American Institute of Mathematical Sciences

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

show all references

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

[1]

Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks & Heterogeneous Media, 2014, 9 (2) : 353-382. doi: 10.3934/nhm.2014.9.353
[2]

Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485
[3]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016
[4]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two-phase flow in porous media. Networks & Heterogeneous Media, 2017, 12 (1) : 147-171. doi: 10.3934/nhm.2017006
[5]

Brahim Amaziane, Mladen Jurak, Leonid Pankratov, Anja Vrbaški. Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 629-665. doi: 10.3934/dcdsb.2018037
[6]

Fang Liu, Aihui Zhou. Localizations and parallelizations for two-scale finite element discretizations. Communications on Pure & Applied Analysis, 2007, 6 (3) : 757-773. doi: 10.3934/cpaa.2007.6.757
[7]

Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223
[8]

Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058
[9]

Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Two-Scale numerical simulation of sand transport problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 151-168. doi: 10.3934/dcdss.2015.8.151
[10]

Theodore Tachim Medjo. A two-phase flow model with delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137
[11]

Jan Prüss, Jürgen Saal, Gieri Simonett. Singular limits for the two-phase Stefan problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5379-5405. doi: 10.3934/dcds.2013.33.5379
[12]

Marianne Korten, Charles N. Moore. Regularity for solutions of the two-phase Stefan problem. Communications on Pure & Applied Analysis, 2008, 7 (3) : 591-600. doi: 10.3934/cpaa.2008.7.591
[13]

Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic & Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707
[14]

Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks & Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143
[15]

Alexandre Mouton. Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel. Kinetic & Related Models, 2009, 2 (2) : 251-274. doi: 10.3934/krm.2009.2.251
[16]

Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic & Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65
[17]

T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665
[18]

Eberhard Bänsch, Steffen Basting, Rolf Krahl. Numerical simulation of two-phase flows with heat and mass transfer. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2325-2347. doi: 10.3934/dcds.2015.35.2325
[19]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1
[20]

Jie Jiang, Yinghua Li, Chun Liu. Two-phase incompressible flows with variable density: An energetic variational approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3243-3284. doi: 10.3934/dcds.2017138

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences