# Fractional Calculus Approach to Reproduce Material Viscoelastic Behavior, including the Time–Temperature Superposition Phenomenon

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fractional Derivative Model in Poles–Zeros Formulation

## 3. Time–Temperature Superposition

_{T}(T, T

_{r}) and b

_{T}(T, T

_{r}) are coefficients that indicate the amount of horizontal and vertical shifting (respectively) to be applied to isotherms of storage and loss moduli measured at a temperature T in order to estimate the material properties at a reference temperature T

_{r}, as qualitatively shown in Figure 3 for explanatory purposes.

_{T}(T, T

_{r}) describe the temperature dependence of the relaxation time and usually follow the empirical Williams–Landel–Ferry (WLF) law (7):

_{1}and C

_{2}are empirical constants whose order of magnitude is about 10 and 100 K, respectively.

_{T}(T, T

_{r}) are related to thermal expansion effects, which for most polymers can be neglected due to their small variation, and that, for this reason, will be neglected in the following. Here, it is worth noting that this hypothesis is highly acceptable in the viscoelastic regions where the frequency/time dependence of material functions is sharp. On the other hand, overlooking thermal vertical shifts in viscoelastic regions with weak frequency/time dependence may lead to different values of horizontal shifting whose accuracy depends on the material under investigation.

_{R}.

_{1}and C

_{2}) have been measured and calculated, respectively. To transform the FDGM in poles–zeros formulation for two different temperatures, T

_{r}an T

_{1}, the standard procedure consists of the following steps:

- Identification of poles and zeros at a reference temperature T
_{r}, applying the scheme represented in Figure 2; - Application of the WLF formulation to the experimental curves referring to a reference temperature T
_{r}to obtain the experimental curves at the new temperature T_{1}, shifting all frequency vector ω_{exp}; - Identification of the poles and zeros starting from the experimental curves referring to the new temperature T
_{1}by means of the procedure summarized in Figure 2.

- Identification of the poles and zeros at reference temperature T
_{r}, with the procedure described in Figure 2; - Application of the WLF law directly on the identified poles and zeros, obtaining the master curves at the new temperature T
_{1}.

_{T}= 1 when T = T

_{rif}.

_{rif}, it is possible to write the WLF function applied to the poles–zeros formulation as (10):

_{1}and C

_{2}and adopting FDGM with three elements (N = 3). It should be noted that for an FDGM with three elements, the subscript k in (10) is three. This means that, for each temperature, three poles and three zeros are identified, and each of them satisfies Equation (10).

## 4. Material Parameters’ Global Identification

#### 4.1. Identification Procedure

_{rif}is the temperature of the isotherm curve that presents the maximum value of the tan δ (i.e., for compound A, T

_{rif}= −20 °C).

- 3N + 1 parameters for the pole, zero, and static modulus (9);
- Two parameters for the WLF (10).

_{1}and C

_{2}in (10), they have been assumed to be 17.44 and 51.60 respectively, which are generally accepted values when T

_{rif}= T

_{g}[26].

#### 4.2. Results

_{rif}= −20° (T − T

_{0}= 0), the procedure is quite accurate, while for temperatures far from this value, some differences appear from the experimental data. These small differences (<10%) are due to the vertical shift factors b

_{T}(T, T

_{r}) that, in this work, were neglected. Figure 9a,b and Figure 10a,b show the results for the compound B and C, respectively. For both compounds, T

_{rif}= −20°.

_{1}and C

_{2}of the WLF law (10).

## 5. Mater Curves and Time–Temperature Superposition Parameters’ Estimation with Partial Experimental Data

_{1}and C

_{2}; however, this difference implies a small difference in the calculation of a

_{T}in the temperature range exanimated, as shown in Figure 12a,b.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Parameter Values Obtained with Pole–Zero Identification Procedure

#### Appendix A.1. Fractal Derivative Generalized Maxwell Model Parameters

Element | Compound A | Compound B | Compound C | ||||||
---|---|---|---|---|---|---|---|---|---|

E0 (MPa) | 13.24 | 25.38 | 17.3 | ||||||

Element | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 |

Pole | 0.30 | 4.13 | 13.00 | −1.29 | 1.05 | 3.12 | −3.94 | 0.30 | 1.83 |

Zero | −6.96 | 0.30 | 4.13 | −6.50 | −1.29 | 1.05 | −8.15 | −2.57 | 0.30 |

Gamma | 0.10 | 0.37 | 0.01 | 0.15 | 0.44 | 0.18 | 0.13 | 0.49 | 0.17 |

**Table A2.**Compounds A, B, and C: parameters of FDGM models composed of 3 elements using a reduced number of DMA experimental data.

