Homogenization of a poro-elasticity model coupled withdiffusive transport and a first order reaction for concrete

Michael Eden; Michael Böhm

doi:10.3934/nhm.2014.9.599

Article Contents

2014, Volume 9, Issue 4: 599-615. Doi: 10.3934/nhm.2014.9.599

This issue Previous Article A review of non local continuum damage: Modelling of failure? Next Article Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity

Homogenization of a poro-elasticity model coupled with diffusive transport and a first order reaction for concrete

Michael Eden^1, and
Michael Böhm^1,

1.
Centre for Industrial Mathematics, FB 3, University of Bremen, Bibliotheksstr., 28201, Bremen

Received: April 2014

Revised: September 2014

Published online: December 2014

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Abstract

We study a two-scale homogenization problem describing the linearized poro-elastic behavior of a periodic two-component porous material exhibited to a slightly compressible viscous fluid flow and a first-order chemical reaction. One material component consists of disconnected parts embedded in the other component which is supposed to be connected. It is shown that a memory effect known from the purely mechanic problem gets inherited by the reaction component of the model.

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Mathematics Subject Classification: Primary: 35B27, 74F25, 74F10; Secondary: 76M50.

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References

[1]	A. Ainouz, Homogenization of a double porosity model in deformable media, Electronic Journal of Differential Equations, 90 (2013), 1-18.
[2]	G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518.doi: 10.1137/0523084.
[3]	G. Allaire, A. Damlamian and U. Hornung, Two-scale convergence on periodic surfaces and applications, in Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media, World Scintific publication, Singapore, (1995), 15-25.
[4]	T. Arbogast, J. Douglas and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM Journal on Mathematical Analysis, 21 (1990), 823-836.doi: 10.1137/0521046.
[5]	M. Biot, General theory of three-dimensional consolidation, Journal of applied physics, 12 (1941), 155-164.doi: 10.1063/1.1712886.
[6]	O. Coussy, Poromechanics, 2nd edition, Wiley, 2005, URL http://amazon.com/o/ASIN/0470849207/.doi: 10.1002/0470092718.
[7]	H. Deresiewicz and R. Skalak, On uniqueness in dynamic poroelasticity, Bulletin of the Seismological Society of America, 53 (1963), 783-788.
[8]	M. Eden, Poroelasticity, Master's thesis, University of Bremen, 2014.
[9]	I. Graf, M. Peter and J. Sneyd, Homogenization of a nonlinear multiscale model of calcium dynamics in biological cells, Journal of Mathematical Analysis and Applications, 419 (2014), 28-47.doi: 10.1016/j.jmaa.2014.04.037.
[10]	U. Hornung and W. Jäger, Diffusion, convection, adsorption, and reaction of chemicals in porous media, Journal of differential equations, 92 (1991), 199-225.doi: 10.1016/0022-0396(91)90047-D.
[11]	A. Meirmanov and R. Zimin, Compactness result for periodic structures and its application to the homogenization of a diffusion-convection equation, Electronic Journal of Differential Equations, 115 (2011), 1-11.
[12]	S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 13 (2003), 43-63.
[13]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Journal on Mathematical Analysis, 20 (1989), 608-623.doi: 10.1137/0520043.
[14]	G. Pavliotis and A. Stuart, Multiscale Methods: Averaging and Homogenization (Texts in Applied Mathematics), Springer, New York, 2008. URL http://amazon.com/o/ASIN/1441925325/.
[15]	M. Peter and M. Böhm, Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium, Mathematical Methods in the Applied Sciences, 31 (2008), 1257-1282.doi: 10.1002/mma.966.
[16]	R. Showalter, Distributed microstructure models of porous media, in Flow in porous media, Springer, 114 (1993), 155-163.
[17]	R. Showalter, Hilbert Space Methods in Partial Differential Equations (Dover Books on Mathematics), Dover Publications, 2010, URL http://amazon.com/o/ASIN/B008SLYENC/.
[18]	R. Showalter and B. Momken, Single-phase Flow in Composite Poro-elastic Media, Technical report, Mathematical Methods in the Applied Sciences, 2002.
[19]	L. Tartar, The General Theory of Homogenization: A Personalized Introduction (Lecture Notes of the Unione Matematica Italiana), 2010th edition, Springer, 2009, URL http://amazon.com/o/ASIN/3642051944/.doi: 10.1007/978-3-642-05195-1.
[20]	F.-J. Ulm, G. Constantinides and F. Heukamp, Is concrete a poromechanics materials? A multiscale investigation of poroelastic properties, Materials and Structures, 37 (2004), 43-58.
[21]	E. Zeidler, Nonlinear Functional Analysis and Its Applications: II/ B: Nonlinear Monotone Operators, Translated from the German by the author and Leo F. Boron. Springer-Verlag, New York, 1990. URL http://amazon.com/o/ASIN/0387968024/.doi: 10.1007/978-1-4612-0985-0.