A macro-micro elasticity-diffusion system modeling absorption-induced swelling in rubber foams: Proof of the strong solvability
Authors:
Toyohiko Aiki, Nils Hendrik Kröger and Adrian Muntean
Journal:
Quart. Appl. Math. 79 (2021), 545-579
MSC (2020):
Primary 35Q74; Secondary 74F25, 74E40
DOI:
https://doi.org/10.1090/qam/1592
Published electronically:
April 8, 2021
MathSciNet review:
4288596
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Additional Information
Abstract:
In this article, we propose a macro-micro (two-scale) mathematical model for describing the macroscopic swelling of a rubber foam caused by the microscopic absorption of some liquid. In our modeling approach, we suppose that the material occupies a one-dimensional domain which swells as described by the standard beam equation including an additional term determined by the liquid pressure. As special feature of our model, the absorption takes place inside the rubber foam via a lower length scale, which is assumed to be inherently present in such a structured material. The liquid’s absorption and transport inside the material is modeled by means of a nonlinear parabolic equation derived from Darcy’s law posed in a non-cylindrical domain defined by the macroscopic deformation (which is a solution of the beam equation).
Under suitable assumptions, we establish the existence and uniqueness of a suitable class of solutions to our evolution system coupling the nonlinear parabolic equation posed on the microscopic non-cylindrical domain with the beam equation posed on the macroscopic cylindrical domain. In order to guarantee the regularity of the non-cylindrical domain, we impose a singularity to the elastic response function appearing in the beam equation.
References
- Toyohiko Aiki and Chiharu Kosugi, Numerical schemes for ordinary differential equations describing shrinking and stretching motion of elastic materials, Adv. Math. Sci. Appl. 29 (2020), no. 2, 459–494. MR 4228003
- Toyohiko Aiki, Weak solutions for Falk’s model of shape memory alloys, Math. Methods Appl. Sci. 23 (2000), no. 4, 299–319. MR 1740317, DOI 10.1002/(SICI)1099-1476(20000310)23:4<299::AID-MMA115>3.3.CO;2-4
- Martin Brokate and Jürgen Sprekels, Hysteresis and phase transitions, Applied Mathematical Sciences, vol. 121, Springer-Verlag, New York, 1996. MR 1411908, DOI 10.1007/978-1-4612-4048-8
- D. Bulut, T. Krups, G. Poll, and U. Giese, Lubricant compatibility of FKM seals in synthetic oils, Industrial Lubrication and Tribology 72(2019), 5.
- S. A. Chester, C. V. Di Leo, and L. Anand, A finite element implementation of a coupled diffusion-deformation theory for elastomeric gels, International Journal of Solids and Structures, 52 (2015), 1–18.
- H. Guo, Y. Chen, J. Tao, B. Jia, D. Li, and Y. Zhai, A viscoelastic constitutive relation for the rate-dependent mechanical behavior of rubber-like elastomers based on thermodynamic theory, Materials and Design, 178(2019), 107876.
- Morton E. Gurtin, Eliot Fried, and Lallit Anand, The mechanics and thermodynamics of continua, Cambridge University Press, Cambridge, 2010. MR 2884384, DOI 10.1017/CBO9780511762956
- N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30(1981), 1–87.
- M. Kvick, D. M. Martinez, D. R. Hewitt, and N. J. Balmforth, Imbibition with swelling: Capillary rise in thin deformable porous media, Physical Review Fluids, 2 (2017), 074001.
- O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822, DOI 10.1090/mmono/023
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
- K. Nagdi, Rubber as an Engineering Material: Guideline for Users, Hanser Publishers, Münich, 1993.
- Frank Neff, Alexander Lion, and Michael Johlitz, Modelling diffusion induced swelling behaviour of natural rubber in an organic liquid, ZAMM Z. Angew. Math. Mech. 99 (2019), no. 3, e201700280, 22. MR 3924306, DOI 10.1002/zamm.201700280
- Marek Niezgódka and Irena Pawłow, A generalized Stefan problem in several space variables, Appl. Math. Optim. 9 (1982/83), no. 3, 193–224. MR 687720, DOI 10.1007/BF01460125
- Ralph E. Showalter, Microstructure models of porous media, Homogenization and porous media, Interdiscip. Appl. Math., vol. 6, Springer, New York, 1997, pp. 183–202, 259–275. MR 1434324, DOI 10.1007/978-1-4612-1920-0_{9}
- N. Sombatsompop, and P. Lertkamolsin, Effects of chemical blowing agents on swelling properties of expanded elastomers, Journal of Elastomers and Plastics, 32(2000), 4, 311–328.
- A. Ricker, N. H. Kröger, M. Ludwig, R. Landgraf, and J. Ihlemann, Validation of a hyperelastic modelling approach for cellular rubber, Constitutive Models for Rubber, XI, 249–254.
