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

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December 2014, 9(4): 599-615. doi: 10.3934/nhm.2014.9.599

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

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

Received April 2014 Revised September 2014 Published December 2014

Abstract
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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.

Keywords: Two-phase concrete, poro-elasticity, two-scale homogenization, chemical reactions..

Mathematics Subject Classification: Primary: 35B27, 74F25, 74F10; Secondary: 76M5.

Citation: Michael Eden, Michael Böhm. Homogenization of a poro-elasticity model coupled with diffusive transport and a first order reaction for concrete. Networks & Heterogeneous Media, 2014, 9 (4) : 599-615. doi: 10.3934/nhm.2014.9.599

References:

[1]	A. Ainouz, Homogenization of a double porosity model in deformable media,, Electronic Journal of Differential Equations, 90 (2013), 1. Google Scholar
[2]	G. Allaire, Homogenization and two-scale convergence,, SIAM Journal on Mathematical Analysis, 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar
[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, (1995), 15. Google Scholar
[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. doi: 10.1137/0521046. Google Scholar
[5]	M. Biot, General theory of three-dimensional consolidation,, Journal of applied physics, 12 (1941), 155. doi: 10.1063/1.1712886. Google Scholar
[6]	O. Coussy, Poromechanics,, 2nd edition, (2005). doi: 10.1002/0470092718. Google Scholar
[7]	H. Deresiewicz and R. Skalak, On uniqueness in dynamic poroelasticity,, Bulletin of the Seismological Society of America, 53 (1963), 783. Google Scholar
[8]	M. Eden, Poroelasticity,, Master's thesis, (2014). Google Scholar
[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. doi: 10.1016/j.jmaa.2014.04.037. Google Scholar
[10]	U. Hornung and W. Jäger, Diffusion, convection, adsorption, and reaction of chemicals in porous media,, Journal of differential equations, 92 (1991), 199. doi: 10.1016/0022-0396(91)90047-D. Google Scholar
[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. Google Scholar
[12]	S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier,, Adv. Math. Sci. Appl., 13 (2003), 43. Google Scholar
[13]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM Journal on Mathematical Analysis, 20 (1989), 608. doi: 10.1137/0520043. Google Scholar
[14]	G. Pavliotis and A. Stuart, Multiscale Methods: Averaging and Homogenization (Texts in Applied Mathematics),, Springer, (2008). Google Scholar
[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. doi: 10.1002/mma.966. Google Scholar
[16]	R. Showalter, Distributed microstructure models of porous media,, in Flow in porous media, 114 (1993), 155. Google Scholar
[17]	R. Showalter, Hilbert Space Methods in Partial Differential Equations (Dover Books on Mathematics),, Dover Publications, (2010). Google Scholar
[18]	R. Showalter and B. Momken, Single-phase Flow in Composite Poro-elastic Media,, Technical report, (2002). Google Scholar
[19]	L. Tartar, The General Theory of Homogenization: A Personalized Introduction (Lecture Notes of the Unione Matematica Italiana),, 2010th edition, (2009). doi: 10.1007/978-3-642-05195-1. Google Scholar
[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. Google Scholar
[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, (1990). doi: 10.1007/978-1-4612-0985-0. Google Scholar

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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. Google Scholar
[2]	G. Allaire, Homogenization and two-scale convergence,, SIAM Journal on Mathematical Analysis, 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar
[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, (1995), 15. Google Scholar
[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. doi: 10.1137/0521046. Google Scholar
[5]	M. Biot, General theory of three-dimensional consolidation,, Journal of applied physics, 12 (1941), 155. doi: 10.1063/1.1712886. Google Scholar
[6]	O. Coussy, Poromechanics,, 2nd edition, (2005). doi: 10.1002/0470092718. Google Scholar
[7]	H. Deresiewicz and R. Skalak, On uniqueness in dynamic poroelasticity,, Bulletin of the Seismological Society of America, 53 (1963), 783. Google Scholar
[8]	M. Eden, Poroelasticity,, Master's thesis, (2014). Google Scholar
[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. doi: 10.1016/j.jmaa.2014.04.037. Google Scholar
[10]	U. Hornung and W. Jäger, Diffusion, convection, adsorption, and reaction of chemicals in porous media,, Journal of differential equations, 92 (1991), 199. doi: 10.1016/0022-0396(91)90047-D. Google Scholar
[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. Google Scholar
[12]	S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier,, Adv. Math. Sci. Appl., 13 (2003), 43. Google Scholar
[13]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM Journal on Mathematical Analysis, 20 (1989), 608. doi: 10.1137/0520043. Google Scholar
[14]	G. Pavliotis and A. Stuart, Multiscale Methods: Averaging and Homogenization (Texts in Applied Mathematics),, Springer, (2008). Google Scholar
[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. doi: 10.1002/mma.966. Google Scholar
[16]	R. Showalter, Distributed microstructure models of porous media,, in Flow in porous media, 114 (1993), 155. Google Scholar
[17]	R. Showalter, Hilbert Space Methods in Partial Differential Equations (Dover Books on Mathematics),, Dover Publications, (2010). Google Scholar
[18]	R. Showalter and B. Momken, Single-phase Flow in Composite Poro-elastic Media,, Technical report, (2002). Google Scholar
[19]	L. Tartar, The General Theory of Homogenization: A Personalized Introduction (Lecture Notes of the Unione Matematica Italiana),, 2010th edition, (2009). doi: 10.1007/978-3-642-05195-1. Google Scholar
[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. Google Scholar
[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, (1990). doi: 10.1007/978-1-4612-0985-0. Google Scholar

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