Homogenization of a fully coupled thermoelasticity problem for a highly heterogeneous medium with <i>a priori</i> known phase transformations

Eden, Michael; Muntean, Adrian

doi:10.1002/mma.4276

Cited by 18 publications

(18 citation statements)

References 33 publications

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“…There, problems are considered on a time interval S and a time dependent domain Ω ε (t) (cf. [8], [9], [10], [11], [5], [6]). The two-scale transformation concept is basically the same for these problems since time is only a parameter in the concept of the two-scale convergence.…”

Section: Direct Homogenisation On the Locally Periodic Domains And Fu...mentioning

confidence: 99%

“…Therefore, the method itself was only proposed and has not been proven until now. Nevertheless, this method found wide application -in the sense that the back-transformations is done formally and only the homogenisation of the substitute problems is proven -since it allows to consider many interesting problems, particularly on domains evolving in time (see [9], [10], [11], [5], [6]).…”

Section: Introductionmentioning

confidence: 99%

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The two-scale transformation method

Wiedemann¹

2021

Preprint

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We prove the two-scale transformation method which allows rigorous homogenisation of problems defined on locally periodic domains by transformation on periodic domains. The idea to consider periodic substitute problems was originally proposed by M. A. Peter for the homogenisation on evolving microstructure and is applied in several works. However, only the homogenisation of the periodic substitute problems was proven, whereas the method itself was just postulated (i.e. the equivalence to the homogenisation of the actual problem had to be assumed). In this work, we develop this idea further and formulate a rigorous two-scale convergence concept for microscopic transformation to prove this method. Moreover, we show a new two-scale transformation rule for gradients which allows to derive new limit problems that are now transformationally independent.

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Section: Direct Homogenisation On the Locally Periodic Domains And Fu...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The two-scale transformation method

Wiedemann¹

2021

Preprint

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“…This method was proposed for the homogenisation of a diffusion problem in [Pet07a]. Later it was applied in several works -in the same sense that the homogenisation of the substitute problem is proven -([Pet07b], [PB09], [EM17], [GNP21]). In [Wie21] a rigorous twoscale convergence concept for this transformation method was developed and the method itself proven, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…The same problem occurs also by the formal back-transformation after the homogenisation of diffusion and elasticity equations in a periodic reference domain (cf. [Pet07a], [EM17]). Using the results of [Wie21], the back-transformation can be done rigorously and a transformation-independent two-scale limit problem can be derived.…”

Section: Introductionmentioning

confidence: 99%

Homogenisation of the Stokes equations for evolving microstructure

Wiedemann¹,

Peter²

2021

Preprint

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We consider the homogenisation of the Stokes equations in a porous medium which is evolving in time. At the interface of the pore space and the solid part, we prescribe an inhomogeneous Dirichlet boundary condition, which enables to model a no-slip boundary condition at the evolving boundary. We pass rigorously to the homogenisation limit with the two-scale transformation method. In order to derive uniform a priori estimates, we show a Korn-type inequality for the two-scale transformation method and construct a family of ε-scaled operators div −1 ε , which are right-inverse to the corresponding divergences. The homogenisation result is a new version of Darcy's law. It features a time-and space-dependent permeability tensor, which accounts for the local pore structure, and a macroscopic compressibility condition, which induces a new source term for the pressure. In the case of a no-slip boundary condition at the interface, this source term relates to the change of the local pore volume.

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“…In Bringedal et al (2016) a formal upscaling of non-isothermal reactive flow in porous media was undertaken, but the solid matrix was assumed rigid. In Eden and Muntean (2017) homogenization of a fully coupled thermoelasticity problem was undertaken, but not in the context of fluid-structure interaction. In the book Coussy (1995) there is also a section on linear thermo-poroelasticity, where the macroscale equations are derived using principles from continuum mechanics and thermodynamics.…”

Section: Introductionmentioning

confidence: 99%

Upscaling of the Coupling of Hydromechanical and Thermal Processes in a Quasi-static Poroelastic Medium

et al. 2018

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We undertake a formal derivation of a linear poro-thermo-elastic system within the framework of quasi-static deformation. This work is based upon the well-known derivation of the quasi-static poroelastic equations (also known as the Biot consolidation model) by homogenization of the fluid-structure interaction at the microscale. We now include energy, which is coupled to the fluid-structure model by using linear thermoelasticity, with the full system transformed to a Lagrangian coordinate system. The resulting upscaled system is similar to the linear poroelastic equations, but with an added conservation of energy equation, fully coupled to the momentum and mass conservation equations. In the end, we obtain a system of equations on the macroscale accounting for the effects of mechanical deformation, heat transfer, and fluid flow within a fully saturated porous material, wherein the coefficients can be explicitly defined in terms of the microstructure of the material. For the heat transfer we consider two different scaling regimes, one where the Péclet number is small, and another where it is unity. We also establish the symmetry and positivity for the homogenized coefficients.

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Homogenization of a fully coupled thermoelasticity problem for a highly heterogeneous medium with a priori known phase transformations

Cited by 18 publications

References 33 publications

The two-scale transformation method

The two-scale transformation method

Homogenisation of the Stokes equations for evolving microstructure

Upscaling of the Coupling of Hydromechanical and Thermal Processes in a Quasi-static Poroelastic Medium

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