Local existence of a solution to a free boundary problem describing migration into rubber with a breaking effect

1.
Faculty of Education, Nagasaki University, Nagasaki, Japan
2.
Department of Mathematics, Faculty of Sciences, Japan Women's University, Tokyo, Japan
3.
Department of Mathematics and Computer Science, Karlstad University, Karlstad, Sweden

Received: 31 December 2021 Revised: 27 July 2022 Accepted: 26 September 2022 Published: 24 October 2022

We consider a one-dimensional free boundary problem describing the migration of diffusants into rubber. In our setting, the free boundary represents the position of the front delimitating the diffusant region. The growth rate of this region is described by an ordinary differential equation that includes the effect of breaking the growth of the diffusant region. In this specific context, the breaking mechanism is should be perceived as a non-dissipative way of describing eventual hyperelastic response to a too fast diffusion penetration. In recent works, we considered a similar class of free boundary problems modeling diffusants penetration in rubbers, but without attempting to deal with the possibility of breaking or accelerating the occurring free boundaries. For simplified settings, we were able to show the global existence and uniqueness as well as the large time behavior of the corresponding solutions to our formulations. Since here the breaking effect is contained in the free boundary condition, our previous results are not anymore applicable. The main mathematical obstacle in ensuring the existence of a solution is the non-monotonic structure of the free boundary. In this paper, we establish the existence and uniqueness of a weak solution to the free boundary problem with breaking effect and give explicitly the maximum value that the free boundary can reach.
Citation: Kota Kumazaki, Toyohiko Aiki, Adrian Muntean. Local existence of a solution to a free boundary problem describing migration into rubber with a breaking effect[J]. Networks and Heterogeneous Media, 2023, 18(1): 80-108. doi: 10.3934/nhm.2023004

Related Papers:

Abstract

We consider a one-dimensional free boundary problem describing the migration of diffusants into rubber. In our setting, the free boundary represents the position of the front delimitating the diffusant region. The growth rate of this region is described by an ordinary differential equation that includes the effect of breaking the growth of the diffusant region. In this specific context, the breaking mechanism is should be perceived as a non-dissipative way of describing eventual hyperelastic response to a too fast diffusion penetration. In recent works, we considered a similar class of free boundary problems modeling diffusants penetration in rubbers, but without attempting to deal with the possibility of breaking or accelerating the occurring free boundaries. For simplified settings, we were able to show the global existence and uniqueness as well as the large time behavior of the corresponding solutions to our formulations. Since here the breaking effect is contained in the free boundary condition, our previous results are not anymore applicable. The main mathematical obstacle in ensuring the existence of a solution is the non-monotonic structure of the free boundary. In this paper, we establish the existence and uniqueness of a weak solution to the free boundary problem with breaking effect and give explicitly the maximum value that the free boundary can reach.

References

[1]	T. Aiki, K. Kumazaki, A. Muntean, A free boundary problem describing migration into rubbers-quest of the large time behavior, Z. Angew. Math. Mech., 102 (2022), e202100134. https://doi.org/10.1002/zamm.202100134 doi: 10.1002/zamm.202100134
[2]	E. Brunier, G. Antonini, Experimental and numerical description of the diffuision of a liquid in a swelling elastomer, In: C. A. Brebbia, G. A. Karamidas, Eds., Computational Methods and Experimental Measurements, Berlin: Springer, 1984,623–632.
[3]	G. Chagnon, E. Verron, L. Gornet, G. Marckmann, P. Charrier, On the relevance of Continuum Damage Mechanics as applied to the Mullins effect in elastomers, J. Mech. phys. Solids, 52 (2004), 1627–1650. https://doi.org/10.1016/j.jmps.2003.12.006 doi: 10.1016/j.jmps.2003.12.006
[4]	N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1–87.
[5]	K. Kumazaki, A. Muntean, Local weak solvability of a moving boundary problem describing swelling along a halfline, Netw. Heterog. Media, 14 (2019), 445–496. https://doi/10.3934/nhm.2019018 doi: 10.3934/nhm.2019018
[6]	K. Kumazaki, A. Muntean, Global weak solvability, continuous dependence on data, and large time growth of swelling moving interfaces, Interfaces Free Bound., 22 (2020), 27–49. https://doi.org/10.4171/ifb/431 doi: 10.4171/ifb/431
[7]	S. Nepal, R. Meyer, N. H. Kröger, T. Aiki, A. Muntean, Y. Wondmagegne, U. Giese, A moving boundary approach of capturing diffusants penetration into rubber: FEM approximation and comparison with laboratory measurements, Kautsch. Gummi, Kunsts., 5 (2020), 61–69.
[8]	Z. Ren, A. S. Verma, Y. Li, J. J.E. Teuwen, Z. Jiang, Offshore wind turbine operations and maintenance: A state-of-the-art review, Renew. Sust. Energ. Rev., 144 (2021), 110886. https://doi.org/10.1016/j.rser.2021.110886 doi: 10.1016/j.rser.2021.110886
[9]	A. Visintin, Introduction to the models of phase transitions, Bollettino dell'Unione Matematica Italiana, 1 (1998), 1–47.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)