Publication Cover
Applicable Analysis
An International Journal
Volume 100, 2021 - Issue 12
64
Views
1
CrossRef citations to date
0
Altmetric
Articles

A problem with viscoelastic mixtures: numerical analysis and computational experiments

ORCID Icon, , & ORCID Icon
Pages 2684-2705 | Received 27 Apr 2018, Accepted 19 Nov 2019, Published online: 06 Dec 2019

References

  • Truesdell C, Toupin RA. The classical field theories. Berlin: Springer-Verlag; 1960. (handbuch der physik; II/3).
  • Kelly P. Reacting continuum. Int J Eng Sci. 1964;2:129–153. doi: 10.1016/0020-7225(64)90001-1
  • Eringen A, Ingram J. A continuum theory of chemically reacting media-i. Int J Eng Sci. 1965;3:197–212. doi: 10.1016/0020-7225(65)90044-3
  • Green AE, Naghdi PM. A dynamical theory of interacting continua. Int J Eng Sci. 1965;3:231–241. doi: 10.1016/0020-7225(65)90046-7
  • Green AE, Naghdi PM. A note on mixtures. Int J Eng Sci. 1968;6:631–635. doi: 10.1016/0020-7225(68)90064-5
  • Müller I. A thermodynamic theory of mixtures of fluids. Arch Ration Mech Anal. 1968;28:1–39. doi: 10.1007/BF00281561
  • Bowen RM, Wiese JC. Diffusion in mixtures of elastic material. Int J Engng Sci. 1969;7:689–722. doi: 10.1016/0020-7225(69)90048-2
  • Bowen RM. Theory of mixtures: continuum physics. Vol. III. New York (NY): Academic Press; 1976. p. 3–120.
  • Atkin RJ, Craine RE. Continuum theories of mixtures: basic theory and historial development. Q J Mech Appl Math. 1972;29:209–245. doi: 10.1093/qjmam/29.2.209
  • Bedford A, Drumhelles DS. Theory of immiscible and structured mixtures. Int J Eng Sci. 1983;21:153–207. doi: 10.1016/0020-7225(83)90071-X
  • Rajagopal KR, Wineman AS. Developments in the mechanics of interactions between a fluid and a highly elastic solid. Essex: Longmans Scientific and Technical; 1990 ( Pitman Research Notes in Mathematics Series; II).
  • Rajagopal KR, Tao L. Mechanics of mixtures. Singapore: World Scientific; 1995.
  • Bedford A, Stern M. A multi-continuum theory for composite elastic materials. Acta Mech. 1972;14:85–102. doi: 10.1007/BF01184851
  • Bedford A, Stern M. Towards a diffusing continuum theory of composite elastic materials. J Appl Mech. 1972;38:8–14. doi: 10.1115/1.3408772
  • Ieşan D, Quintanilla R. Existence and continuous dependence results in the theory of interacting continua. J Elast. 1994;36:85–98. doi: 10.1007/BF00042493
  • Martinez F, Quintanilla R. Some qualitative results for the linear theory of binary mixtures of thermoelastic solids. Collect Math. 1995;46:263–277.
  • Vinson JR, Sierakowski RL. The behaviour of structures composed of composite materials. Dordrecht: Kluwer Academics Publishing; 2002.
  • Elangovan S, Altan BS, Odegard GM. An elastic micropolar mixture theory for predicting elastic properties of cellular materials. Mech Mater. 2008;40:602–614. doi: 10.1016/j.mechmat.2008.02.002
  • Alves MS, Muñoz Rivera JE, Quintanilla R. Exponential decay in a thermoelastic mixture of solids. Int J Solids Struct. 2009;46:1659–1666. doi: 10.1016/j.ijsolstr.2008.12.005
  • Alves MS, Muñoz Rivera JE, Sepúlveda M, et al. Exponential stability in thermoviscoelastic mixtures of solids. Int J Solids Struct. 2009;46:4151–4162. doi: 10.1016/j.ijsolstr.2009.07.026
  • Alves MS, Muñoz Rivera JE, Sepúlveda M, et al. Stabilization in three-dimensional porous thermoviscoelastic mixtures. Q J Math. 2013;64:37–57. doi: 10.1093/qmath/har029
  • Fernández-Sare H, Muñoz Rivera J, Quintanilla R. On the rate of decay in interacting continua with memory. J Differ Equ. 2011;251:3583–3605. doi: 10.1016/j.jde.2011.08.015
  • Ieşan D. A theory of mixtures of nonsimple elastic solids. Int J Eng Sci. 1992;30:317–328. doi: 10.1016/0020-7225(92)90076-S
  • Ieşan D. Continuous dependence in a nonlinear theory of viscoelastic porous mixtures. Int J Eng Sci. 2006;44:1127–1145. doi: 10.1016/j.ijengsci.2006.07.001
  • Ieşan D, Quintanilla R. A theory of porous thermoviscoelastic mixtures. J Therm Stress. 2007;30:693–714. doi: 10.1080/01495730701212880
  • Muñoz Rivera J, Naso MG, Quintanilla R. Decay of solutions for a mixture of thermoelastic one-dimensional solids. Comput Math Appl. 2013;66:263–277. doi: 10.1016/j.camwa.2013.03.022
  • Muñoz Rivera J, Naso MG, Quintanilla R. Decay of solutions for a mixture of thermoelastic solids with different temperatures. Comput Math Appl. 2016;71:991–1009. doi: 10.1016/j.camwa.2016.01.010
  • Quintanilla R. Existence and exponential decay in the linear theory of viscoelastic mixtures. Eur J Mech A Solids. 2005;24:311–324. doi: 10.1016/j.euromechsol.2004.11.008
  • Copetti MIM, Fernández JR, Masid M. Numerical analysis of a viscoelastic mixture problem. Int J Solids Struct. 2016;80:393–404. doi: 10.1016/j.ijsolstr.2015.09.014
  • Fernández JR, Magaña A, Masid M. On the viscoelastic mixtures of solids. Appl Math Optim. 2019;79(2):309–326. doi: 10.1007/s00245-017-9439-8
  • Ciarlet PG. The finite element method for elliptic problems: handbook of numerical analysis. North-Holland; 1991. p. 17–352. (Handbook of numerical analysis; II).
  • Campo M, Fernández JR, Kuttler KL, et al. Numerical analysis and simulations of a dynamic frictionless contact problem with damage. Comput Methods Appl Mech Eng. 2006;196:476–488. doi: 10.1016/j.cma.2006.05.006
  • Barboteu M, Fernández JR, Hoarau-Mantel T-V. A class of evolutionary variational inequalities with applications in viscoelasticity. Math Models Methods Appl Sci. 2005;15(10):1595–1617. doi: 10.1142/S0218202505000820

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.