coefficients ${\mathsf{\alpha}}_{i}$ on relaxation time ${\mathsf{\beta}}_{i}$. Red dots are the value of the weight coefficient ${\mathsf{\alpha}}_{i}$ at a certain relaxation time ${\mathsf{\beta}}_{i}$.

**Figure 7.**Stress–strain diagram: the red line is experimental data; the black solid line is the elastic body; the dashed line is the elastic–plastic body; and the dots are the viscoelastic body.

**Figure 8.**Stress dependence on time: the red line is experimental data; the black solid line is the elastic body; the dashed line is the elastic–plastic body; the dots are the viscoelastic body.

**Figure 9.**Stress dependence on deformation: the red line is experimental data; the black solid line is the elastic body; the dashed line is the elastic–plastic body; and the dots are the viscoelastic body.

**Figure 11.**Dependence of contact pressure on the radius of indenter insertion: the red line is the analytical solution; black lines are maximum load; gray lines are 1 h exposure time; the dashed line is the elastic–plastic body; the dots are the viscoelastic body.

**Figure 12.**Dependence of maximum values of strain intensity on time: the dotted line is the elastic–plastic body; the dots are the viscoelastic body.

**Figure 13.**Dependence of maximum values of stress intensity on time: the dotted line is the elastic–plastic body, the dots are the viscoelastic body.

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**MDPI and ACS Style**

Kamenskikh, A.A.; Nosov, Y.O.; Bogdanova, A.P.
The Study Influence Analysis of the Mathematical Model Choice for Describing Polymer Behavior. *Polymers* **2023**, *15*, 3630.
https://doi.org/10.3390/polym15173630

**AMA Style**

Kamenskikh AA, Nosov YO, Bogdanova AP.
The Study Influence Analysis of the Mathematical Model Choice for Describing Polymer Behavior. *Polymers*. 2023; 15(17):3630.
https://doi.org/10.3390/polym15173630

**Chicago/Turabian Style**

Kamenskikh, Anna A., Yuriy O. Nosov, and Anastasia P. Bogdanova.
2023. "The Study Influence Analysis of the Mathematical Model Choice for Describing Polymer Behavior" *Polymers* 15, no. 17: 3630.
https://doi.org/10.3390/polym15173630