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SUMMARY Seismic attenuation (quantified by the quality factor Q) has a significant impact on the seismic waveforms, especially in the fluid-saturated rocks. This dissipative process can be phenomenologically represented by viscoelastic models. Previous seismological studies show that the Q value of Earth media exhibits a nearly frequency-independent behaviour (often referred to as constant-Q in literature) in the seismic frequency range. Such attenuation can be described by the mathematical Kjartansson constant-Q model, which lacks of a physical representation in the viscoelastic sense. Inspired by the fractal nature of the pore fluid distribution in patchy-saturated rocks, here we propose two fractal mechanical network (FMN) models, that is, a fractal tree model and a quasi-fractal ladder model, to phenomenologically represent the frequency-independent Q behaviour. As with the classic viscoelastic models, the FMN models are composed of mechanical elements (spring and dashpots) arranged in different hierarchical patterns. A particular parametrization of each model can produce the same complex modulus as in the Kjartansson model, which leads to the constant-Q. Applying the theory to several typical rock samples, we find that the seismic attenuation signature of these rocks can be accurately represented by either one of the FMN models. Besides, we demonstrate that the ladder model in particular exhibits the realistic multiscale fractal structure of the saturated rocks. Therefore, the FMN models as a proxy could provide a new way to estimate the microscopic rock structure property from macroscopic seismic attenuation observation.
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Physically motivated models of polymer networks with dynamic cross-links: comparative study and future outlook
Polymer networks consisting of a mixture of chemical and physical cross-links are known to exhibit complex time-dependent behaviour due to the kinetics of bond association and dissociation. In this article, we highlight and compare two recent physically based constitutive models that describe the nonlinear viscoelastic behaviour of such transient networks. These two models are developed independently by two groups of researchers using different mathematical formulations. Here, we show that this difference can be attributed to different viewpoints: Lagrangian versus Eulerian. We establish the equivalence of the two models under the special situation where chains obey Gaussian statistics and steady-state bond dynamics. We provide experimental data demonstrating that both models can accurately predict the time-dependent uniaxial behaviour of a poly(vinylalcohol) dual cross-link hydrogel. We review the advantages and disadvantages of both approaches in applications and close by discussing a list of open challenges and questions regarding the mathematical modelling of soft, viscoelastic networks.
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- PAR ID:
- 10328247
- Date Published:
- Journal Name:
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Volume:
- 477
- Issue:
- 2255
- ISSN:
- 1364-5021
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1098/rspa.2021.0608
