# A New Multiparameter Model for Multiaxial Fatigue Life Prediction of Rubber Materials

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}coefficient by comparing the predicted values of every model, with the experimental ones. The obtained results show a significant improvement in the fatigue life prediction. The proposed model does not aim to be a universal and definitive approach for elastomer fatigue, but it provides a reliable general tool that can be used for processing data obtained from experimental tests carried out under different conditions.

## 1. Introduction

## 2. State-of-the-Art Review of Fatigue Damage Parameters for Rubber Materials

#### 2.1. Fatigue Damage Parameters Based on Stresses, Strains, and Energy

#### 2.2. Fatigue Damage Parameters Based on the Critical Plane

#### 2.2.1. Cracking Energy Density (CED)

_{c}in the plane of failure, as shown in Equation (1):

**ε**is the strain increment tensor.

#### 2.2.2. Fatemi-Socie Parameter

_{n,max}is the maximum normal stress in that plane, σ

_{y}is the elastic limit of the material, and k

_{FS}is a constant of the material.

#### 2.2.3. Smith–Watson–Topper Parameter

_{n}is the normal stress in a plane θ, and ε

_{1}is the maximum principal strain range.

#### 2.2.4. Liu I and Liu II Parameters

#### 2.2.5. Findley Parameter

_{a}is the amplitude of the shear stress in a plane θ, σ

_{n,max}is the maximum normal stress in that plane, and k

_{F}is a constant parameter property of the material.

#### 2.2.6. Brown –Miller Parameter

_{max}is the maximum angular distortion range, and ε

_{n}is the normal strain range in the plane experienced by the angular distortion range γ

_{max.}

#### 2.2.7. Wang –Brown Parameter

_{n}is the normal strain range in the same plane θ, and S is a property of the material.

#### 2.2.8. McDiarmid Parameter

_{max}is the maximum shear stress range, σ

_{n,max}is the maximum normal stress in the direction perpendicular to the Δτ

_{max}plane, τ

_{f}is the limit of torsional fatigue, and σ

_{u}is the ultimate tensile strength of the material.

## 3. Proposed Fatigue Damage Multi-Parameter (FDMP) for Multiaxial Fatigue Analysis

#### 3.1. Proposed Fatigue Multi-Parameter

^{2}with the real experimental data. The calculated weights maximize the value of R

^{2}for a line defined in Equation (11):

- α
_{i}: Weights for each variable (amplitude and mean value) taken into account - β
_{i}: Weights for each variable (maximum value) taken into account - N
_{cycles-var,i}: Number of cycles experienced by variable i during one load cycle. - The value of var
_{eq,i}is defined in Equations (13) and (14):$$va{r}_{eq,i}=va{r}_{amp,\text{}i}\xb7{\left(1-{R}_{var,\text{}i}\right)}^{{\gamma}_{i}-1}\text{}$$$$\text{}{R}_{var,\text{}i}=\frac{va{r}_{min,\text{}i}}{va{r}_{max,i}}\text{}$$ - γ
_{i}: Coefficient in order to take into account the mean value of each variable

_{cycles}will be equal to 1. For the cases in which, N = 1, β

_{1}= 0, and γ

_{1}= 1, they will be like a classic fatigue model with only one variable and only taking into account its amplitude.

#### 3.2. Methodology for Validation of the Proposed Model (FDMP)

^{2}, it was established which of all the models correlated best with the fatigue life of each of the experiments.

- Numerical simulation of each of the tests for each of the specimens and each of the materials.
- Obtaining and calculation of the evolution of 20 mechanical variables throughout a load cycle.
- Calculation of the number of cycles, amplitude, maximum value, and average value of each variable for each load cycle.
- Correlation of the calculated values with the fatigue life of each material and obtaining the parameter R
^{2}. - Calculation of weights and coefficients for the model proposed in this work for each of the test batteries.
- Application of the proposed model, correlation with the fatigue life of each material and obtaining the parameter R
^{2}. - Comparison of the R
^{2}parameters of all the fatigue variables considered. - The obtained results are analysed in detail in Section 7: Discussion.

#### 3.3. Experimental Data for Model Validation

#### 3.3.1. Mars and Fatemi Experimental Data Description (NBR Tests)

#### 3.3.2. Ayoub Experimental Data Description (SBR Tests)

## 4. Numerical Simulations and Results

#### 4.1. NBR Numerical Simulations and Results

_{1}= 1.5 and J = 1.

