# 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 |

## References

- Dufton, P.W. Thermoplastic Elastomers; Smithers Rapra Publishing: Shrewsbury, UK, 2001. [Google Scholar]
- Delogu, M.; Del Pero, F.; Pierini, M. Lightweight design solutions in the automotive field: Environmental modelling based on fuel reduction value applied to diesel turbocharged vehicles. Sustainability
**2016**, 8, 1167. [Google Scholar] [CrossRef][Green Version] - Koltzenburg, S.; Maskos, M.; Nuyken, O. Elastomers. In Polymer Chemistry; Springer: Berlin/Heidelberg, Germany, 2017; pp. 477–491. ISBN 978-3-662-49279-6. [Google Scholar]
- Ciampa, F.; Mahmoodi, P.; Pinto, F.; Meo, M. Recent advances in active infrared thermography for non-destructive testing of aerospace components. Sensors
**2018**, 18, 609. [Google Scholar] [CrossRef] [PubMed][Green Version] - Markl, E.; Lackner, M. Devulcanization technologies for recycling of tire-derived rubber: A review. Materials
**2020**, 13, 1246. [Google Scholar] [CrossRef] [PubMed][Green Version] - Araujo-Morera, J.; Santana, M.H.; Verdejo, R.; López-Manchado, M.A. Giving a second opportunity to tire waste: An alternative path for the development of sustainable self-healing styrene-butadiene rubber compounds overcoming the magic triangle of tires. Polymers
**2019**, 11, 2122. [Google Scholar] [CrossRef] [PubMed][Green Version] - Buss, A.H.; Kovaleski, J.L.; Pagani, R.N.; da Silva, V.L.; de Silva, J.M. Proposal to reuse rubber waste from end-of-life tires using thermosetting resin. Sustainability
**2019**, 11, 6997. [Google Scholar] [CrossRef][Green Version] - Rodgers, B. Rubber Compounding: Chemistry and Applications; CRC Press: Boca Raton, FL, USA, 2015. [Google Scholar]
- Szczypinski-Sala, W.; Lubas, J. Tribological characteristic of a ring seal with graphite filler. Materials
**2020**, 13, 311. [Google Scholar] [CrossRef][Green Version] - Liang, B.; Yang, X.; Wang, Z.; Su, X.; Liao, B.; Ren, Y.; Sun, B. Influence of randomness in rubber materials parameters on the reliability of rubber O-ring seal. Materials
**2019**, 12, 1566. [Google Scholar] [CrossRef][Green Version] - Yoon, S.H.; Winters, M.; Siviour, C.R. High strain-rate tensile characterization of EPDM rubber using non-equilibrium loading and the virtual fields method. Exp. Mech.
**2016**, 56, 25–35. [Google Scholar] [CrossRef] - Arghavan, A.; Kashyzadeh, K.R.; Asfarjani, A.A. Investigating effect of industrial coatings on fatigue damage. In Applied Mechanics and Materials; Trans Tech Publications Ltd.: Kapellweg, Switzerland, 2011. [Google Scholar]
- Gobbato, M.; Kosmatka, J.B.; Conte, J.P. A recursive bayesian approach for fatigue damage prognosis: An experimental validation at the reliability component level. Mech. Syst. Signal Process.
**2014**. [Google Scholar] [CrossRef] - Pfingstl, S.; Steiner, M.; Tusch, O.; Zimmermann, M. Crack detection zones: Computation and validation. Sensors
**2020**, 20, 2568. [Google Scholar] [CrossRef] - Seichter, S.; Archodoulaki, V.M.; Koch, T.; Holzner, A.; Wondracek, A. Investigation of different influences on the fatigue behaviour of industrial rubbers. Polym. Test.
**2017**. [Google Scholar] [CrossRef] - Abdelaziz, N.M.; Ayoub, G.; Colin, X.; Benhassine, M.; Mouwakeh, M. New developments in fracture of rubbers: Predictive tools and influence of thermal aging. Int. J. Solids Struct.
**2019**. [Google Scholar] [CrossRef] - Neuhaus, C.; Lion, A.; Johlitz, M.; Heuler, P.; Barkhoff, M.; Duisen, F. Fatigue behaviour of an elastomer under consideration of ageing effects. Int. J. Fatigue
**2017**. [Google Scholar] [CrossRef] - Duan, X.; Shangguan, W.B.; Li, M.; Rakheja, S. Measurement and modelling of the fatigue life of rubber mounts for an automotive powertrain at high temperatures. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2016**. [Google Scholar] [CrossRef] - Loo, M.S.; Le Cam, J.B.; Andriyana, A.; Robin, E.; Coulon, J.F. Effect of swelling on fatigue life of elastomers. Polym. Degrad. Stab.