Element | Compound A | Compound B | Compound C | ||||||
---|---|---|---|---|---|---|---|---|---|

E0 (MPa) | 12.62 | 22.99 | 15.89 | ||||||

Element | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 |

Pole | −1.50 | 3.59 | 5.27 | −1.22 | 0.96 | 2.62 | −2.92 | 0.35 | 1.32 |

Zero | −8.62 | 0.39 | 4.93 | −8.30 | −1.22 | 0.96 | −8.02 | −2.92 | 0.35 |

Gamma | 0.11 | 0.40 | 0.41 | 0.13 | 0.45 | 0.23 | 0.11 | 0.46 | 0.21 |

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**Figure 2.**Constrained nonlinear optimization procedure to identify poles–zeros coefficients: functional scheme.

**Figure 3.**Determination of the frequency–temperature superposition shifting factor a

_{T}and b

_{T}: (

**a**) Isotherms of storage modulus on the frequency range measurable by DMA, at temperatures T

_{1}, T

_{2}, and T

_{r}, with T

_{1}< T

_{r}< T

_{2}; (

**b**) Isotherms of storage modulus after application of the horizontal shift factors, taking T

_{r}as the reference temperature; (

**c**) Isotherms of storage modulus after application of both horizontal and vertical shift factors, taking T

_{r}as the reference temperature [28].

**Figure 8.**Compound A: Storage modulus and loss tangent (

**a**), WLF law (

**b**)—experimental data (dashed line) vs. FDGM model.

**Figure 9.**Compound B: Storage modulus and loss tangent (

**a**), WLF law (

**b**)—experimental data (dashed line) vs. FDGM model.

**Figure 10.**Compound C: Storage modulus and loss tangent (

**a**), WLF law (

**b**)—experimental data (dashed line) vs. FDGM model.

**Figure 11.**Example of the experimental starting set for the identification with partial DMA for compound C.

**Figure 12.**Compound C: Storage modulus and loss tangent (

**a**), WLF law (

**b**)—experimental data (dashed line) vs. FDGM model complete (light blue) and FDGM obtained with partial staring experimental dataset (red).

I T_{g}(°C) | II T_{g}(°C) | |
---|---|---|

Compound A | - | 9 |

Compound B | −33 | 8 |

Compound C | −20 | 9 |

NRMSE E′ (MPa) | NRMSE tanδ (-) | NRMSE Total | T_{rif}(°C) | C_{1} | C_{2} | |
---|---|---|---|---|---|---|

Compound A | 0.0108 | 0.0637 | 0.0372 | −20 | 23.36 | 158.21 |

Compound B | 0.0486 | 0.0442 | 0.0464 | −20 | 13.19 | 55.12 |

Compound C | 0.0402 | 0.0406 | 0.0404 | −20 | 17.31 | 56.24 |

**Table 3.**NRMSEs and identified WLF coefficients for compounds A, B, and C in case of partial DMA data.

NRMSE E′ (MPa) | NRMSE tanδ (-) | NRMSE Total | T_{rif}(°C) | C_{1} | C_{2} | |
---|---|---|---|---|---|---|

Compound A | 0.0339 | 0.0568 | 0.0454 | −20 | 30.24 | 203.53 |

Compound B | 0.012 | 0.057 | 0.0345 | −20 | 16.13 | 65.04 |

Compound C | 0.011 | 0.11 | 0.0617 | −20 | 12.15 | 53.64 |

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**MDPI and ACS Style**

Genovese, A.; Farroni, F.; Sakhnevych, A.
Fractional Calculus Approach to Reproduce Material Viscoelastic Behavior, including the Time–Temperature Superposition Phenomenon. *Polymers* **2022**, *14*, 4412.
https://doi.org/10.3390/polym14204412

**AMA Style**

Genovese A, Farroni F, Sakhnevych A.
Fractional Calculus Approach to Reproduce Material Viscoelastic Behavior, including the Time–Temperature Superposition Phenomenon. *Polymers*. 2022; 14(20):4412.
https://doi.org/10.3390/polym14204412

**Chicago/Turabian Style**

Genovese, Andrea, Flavio Farroni, and Aleksandr Sakhnevych.
2022. "Fractional Calculus Approach to Reproduce Material Viscoelastic Behavior, including the Time–Temperature Superposition Phenomenon" *Polymers* 14, no. 20: 4412.
https://doi.org/10.3390/polym14204412