Modelado multiescala de medios porosos heterogéneos considerando interacción poro-estructura.

Abstract

En esta tesis se desarrollan formulaciones multiescala variacionalmente consistentes aplicadas a problemas hidromecánicos, empleando el concepto de Elemento de Volumen Representativo (RVE: Representative Volume Element) junto con técnicas específicas de homogeneización computacional. Si bien el empleo de metodologías multiescala se encuentra ampliamente consolidado y ha demostrado gran eficacia en el análisis de materiales complejos y fenómenos multifísicos, existe una dificultad particular cuando se modelan, bajo un esquema de homogeneización de primer orden, problemas gobernados por ecuaciones de balance de una cantidad escalar con término fuente, como es el caso del balance de masas o de determinadas particularizaciones de las ecuaciones de difusión-reacción. En estos casos, la presencia de un término transitorio induce una dependencia de la respuesta macroscópica con respecto al tamaño de la microescala. Debido a las implicaciones que se tienen de homogeneizar dicho término, comúnmente se lo ha denominado ``efecto de segundo orden''. En consecuencia, surge la problemática de que no es posible encontrar un tamaño físicamente admisible de la microestructura a partir del cual la respuesta homogeneizada resulte insensible a dicho tamaño, perdiéndose así la noción básica de existencia de RVE. Esto representa un tópico no tratado aún apropiadamente en la literatura, que plantea desafíos teóricos y numéricos de gran interés ingenieril.In this thesis, variationally consistent multiscale formulations are developed and applied to hydro-mechanical problems, employing the concept of Representative Volume Element (RVE) together with specific computational homogenization techniques. Although the use of multiscale methodologies is widely established and has proven highly effective in the analysis of complex materials and multiphysics phenomena, a particular difficulty arises when modeling, within a first-order homogenization framework, problems governed by balance equations for a scalar quantity with a source term, such as the mass balance equation or certain specific forms of diffusion–reaction equations. In such cases, the presence of a transient term induces a dependence of the macroscopic response on the size of the micro-scale. Due to the implications of homogenizing this term, it has commonly been referred to as the ``second-order effect''. Consequently, the problem arises that no physically admissible size of the micro-structure can be identified for which the homogenized response becomes insensitive to that size, thus losing the basic notion of RVE existence. This represents a topic not yet properly addressed in the literature, which poses theoretical and numerical challenges of great engineering interest. There are two fundamental ideas in this thesis. First, to obtain a macroscopic model that preserves the objectivity of the homogenized response when addressing this type of phenomenon, with special emphasis on saturated porous media. Subsequently, to extend the treatment to micro-architectures formed by a matrix of saturated porous materials containing impermeable solid inclusions. This type of heterogeneous media, with an internal micro-structure consisting of components that require a different number or nature of primary fields to describe their physical behavior, requires detailed revisions and novel adjustments to RVE-based multiscale theories, constituting the second key point of this contribution. In this context, special attention is devoted to the homogenization rules applied to the pore-pressure field and its gradient, with the aim of building a Minimally Constrained Multiscale Model (MCMM). On this basis, alternative models are also proposed that incorporate additional constraints defined over appropriately established partitions on the micro-cell (MC), generating a family of sub-models applicable for more complex scenarios. The proposed models are formulated axiomatically and variationally using the Method of Multiscale Virtual Power (MMVP), and their numerical solution is addressed through the FE$^2$ (Finite Element squared) approach. To this end, a general computational code was developed, whose data structure is specifically designed for a multiscale modeling environment. Finally, several examples verify the proposed formulation, including comparisons with Direct Numerical Simulation (DNS) methodologies. This thesis is based on three published articles and two manuscripts currently under review in international scientific journals.