A physical interpretation of fractional viscoelasticity based on the fractal structure of media: Theory and experimental validation

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Mashayekhi, S. and Hussaini, M.Y.
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In this work, a physical connection between the fractional time derivative and fractal geometry of fractal media is developed and applied to viscoelasticity and thermal diffusion in elastomers. Integral to this formulation is the application of both the fractal dimension and the spectral dimension which characterizes diffusion in fractal media. The methodology extends the generalized molecular theory of Rouse and Zimm where generalized Gaussian structures (GGSs) replace the Rouse matrix with the generalized Gaussian Rouse matrix (GRM). Importantly, the Zimm model is extended to fractal media where the new relaxation formulation contains internal state variables that naturally depend on the fractional time derivative of deformation. Through the use of thermodynamic laws in fractal media, we derive the linear fractional model of viscoelasticity based on both spectral and fractal dimensions. This derivation shows how the order of the fractional derivative in the linear fractional model of viscoelasticity is a rate dependent material property that is strongly correlated with fractal and spectral dimensions in fractal media. To validate the correlation between fractional rates and fractal material structure, we measure the viscoelasticity and thermal diffusion of two different dielectric elastomers: Very High Bond (VHB) 4910 and VHB 4949. Using Bayesian uncertainty quantification (UQ) based on uniaxial stress–strain measurements, the fractional order of the derivative in the linear fractional model of viscoelasticity is quantified. Two dimensional fractal dimensions are also independently quantified using the box counting method. Lastly, the diffusion equation in fractal media is inferred from experiments using Bayesian UQ to quantify the spectral dimension by heating the polymer locally with a laser beam and quantifying thermal diffusion. Comparing theory to experiments, a strong correlation is found between the viscoelastic fractional order obtained from stress–strain measurements in comparisons with independent predictions of fractional viscoelasticity based on the fractal structure and fractional thermal diffusion rates.
Journal of Mechanics and Physics of Science
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