**2016**. [Google Scholar] [CrossRef] - Ruellan, B.; Le Cam, J.B.; Jeanneau, I.; Canévet, F.; Mortier, F.; Robin, E. Fatigue of natural rubber under different temperatures. Int. J. Fatigue
**2019**. [Google Scholar] [CrossRef] - Zhou, Y.; Jerrams, S.; Betts, A.; Farrell, G.; Chen, L. The influence of particle content on the equi-biaxial fatigue behaviour of magnetorheological elastomers. Mater. Des.
**2015**. [Google Scholar] [CrossRef][Green Version] - Spagnoli, A.; Terzano, M.; Brighenti, R.; Artoni, F.; Carpinteri, A. How soft polymers cope with cracks and notches. Appl. Sci.
**2019**, 9, 1086. [Google Scholar] [CrossRef][Green Version] - Guo, H.; Li, F.; Wen, S.; Yang, H.; Zhang, L. Characterization and quantitative analysis of crack precursor size for rubber composites. Materials
**2019**, 12, 3442. [Google Scholar] [CrossRef][Green Version] - Li, F.; Liu, J.; Mars, W.V.; Chan, T.W.; Lu, Y.; Yang, H.; Zhang, L. Crack precursor size for natural rubber inferred from relaxing and non-relaxing fatigue experiments. Int. J. Fatigue
**2015**. [Google Scholar] [CrossRef] - Wada, S.; Zhang, R.; Mannava, S.R.; Vasudevan, V.K.; Qian, D. Simulation-based prediction of cyclic failure in rubbery materials using nonlinear space-time finite element method coupled with continuum damage mechanics. Finite Elem. Anal. Des.
**2018**. [Google Scholar] [CrossRef] - Nyaaba, W.; Frimpong, S.; Anani, A. Fatigue damage investigation of ultra-large tire components. Int. J. Fatigue
**2019**. [Google Scholar] [CrossRef] - Feng, X.; Li, Z.; Wei, Y.; Chen, Y.; Kaliske, M.; Zopf, C.; Behnke, R. A novel method for constitutive characterization of the mechanical properties of uncured rubber. J. Elastomers Plast.
**2016**, 48, 523–534. [Google Scholar] [CrossRef] - Carleo, F.; Barbieri, E.; Whear, R.; Busfield, J.J.C. Limitations of viscoelastic constitutive models for carbon-black reinforced rubber in medium dynamic strains and medium strain rates. Polymers
**2018**, 10, 988. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mars, W.V.; Ellul, M.D. Fatigue characterization of a thermoplastic elastomer. Rubber Chem. Technol.
**2017**. [Google Scholar] [CrossRef] - Cruanes, C.; Lacroix, F.; Berton, G.; Méo, S.; Ranganathan, N. Study of the fatigue behavior of a synthetic rubber undergoing cumulative damage tests. Int. J. Fatigue
**2016**. [Google Scholar] [CrossRef] - Zhang, B.; Yu, X.; Gu, B. Modeling and experimental validation of interfacial fatigue damage in fiber-reinforced rubber composites. Polym. Eng. Sci.
**2018**. [Google Scholar] [CrossRef] - Luo, P.; Yao, W.; Wang, Y.; Li, P. A survey on fatigue life analysis approaches for metallic notched components under multi-axial loading. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng.
**2019**, 233, 3870–3890. [Google Scholar] [CrossRef] - Narynbek, U.K.; Huneau, B.; Verron, E.; Béranger, A.S.; Heuillet, P. True stress controlled fatigue life experiments for elastomers. Int. J. Fatigue
**2017**. [Google Scholar] [CrossRef] - Sun, C.; Du, Z.; Nagarajan, S.; Zhao, H.; Wen, S.; Zhao, S.; Zhang, P.; Zhang, L. Impact of uniaxial tensile fatigue on the evolution of microscopic and mesoscopic structure of carbon black filled natural rubber. R. Soc. Open Sci.
**2019**. [Google Scholar] [CrossRef][Green Version] - Gehrmann, O.; El Yaagoubi, M.; El Maanaoui, H.; Meier, J. Lifetime prediction of simple shear loaded filled elastomers based on the probability distribution of particles. Polym. Test.
**2019**. [Google Scholar] [CrossRef] - Mars, W.V.; Fatemi, A. A literature survey on fatigue analysis approaches for rubber. Int. J. Fatigue
**2002**, 24, 949–961. [Google Scholar] [CrossRef] - Moon, B.; Lee, J.; Park, S.; Seok, C.S. Study on the aging behavior of natural rubber/butadiene rubber (NR/BR) blends using a parallel spring model. Polymers
**2018**, 10, 658. [Google Scholar] [CrossRef][Green Version] - Zhang, J.; Xue, F.; Wang, Y.; Zhang, X.; Han, S. Strain energy-based rubber fatigue life prediction under the influence of temperature. R. Soc. Open Sci.
