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TPM²

A two-scale homogenisation approach for fluid saturated porous media

In future applications it will be essential to consider the influences of discrete microstructures. Today it is possible, to receive the discrete topology of the microstructure of a material by CT-scanning. The information can be transferred to a finite element model. Therefore we are preparing the TPM for a two-scale homogenisation technique where the macroscopic body as well as the micostructure is analysed with the Finite Element Method.

Microscopic structure of the material  (c) https://ifam.fraunhofer.de
Microscopic structure of the material

Representative Volume Element (RVE) of a microstructure meshed with finite elements (c) https://simpleware.com
Representative Volume Element (RVE) of a microstructure meshed with finite elements

The poro-elastic material description is based on the Theory of Porous Media (TPM), see [1], [2]. We derived a two-phase material model regarding a solid and a fluid phase.

For the two-scale approach the FE²-Method is applied, where both the global macroscopic structure and the underlying micro-structure are analysed with the Finite Element Method, see [3], [4].

Therefore, the weak forms for the macroscopic and the microscopic scale are formulated and an averaged macroscopic tangent moduli considering the micro-scale is derived. Additionally, the formulation of lower level boundary conditions, which refer to the quantities that are transmitted from the macro- to the micro-scale, under strict compliance of the Hill-Mandel homogeneity condition is taken into account.

Mathematical setup (c)
Mathematical setup

Numerical example (c)
A numerical example using the TPM²-Method shows the influences between two different microstructures for the displacements and pressure distribution on a compression test.

RVE displacements and pressure (c)
Displacements and pressure in representative volume elements

Literature

[1] R. de Boer and W. Ehlers.
Theorie der Mehrkomponentenkontinua mit Anwendung auf bodenmechanische Probleme. Technical report, Teil I, Forschungsberichte aus dem Fachbereich 
Bauwesen, Heft 40, Universität-GH-Essen, 1986.

[2] W. Ehlers.
Poröse Medien -- ein kontinuumsmechanisches Modell auf der Basis der Mischungstheorie.
Technical Report 47, Universität-GH Essen, 1989.

[3] Christian Miehe.
Computational micro-to-macro transitions for discretised micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy.
Computer methods in applied mechanics and engineering, 2002.

[4] Jörg Schröder.
CISM Course "Plasticity and Beyond: Microstructures. Crystal-Plasticity and Phase Transitions", Chapter A numerical two-scale
homogenization scheme: the FE²-method.
Springer Verlag, 2013.

Publications

Bartel, F., Ricken, T., Schröder, J. and Bluhm, J.
Microstructural influence on macroscopic response regarding fluid flow through porous media applying TPM² Method.
Proc. Appl. Math. Mech., 17, 577-578 (2017) / DOI 10.1002/pamm.201710258

Conference Contributions

2018

A two-scale homogenisation approach for fluid saturated porous media based
on TPM and FE2-Method
89rd GAMM Annual Meeting, Munich, Germany, 2018

A two-scale homogenisation approach for fluid saturated porous media based
on TPM and FE2-Method
9. Workshop junger Nachwuchswissenschaftler in der Mechanik, Zell am See,
Austria, 2018

2017

A two-scale homogenisation approach for fluid saturated porous media based
on TPM and FE2-Method
7th GACM, Stuttgart, Germany, 2017

A two-scale homogenisation approach for fluid saturated porous media based
on TPM and FE2-Method
30th COMPLAS, Barcelona, Spain, 2017

A two-scale homogenisation approach for fluid saturated porous media based
on TPM and FE2-Method
88rd GAMM Annual Meeting, Weimar, Germany, 2017

A two-scale homogenisation approach for fluid saturated porous media based
on TPM and FE2-Method
8. Workshop junger Nachwuchswissenschaftler in der Mechanik, Zell am See,
Austria, 2017

2016

A two-scale homogenisation approach for fluid saturated porous media based on TPM and FE2-Method
SEMC 2016, Cape Town, South Afrika, 2016

A two-scale homogenisation approach for fluid saturated porous media based on TPM and FE2-Method
87rd GAMM Annual Meeting, Braunschweig, Germany, 2016

A two-scale homogenisation approach for fluid saturated porous media based on TPM and FE2-Method
7. Workshop junger Nachwuchswissenschaftler in der Mechanik, Zell am See, Austria, 2016

2015

A two-scale homogenisation approach for fluid saturated porous media based on TPM and FE2-Method
Evolutionary Solid Bodies, Bollendorf, Germany, 2015

A two-scale homogenisation approach for fluid saturated porous media based on TPM and FE2-Method
XXV International Workshop on Computational Micromechanics of Materials, Bochum, Germany,2015.

A two-scale homogenisation approach for fluid saturated porous media based on TPM and FE2-Method
Forschungskolloquium Baustatik - Baupraxis 2015, Döllnsee, Germany, 2015

A two-scale homogenisation approach for fluid saturated porous media based on TPM and FE2-Method
86rd GAMM Annual Meeting, Lecce, Italy, 2015

A two-scale homogenisation approach for fluid saturated porous media based on TPM and FE2-Method
6. Workshop junger Nachwuchswissenschaftler in der Mechanik, Zell am See, Austria, 2015

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Univ.-Prof. Dr.-Ing.

Tim Ricken

Head of Department