_{1}and D are material constants and J is the determinant of the gradient strain tensor F.

_{1}, σ

_{2}and σ

_{3}), Von Mises stress (σ

_{VM}), Tresca stress (τ

_{max}). (c) Strain variables: principal strains (ε

_{1}, ε

_{2}and ε

_{3}), octahedral strain (ε

_{oct}), octahedral angular strain (γ

_{oct}). (d) Energy variables [MPa]: Strain energy density (W), crack energy density (W

_{c}). In addition, the evolution of the critical plane parameters according to the angle of this plane concerning the maximum main direction are plotted: (e) Dimensionless parameters: Brown–Miller (P

_{BM}) and Fatemi–Socie (P

_{FS}). (f) Dimensionless parameter: Wang–Brown (P

_{WB}). (g) Parameters with stress dimensions: Findley and McDiarmid [MPa]. (h) Parameters with energy density dimensions: Smith–Watson–Topper (SWT), and Liu WI and Liu WII [MPa].

#### 4.2. SBR Numerical Simulations and Results

_{1}= 5.25 MPa, μ

_{2}= 1.52 × 10

^{−2}MPa, α

_{1}= 2.14 × 10

^{−1}, α

_{2}= 4.06 and λ

_{1}, λ

_{2}and λ

_{3}are the principal stretches.

_{1}, σ

_{2}and σ

_{3}), Von Mises stress (σ

_{VM}), Tresca stress (τ

_{max}). (c) Strain variables: principal strains (ε

_{1}, ε

_{2}and ε

_{3}), octahedral strain (ε

_{oct}), octahedral angular strain (γ

_{oct}). (d) Strain energy density (W) [MPa], Crack energy density (W

_{c}) [MPa]. In addition, the evolution of the critical plane parameters according to the angle of this plane with respect to the maximum main direction are plotted: (e) Dimensionless parameters: Brown-Miller (P

_{BM}) and Fatemi-Socie (P

_{FS}). (f) Dimensionless parameter: Wang-Brown (P

_{WB}). (g) Parameters with stress dimensions: Findley and McDiarmid [Mpa]. (h) Parameters with Energy Density dimensions: Smith-Watson-Topper (SWT), Liu WI and Liu WII [ MPa].

## 5. Fatigue Lifetime Correlation

#### 5.1. NBR–Fatigue Parameters Correlation

#### 5.2. SBR–Fatigue Parameters Correlation

## 6. Application of the Proposed Fatigue Damage Multi-Parameter (FDMP)

#### 6.1. NBR – FDMP Results

^{2}, the values obtained for each of the weights are included in Table 3:

^{2}value is maximized (Equation (11)) are A = −0.210, B = 3.200:

^{2}is 0.934.

#### 6.2. SBR–FDMP Results

^{2}, the values obtained for each of the weights are included in Table 4.

^{2}value is maximized (Equation (11)) are A = −0.043, B = 0.803. The values of the fatigue multiparameter for each test versus lifetime are plotted in Figure 14, obtaining an R

^{2}value of 0.940.

## 7. Discussion

^{2}coefficient (calculated by comparing the predicted values with the experimental ones) is considerably higher for the proposed multiparameter in both types of test.

## 8. Conclusions

^{2}with real experimental results.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CED | Cracking Energy Density |

FDMP | Fatigue Damage Multi-Parameter |

FEM | Finite Element Method |

HCF | High Cycle Fatigue |

NBR | Acrylonitrile Butadiene Rubber |

NR | Natural Rubber |

P_{BM} | Brown-Miller parameter |

P_{FS} | Fatemi-Socie parameter |

P_{WB} | Wang-Brown parameter |

R^{2} | Coefficient of determination |

SBR | Styrene-Butadiene Rubber |

SED | Strain Energy Density |

SWT | Smith-Watson-Topper |

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**Figure 1.**3D model for the calculation of stresses, strains and energies, based on Mars and Fatemi tests [45].

**Figure 2.**3D model for the calculation of stresses, strains and energies, based on Ayoub tests [46].

**Figure 3.**Original and deformed configuration of a material element subjected to axial and shear strains resulting from axial and torsional displacements δ and θ. Adapted from [45].

**Figure 5.**Result of the mechanical simulation of test A1 at the failure point of the sample caused by fatigue. Initial data from [45].

**Figure 6.**Result of the mechanical simulation of test B1 at the failure point of the sample caused by fatigue. Initial data from [45].

