## Abstract

In this paper, we present a formulation of fractional viscoelasticity of nonlinear time-dependent responses of isotropic materials undergoing small deformation gradients. The model considers the separation of functions of the time-dependent kernel and stress-dependent nonlinear elastic strain measure. The Riemann–Liouville fractional integral is considered for the time-dependent kernel function. Characterization of material parameters in the fractional viscoelasticity model is presented using experimental data on a polymer. The nonuniqueness of the calibrated material parameters from the fractional power terms is discussed. A numerical method is also presented to solve the nonlinear fractional viscoelastic constitutive model. The response characteristics and convergence behaviors of the presented nonlinear fractional viscoelastic constitutive model are compared to the corresponding nonlinear model derived based on classical viscoelasticity. The presented nonlinear viscoelastic fractional model is shown to be capable of describing multiaxial responses of polymers under various loading histories. The fractional model has significantly fewer material parameters, which can offer an advantage when a relatively long-term response of materials is of interest. However, the model is computationally more expensive when compared to the classical viscoelastic model based on the Prony series kernel function, which can hinder its practical use in solving rather complex boundary value problems.

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## Data Availability

Data available on request from the authors.

## Notes

The time-dependent kernel function must be a positive, continuous, and increasing function of time in case of creep function whereas a decreasing function of time for the relaxation function.

In a linear viscoelastic material, there are two sources of linearity. A linearized strain measure is used, i.e., only the linear term of the displacement gradients is accounted for. The responses are proportional to the inputs and can be obtained by superimposing the responses of several different inputs. In a linear viscoelastic model the expression for the time-dependent stress and strain is interchangeable.

If we define the normalized time history \(\hat{s} = s/\tau _{c}\), then the second term of Eq. (2.13) becomes \(\int _{0}^{t^{*}/\tau _{c}} D\left ( \frac{t}{\tau _{c}} - \hat{s} \right )\frac{df}{d\sigma } \frac{d\sigma }{d\hat{s}}\tau _{c}d\hat{s} \approx 0\).

In the limit of the denominator going to zero, \(\lim _{x \to 0}\frac{e^{\lambda x} - 1}{x} = \lim _{x \to 0}\frac{\lambda e^{\lambda x}}{1} = \lambda \). Since \(\lambda \) is constant, at zero stresses, \(\lambda 0=0\).

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## Acknowledgements

The research work of this paper was performed within the framework of the COMET-program of the Austrian Ministry of Traffic, Innovation, and Technology. We would like to thank Dr. Daniel Tscharnuter for carrying out the experimental test. This work was partly supported by the National Science Foundation under grant CMMI-1761015.

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## Appendix: Experimental study on polyoxymethylene (POM)

### Appendix: Experimental study on polyoxymethylene (POM)

The investigated polyoxymethylene (POM) is Tenac 3010 produced by Asahi Kasei Corporation (Tokyo, Japan). ISO 3176 type B specimens were injection-molded (Tscharnuter and Muliana 2013). The POM specimens were subjected to uniaxial tensile tests under various histories: quasistatic ramp loadings with constant strain rates and creep under constant stresses as well as recovery tests. Tensile tests at room temperature with constant rates of 2.5 and 25 MPa/sec were performed using an MTS servohydraulic testing machine. The creep tests were performed using force control with initial loading at a prescribed force rate. The recovery from monotonic tension was measured directly on the testing machine. Due to the extensive duration of the recovery, which was monitored for 2.5 months after 24 and 72 hours of creep, the recovery of the strain associated with the viscoelastic response was measured on a separate device. Axial and transverse strains were measured using the digital image correlation system ARAMIS (GOM mbH, Germany. For this study, the measurement was performed in three-dimensional (3D) mode using 105-mm lenses. A detailed discussion on the experimental tests can be found in Tscharnuter and Muliana (2013).

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Muliana, A. A fractional model of nonlinear multiaxial viscoelastic behaviors.
*Mech Time-Depend Mater* **27**, 1187–1207 (2023). https://doi.org/10.1007/s11043-022-09542-3

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DOI: https://doi.org/10.1007/s11043-022-09542-3