**2018**. [Google Scholar] [CrossRef] [PubMed][Green Version] - Poulain, X.; Lefèvre, V.; Lopez-Pamies, O.; Ravi-Chandar, K. Damage in elastomers: Nucleation and growth of cavities, micro-cracks, and macro-cracks. Int. J. Fract.
**2017**. [Google Scholar] [CrossRef] - Huneau, B.; Masquelier, I.; Marco, Y.; Le Saux, V.; Noizet, S.; Schiel, C.; Charrier, P. Fatigue crack initiation in a carbon black-filled natural rubber. Rubber Chem. Technol.
**2016**. [Google Scholar] [CrossRef][Green Version] - Rubio-Mateos, A.; Rivero, A.; Ukar, E.; Lamikiz, A. Influence of elastomer layers in the quality of aluminum parts on finishing operations. Metals
**2020**, 10, 289. [Google Scholar] [CrossRef][Green Version] - Lei, T.; Zhang, Y.-W.; Kuang, D.-L.; Yang, Y.-R. Preparation and properties of rubber blends for high-damping-isolation bearings. Polymers
**2019**, 11, 1374. [Google Scholar] [CrossRef][Green Version] - Zanchet, A. Elastomeric composites containing SBR industrial scraps devulcanized by microwaves: Raw material, not a trash. Recycling
**2020**, 5, 3. [Google Scholar] [CrossRef][Green Version] - Shen, M.-Y.; Chiou, Y.-C.; Tan, C.-M.; Wu, C.-C.; Chen, W.-J. Effect of wall thickness on stress-strain response and buckling behavior of hollow-cylinder rubber fenders. Materials
**2020**, 13, 1170. [Google Scholar] [CrossRef][Green Version] - Fatemi, A.; Mars, W.V. Multiaxial fatigue of rubber: Part II: Experimental observations and life predictions. Fatigue Fract. Eng. Mater. Struct.
**2005**, 28, 523–538. [Google Scholar] - Ayoub, G.; Naït-Abdelaziz, M.; Zaïri, F.; Gloaguen, J.M.; Charrier, P. Fatigue life prediction of rubber-like materials under multiaxial loading using a continuum damage mechanics approach: Effects of two-blocks loading and R ratio. Mech. Mater.
**2012**. [Google Scholar] [CrossRef] - Lu, C. Etude du Comportement Mécanique et des Mécanismes D’endommagement des Élastomères en Fatigue et en Fissuration Par Fatigue. Ph.D. Thesis, CNAM, Paris, France, 1991. [Google Scholar]
- Abraham, F.; Alshuth, T.; Jerrams, S. The effect of minimum stress and stress amplitude on the fatigue life of non strain crystallising elastomers. Mater. Des.
**2005**, 26, 239–245. [Google Scholar] [CrossRef] - André, N.; Cailletaud, G.; Piques, R. Others haigh diagram for fatigue crack initiation prediction of natural rubber components. Kautsch. Gummi Kunstst.
**1999**, 52, 120–123. [Google Scholar] - Andre, N. Critère local d’amorçage de fissure en fatigue dans un élastomère de type NR. Ph.D. Thesis, Ecole Nationale Supérieure des Mines de Paris, Paris, France, 1999. [Google Scholar]
- Shaker, R.; Rodrigue, D. Rotomolding of thermoplastic elastomers based on low-density polyethylene and recycled natural rubber. Appl. Sci.
**2019**, 9, 5430. [Google Scholar] [CrossRef][Green Version] - Wang, W.T.; Xiao, S.H.; Huang, J.L.; Xie, X.X. Investigation on rubber isolator’s fatigue life prediction under uniaxial tensile load. Zhendong Yu Chongji J. Vib. Shock
**2014**. [Google Scholar] [CrossRef] - Woo, C.S.; Kim, W.D.; Kwon, J. Do a study on the material properties and fatigue life prediction of natural rubber component. Mater. Sci. Eng. A
**2008**. [Google Scholar] [CrossRef] - Suryatal, B.; Phakatkar, H.; Rajkumar, K.; Thavamani, P. Fatigue life estimation of an elastomeric pad by ε-N curve and FEA. J. Surf. Eng. Mater. Adv. Technol.
**2015**. [Google Scholar] [CrossRef][Green Version] - Li, Q.; Wen, Z.-W.; He, G.; Yuan, M.-H.; Zhu, W.-D. Fatigue life prediction of a rubber mount based on the continuum damage mechanics. J. Macromol. Sci. Part B
**2019**, 58, 947–958. [Google Scholar] [CrossRef] - Roberts, B.J.; Benzies, J.B. The relationship between uniaxial and equibiaxial fatigue in gum and carbon black filled vulcanizates. Proc. Rubbercon
**1977**, 77, 1–2. [Google Scholar] - Greensmith, H.W.; Mullins, L.; Thomas, A.G. The Chemistry and Physics of Rubber Like Substances; Wiley: New York, NY, USA, 1963. [Google Scholar]
- Mars, W.V.; Fatemi, A. Multiaxial fatigue of rubber: Part I: Equivalence criteria and theoretical aspects. Fatigue Fract. Eng. Mater. Struct.