**Figure 8.**Result of the mechanical simulation of test A1 at the failure point of the sample caused by fatigue. Initial data from [46].

**Figure 9.**Result of the mechanical simulation of test B9 at the failure point of the sample caused by fatigue. Initial data from [46].

**Figure 10.**Fatigue parameters and tests lifetime [cycles] correlation for NBR tests. Initial data from [45].

**Figure 11.**Fatigue parameters and tests lifetime (cycles) correlation for SBR tests. Initial data from [46].

**Figure 12.**Proposed FDMP and tests lifetime (cycles) correlation for NBR tests. Initial data from [45].

**Figure 13.**Predicted life with FDMP (cycles) versus real life (cycles) for NBR. Initial data from [45].

**Figure 14.**Proposed FDMP and tests lifetime (cycles) correlation for SBR tests. Initial data from [46].

**Figure 15.**Predicted life (cycles) with FDMP versus real life (cycles) for SBR tests. Initial data from [46].

**Table 1.**Information of Mars and Fatemi battery tests. Initial test data adapted from [45].

No | Test (Type-Number) | δ_{max}(mm) | δ_{min}(mm) | θ_{max} (°) | θ_{min} (°) | P_{a}(N) | P_{m}(N) | T_{a}(Nm) | T_{m}(Nm) | Offset (°) | Lifetime (Cycles) |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | A1 | 1.27 | 0 | 0 | 0 | 1136 | 581 | 0 | 1 | 0 | 339,167 |

83 | I4 | 3.55 | 0 | 14 | 0 | 1645 | 638 | 60 | 47 | 180 | 12,408 |

**Table 2.**Information of the Ayoub battery tests. Initial test data adapted from [46].

Test | Test Type | F (Hz) | d_{max}(mm) | d_{min}(mm) | θ_{max}(°) | θ_{min}(°) | Lifetime (Cycles) |
---|---|---|---|---|---|---|---|

1 | A | 5 | 2.25 | 0 | 0 | 0 | 760,000 |

128 | E7 | 5 | 3.8 | 1.75 | 0 | 0 | 363,500 |

**Table 3.**Model weights of FDMP model for NBR tests. Initial data from [45].

n | Variable | γ_{n} | α_{n} | β_{n} |
---|---|---|---|---|

1 | σ_{1} | 0.550 | 0.017 | −0.188 |

2 | σ_{2} | 0.932 | −0.139 | 0.253 |

3 | σ_{3} | 1.000 | −0.068 | −0.134 |

4 | ε_{1} | 12.152 | −0.153 | 2.580 |

5 | ε_{2} | 1.000 | 1.500 | −0.912 |

6 | ε_{3} | 1.000 | 0.283 | 4.470 |

7 | SED | −1.715 | 0.151 | 0.071 |

**Table 4.**Model weights of FDMP model for SBR tests. Initial data from [46].

n | Variable | γ_{n} | α_{n} | β_{n} |
---|---|---|---|---|

1 | σ_{1} | 0.006 | 319.264 | −160.034 |

2 | σ_{2} | 1.000 | −1.371 | 0.962 |

3 | σ_{3} | 1.000 | −0.563 | 1.188 |

4 | ε_{1} | 0.006 | −25.018 | 13.890 |

5 | ε_{2} | 1.000 | −0.412 | −0.297 |

6 | ε_{3} | 1.000 | −0.144 | −1.452 |

7 | SED | −0.093 | 16.546 | −8115 |

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## Share and Cite

**MDPI and ACS Style**

Tobajas, R.; Elduque, D.; Ibarz, E.; Javierre, C.; Gracia, L.
A New Multiparameter Model for Multiaxial Fatigue Life Prediction of Rubber Materials. *Polymers* **2020**, *12*, 1194.
https://doi.org/10.3390/polym12051194

**AMA Style**

Tobajas R, Elduque D, Ibarz E, Javierre C, Gracia L.
A New Multiparameter Model for Multiaxial Fatigue Life Prediction of Rubber Materials. *Polymers*. 2020; 12(5):1194.
https://doi.org/10.3390/polym12051194

**Chicago/Turabian Style**

Tobajas, Rafael, Daniel Elduque, Elena Ibarz, Carlos Javierre, and Luis Gracia.
2020. "A New Multiparameter Model for Multiaxial Fatigue Life Prediction of Rubber Materials" *Polymers* 12, no. 5: 1194.
https://doi.org/10.3390/polym12051194