References
- Toyohiko Aiki and Chiharu Kosugi, Numerical schemes for ordinary differential equations describing shrinking and stretching motion of elastic materials, Adv. Math. Sci. Appl. 29 (2020), no. 2, 459–494. MR 4228003
- Toyohiko Aiki, Weak solutions for Falk’s model of shape memory alloys, Math. Methods Appl. Sci. 23 (2000), no. 4, 299–319. MR 1740317, DOI 10.1002/(SICI)1099-1476(20000310)23:4$\langle$299::AID-MMA115$\rangle$3.3.CO;2-4
- Martin Brokate and Jürgen Sprekels, Hysteresis and phase transitions, Applied Mathematical Sciences, vol. 121, Springer-Verlag, New York, 1996. MR 1411908, DOI 10.1007/978-1-4612-4048-8
- D. Bulut, T. Krups, G. Poll, and U. Giese, Lubricant compatibility of FKM seals in synthetic oils, Industrial Lubrication and Tribology 72(2019), 5.
- S. A. Chester, C. V. Di Leo, and L. Anand, A finite element implementation of a coupled diffusion-deformation theory for elastomeric gels, International Journal of Solids and Structures, 52 (2015), 1–18.
- H. Guo, Y. Chen, J. Tao, B. Jia, D. Li, and Y. Zhai, A viscoelastic constitutive relation for the rate-dependent mechanical behavior of rubber-like elastomers based on thermodynamic theory, Materials and Design, 178(2019), 107876.
- Morton E. Gurtin, Eliot Fried, and Lallit Anand, The mechanics and thermodynamics of continua, Cambridge University Press, Cambridge, 2010. MR 2884384, DOI 10.1017/CBO9780511762956
- N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30(1981), 1–87.
- M. Kvick, D. M. Martinez, D. R. Hewitt, and N. J. Balmforth, Imbibition with swelling: Capillary rise in thin deformable porous media, Physical Review Fluids, 2 (2017), 074001.
- O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). MR 0241822
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
- K. Nagdi, Rubber as an Engineering Material: Guideline for Users, Hanser Publishers, Münich, 1993.
- Frank Neff, Alexander Lion, and Michael Johlitz, Modelling diffusion induced swelling behaviour of natural rubber in an organic liquid, ZAMM Z. Angew. Math. Mech. 99 (2019), no. 3, e201700280, 22. MR 3924306, DOI 10.1002/zamm.201700280
- Marek Niezgódka and Irena Pawłow, A generalized Stefan problem in several space variables, Appl. Math. Optim. 9 (1982/83), no. 3, 193–224. MR 687720, DOI 10.1007/BF01460125
- Ralph E. Showalter, Microstructure models of porous media, Homogenization and porous media, Interdiscip. Appl. Math., vol. 6, Springer, New York, 1997, pp. 183–202, 259–275. MR 1434324, DOI 10.1007/978-1-4612-1920-0_9
- N. Sombatsompop, and P. Lertkamolsin, Effects of chemical blowing agents on swelling properties of expanded elastomers, Journal of Elastomers and Plastics, 32(2000), 4, 311–328.
- A. Ricker, N. H. Kröger, M. Ludwig, R. Landgraf, and J. Ihlemann, Validation of a hyperelastic modelling approach for cellular rubber, Constitutive Models for Rubber, XI, 249–254.
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Additional Information
Toyohiko Aiki
Affiliation:
Department of Mathematical and Physical Sciences, Japan Women’s University, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo 112-8681, Japan
Address at time of publication:
Department of Mathematics and Computer Science, Karlstad, Sweden
MR Author ID:
267313
ORCID:
0000-0002-8172-7134
Email:
aikit@fc.jwu.ac.jp
Nils Hendrik Kröger
Affiliation:
Deutsches Institut für Kautschuktechnologie e. V. (German Institute of Rubber Technology e. V.) Eupener Straße 33, 30519 Hannover, Germany
Address at time of publication:
material prediction GmbH, Nordkamp 24, 26203 Wardenburg, Germany
ORCID:
0000-0001-5336-4869
Email:
n.kroeger@materialprediction.de
Adrian Muntean
Affiliation:
Department of Mathematics and Computer Science, Karlstad, Sweden
MR Author ID:
684703
ORCID:
0000-0002-1160-0007
Email:
adrian.muntean@kau.se
Received by editor(s):
October 6, 2020
Received by editor(s) in revised form:
February 20, 2021
Published electronically:
April 8, 2021
Additional Notes:
This work was partially supported by JSPS KAKENHI Grant Number JP19K03572. Thanks are due to the Knowledge Foundation (project nr. KK 2019-0213) for financially supporting this research
Article copyright:
© Copyright 2021
Brown University
