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Some Remarks on the Effects of Inertia and Viscous Dissipation in the Onset of Cavitation in Rubber

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Abstract

Through direct comparisons with experiments, Lefèvre et al. (Int. J. Frac. 192:1–23, 2015) have recently confirmed the prevailing belief that the nonlinear elastic properties of rubber play a significant role in the so-called phenomenon of cavitation—that is, the sudden growth of inherent defects in rubber into large enclosed cavities/cracks in response to external stimuli. These comparisons have also made it plain that cavitation in rubber is first and foremost a fracture process that may possibly depend, in addition to the nonlinear elastic properties of the rubber, on inertial effects and/or on the viscous dissipation innate to rubber. This is because the growth of defects into large cavities/cracks is locally in time an extremely fast process.

The purpose of this Note is to provide insight into the relevance of inertial and viscous dissipation effects on the onset of cavitation in rubber. To this end, leaving fracture properties aside, we consider the basic problem of the radially symmetric dynamic deformation of a spherical defect embedded at the center of a ball made up of an isotropic incompressible nonlinear viscoelastic solid that is subjected to external hydrostatic loading. Specifically, the defect is taken to be vacuous and the viscoelastic behavior of the solid is characterized by a fairly general class of constitutive relations given in terms of two thermodynamic potentials—namely, a free energy function describing the nonlinear elasticity of the solid and a dissipation potential describing its viscous dissipation—which has been shown to be capable to describe the mechanical response of a broad variety of rubbers over wide ranges of deformations and deformations rates. It is found that the onset of cavitation is not affected by inertial effects so long as the external loads are not applied at a high rate. On the other hand, even when the external loads are applied quasi-statically, viscous dissipation can greatly affect the critical values of the applied loads at which cavitation ensues.

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Notes

  1. The focus of this Note is on homogeneous solids thus the choice of constant mass density \(\rho_{0}\).

  2. As is the case in elastodynamics, accounting for a pressurized cavity would not pose any additional difficulty.

  3. Throughout this work we consider isothermal conditions.

  4. The function (14) can be alternatively written as \(r(R,t)= ( 1+ \frac{a^{3}(t)-A^{3}}{R^{3}} ) ^{1/3}R\) in terms of the inner radius \(a(t)\). For our purposes here, we find dealing with the form (14) in terms of the outer radius \(b(t)\) more convenient.

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Acknowledgements

This work was supported by the National Science Foundation through the Grant CMMI–1235352. D. A.-I. is also indebted to the Spanish Ministry of Economy and Competitiveness (Projects EUIN2015-62556 and DPI2014-57989-P) for the financial support received which allowed him to conduct part of this work.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Illinois, Urbana–Champaign, IL, 61801-2352, USA

    Aditya Kumar & Oscar Lopez-Pamies

  2. Department of Continuum Mechanics and Structural Analysis, University Carlos III of Madrid, 28911, Leganés, Spain

    Damian Aranda-Iglesias

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  1. Aditya Kumar
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  2. Damian Aranda-Iglesias
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  3. Oscar Lopez-Pamies
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Correspondence to Oscar Lopez-Pamies.

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Kumar, A., Aranda-Iglesias, D. & Lopez-Pamies, O. Some Remarks on the Effects of Inertia and Viscous Dissipation in the Onset of Cavitation in Rubber. J Elast 126, 201–213 (2017). https://doi.org/10.1007/s10659-016-9589-y

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