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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

Michael Eden 1, and  Michael Böhm 1,

1. 

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

Received  April 2014 Revised  September 2014 Published  December 2014

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), 1-18.  Google Scholar

[2]

G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518. 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, World Scintific publication, Singapore, (1995), 15-25. 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), 823-836. doi: 10.1137/0521046.  Google Scholar

[5]

M. Biot, General theory of three-dimensional consolidation, Journal of applied physics, 12 (1941), 155-164. doi: 10.1063/1.1712886.  Google Scholar

[6]

O. Coussy, Poromechanics, 2nd edition, Wiley, 2005, URL http://amazon.com/o/ASIN/0470849207/. 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), 783-788. Google Scholar

[8]

M. Eden, Poroelasticity, Master's thesis, University of Bremen, 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), 28-47. 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), 199-225. 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), 1-11.  Google Scholar

[12]

S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 13 (2003), 43-63.  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), 608-623. doi: 10.1137/0520043.  Google Scholar

[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/.  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), 1257-1282. doi: 10.1002/mma.966.  Google Scholar

[16]

R. Showalter, Distributed microstructure models of porous media, in Flow in porous media, Springer, 114 (1993), 155-163.  Google Scholar

[17]

R. Showalter, Hilbert Space Methods in Partial Differential Equations (Dover Books on Mathematics), Dover Publications, 2010, URL http://amazon.com/o/ASIN/B008SLYENC/. Google Scholar

[18]

R. Showalter and B. Momken, Single-phase Flow in Composite Poro-elastic Media, Technical report, Mathematical Methods in the Applied Sciences, 2002. Google Scholar

[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.  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), 43-58. 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, New York, 1990. URL http://amazon.com/o/ASIN/0387968024/. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

show all references

References:
[1]

A. Ainouz, Homogenization of a double porosity model in deformable media, Electronic Journal of Differential Equations, 90 (2013), 1-18.  Google Scholar

[2]

G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518. 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, World Scintific publication, Singapore, (1995), 15-25. 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), 823-836. doi: 10.1137/0521046.  Google Scholar

[5]

M. Biot, General theory of three-dimensional consolidation, Journal of applied physics, 12 (1941), 155-164. doi: 10.1063/1.1712886.  Google Scholar

[6]

O. Coussy, Poromechanics, 2nd edition, Wiley, 2005, URL http://amazon.com/o/ASIN/0470849207/. 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), 783-788. Google Scholar

[8]

M. Eden, Poroelasticity, Master's thesis, University of Bremen, 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), 28-47. 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), 199-225. 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), 1-11.  Google Scholar

[12]

S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 13 (2003), 43-63.  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), 608-623. doi: 10.1137/0520043.  Google Scholar

[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/.  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), 1257-1282. doi: 10.1002/mma.966.  Google Scholar

[16]

R. Showalter, Distributed microstructure models of porous media, in Flow in porous media, Springer, 114 (1993), 155-163.  Google Scholar

[17]

R. Showalter, Hilbert Space Methods in Partial Differential Equations (Dover Books on Mathematics), Dover Publications, 2010, URL http://amazon.com/o/ASIN/B008SLYENC/. Google Scholar

[18]

R. Showalter and B. Momken, Single-phase Flow in Composite Poro-elastic Media, Technical report, Mathematical Methods in the Applied Sciences, 2002. Google Scholar

[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.  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), 43-58. 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, New York, 1990. URL http://amazon.com/o/ASIN/0387968024/. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

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