Hausdorff Calculus by Yingjie Liang Wen Chen Wei Cai

Author:Yingjie Liang, Wen Chen, Wei Cai
Language: eng
Format: epub
Publisher: Walter de Gruyter
Published: 2019-06-15T00:00:00+00:00

6.3.6 Remarks

Studies find that the normal diffusion can be linked to the anomalous one with scaling transformation. Enlightened by such a relationship, the relaxation modulus of the classical Maxwell model is modified with law. It is observed that the proposed formulation well fits the experimental data with high accuracy, and appears similar to the well-known KWW stretched exponential function. Such function has also been generalized to characterize complex relaxation behaviors whose limits deviate from 0. The efficiency of the present reformulation has been validated by experimental data at different time scales.

The same modification is also applied to the creep compliance. In comparison with the traditional models, the proposed models are observed to achieve higher fitting accuracy. In addition, the modified formulation only requires one more parameter α without cumbersome parameters. Thus, the classical Maxwell, Kelvin, and Zener viscoelastic models have been reactivated to describe complex rheological behaviors with the proposed methodology of scaling transformation.

The generalized relaxation and creep functions can also rigorously be derived from the fractal derivative Zener model, which is found capable to characterize either the creep behavior with an initial value, or the relaxation process whose limit deviates from 0. The rigorously derived creep compliance and relaxation modulus are also found in the same form as those modified with . It has been further observed that the modified relaxation modulus of the Maxwell model and the creep compliance of the Kelvin model can be directly obtained from the corresponding fractal derivative Maxwell and Kelvin models. Thus, it can be concluded that the fractal derivative operator is inherently consistent with scaling transformation in time and is capable to describe anomalous rheological behaviors of complex viscoelastic media.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.