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A study on the Gurtin–Murdoch model for spherical solids with surface tension

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Abstract

An investigation is made on the Gurtin–Murdoch (GM) model for soft spherical elastic solids with surface tension-induced bulk residual stress. Unlike the original GM model which treats the surface tension as a finite value but the surface tension-induced bulk residual stress as infinitesimal, the present model treats both consistently as finite values. Detailed comparisons between the present model and the GM model show that for a solid sphere (for which the surface tension-induced bulk residual stress remains uniform in the entire body), the GM model deviates considerably from the present model. Conversely, for a small cavity in an infinite body (for which the surface tension-induced bulk residual stress is significant only nearby the cavity’s surface but quickly diminishes with radial distance), the deviation between the present model and the original GM model is negligible. It is concluded that the surface tension-induced bulk residual stress, which is addressed in the present paper but ignored in the GM model, can have a significant effect on deformation and vibration of soft elastic solids when its magnitude does not vanishingly diminish within the body.

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Acknowledgements

Yang thanks the support of the China Scholarship Council, and Gao acknowledges the support of the National Natural Science Foundation of China (No.11872203), Joint Fund of Advanced Aerospace Manufacturing Technology Research (U1937601), National Natural Science Foundation of China for Creative Research Groups (No. 51921003). Ru thanks the support of Natural Science and Engineering Research Council of Canada (NSERC-RGPIN204992).

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Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Guang Yang & Cun-Fa Gao

  2. Department of Mechanical Engineering, University of Alberta, Edmonton, AL, T6G 1H9, Canada

    Guang Yang & C. Q. Ru

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  1. Guang Yang
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  3. C. Q. Ru
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Correspondence to C. Q. Ru.

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Yang, G., Gao, CF. & Ru, C.Q. A study on the Gurtin–Murdoch model for spherical solids with surface tension. Z. Angew. Math. Phys. 72, 95 (2021). https://doi.org/10.1007/s00033-021-01502-0

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