**2005**, 28, 515–522. [Google Scholar] [CrossRef] - Mars, W.V.; Fatemi, A. Criteria for fatigue crack nucleation in rubber under multiaxial loading. Const. Model. Rubber
**2001**, 2, 213–222. [Google Scholar] - Peng, Y.; Liu, G.; Quan, Y.; Zeng, Q. Cracking energy density calculation of hyperelastic constitutive model and its application in rubber fatigue life estimations. J. Appl. Polym. Sci.
**2016**. [Google Scholar] [CrossRef] - Verron, E. Prediction of fatigue crack initiation in rubber with the help of configurational mechanics. In Constitutive Models for Rubber-Proceedings; CRC Press: Boca Raton, FL, USA, 2005; Volume 4, p. 3. [Google Scholar]
- Verron, E.; Le Cam, J.-B.; Gornet, L. A multiaxial criterion for crack nucleation in rubber. Mech. Res. Commun.
**2006**, 33, 493–498. [Google Scholar] [CrossRef][Green Version] - Barbash, K.P.; Mars, W.V. Critical plane analysis of rubber bushing durability under road loads. In Proceedings of the SAE World Congress and Exhibition, Detroit, MI, USA, 12–14 April 2016. [Google Scholar]
- Mars, W.V.; Wei, Y.; Hao, W.; Bauman, M.A. Computing tire component durability via critical plane analysis. Tire Sci. Technol.
**2019**. [Google Scholar] [CrossRef] - Fatemi, A.; Socie, D.F. A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fatigue Fract. Eng. Mater. Struct.
**1988**, 11, 149–165. [Google Scholar] [CrossRef] - Smith, K.; Topper, T.H.; Watson, P. A stress-strain function for the fatigue of metals (Stress-strain function for metal fatigue including mean stress effect). J. Mater.
**1970**, 5, 767–778. [Google Scholar] - Liu, K.C. A method based on virtual strain-energy parameters for multiaxial fatigue life prediction. In Advances in Multiaxial Fatigue; ASTM International: West Conshohocken, PA, USA, 1993. [Google Scholar]
- Findley, W.N. Fatigue of Metals Under Combinations of Stresses; Division of Engineering, Brown University: Providence, RI, USA, 1956. [Google Scholar]
- Brown, M.W.; Miller, K.J. A theory for fatigue failure under multiaxial stress-strain conditions. Proc. Inst. Mech. Eng.
**1973**, 187, 745–755. [Google Scholar] [CrossRef] - Wang, C.H.; Brown, M.W. Life prediction techniques for variable amplitude multiaxial fatigue—Part 1: Theories. J. Eng. Mater. Technol.
**1996**, 118, 367–370. [Google Scholar] [CrossRef] - Kandil, F.A.; Brown, M.W.; Miller, K.J. Biaxial low-cycle fatigue failure of 316 stainless steel at elevated temperatures. In Mechanical Behaviour and Nuclear Applications of Stainless Steel at Elevated Temperatures; Maney Pub.: London, UK, 1982. [Google Scholar]
- McDiarmid, D.L. A general criterion for high cycle multiaxial fatigue failure. Fatigue Fract. Eng. Mater. Struct.
**1991**, 14, 429–453. [Google Scholar] [CrossRef] - Lasdon, L.S.; Fox, R.L.; Ratner, M.W. Nonlinear optimization using the generalized reduced gradient method. Rev. Fr. Autom Inf Rech Oper
**1974**. [Google Scholar] [CrossRef][Green Version] - Brown, A.M. A step-by-step guide to non-linear regression analysis of experimental data using a Microsoft Excel spreadsheet. Comput. Methods Programs Biomed.
**2001**. [Google Scholar] [CrossRef] - Baschnagel, F.; Härdi, R.; Triantafyllidis, Z.; Meier, U.; Terrasi, G. Pietro Fatigue and durability of laminated carbon fibre reinforced polymer straps for bridge suspenders. Polymers
**2018**, 10, 169. [Google Scholar] [CrossRef] [PubMed][Green Version] - Tobajas, R.; Elduque, D.; Ibarz, E.; Javierre, C.; Canteli, A.F.; Gracia, L. Visco-hyperelastic model with damage for simulating cyclic thermoplastic elastomers behavior applied to an industrial component. Polymers
**2018**, 10, 668. [Google Scholar] [CrossRef][Green Version]

**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 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## 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