Variational Foundations and Generalized Unified Theory of RVE-Based Multiscale Models

Abstract

A unified variational theory is proposed for a general class of multiscale models based on the concept of Representative Volume Element. The entire theory lies on three fundamental principles: (1) kinematical admissibility, whereby the macro-and micro-scale kinematics are defined and linked in a physically meaningful way; (2) duality, through which the natures of the force-and stress-like quantities are uniquely identified as the duals (power-conjugates) of the adopted kinematical variables; and (3) the Principle of Multiscale Virtual Power, a generalization of the wellknown Hill-Mandel Principle of Macrohomogeneity, from which equilibrium equations and homogenization relations for the force-and stress-like quantities are unequivocally obtained by straightforward variational arguments. The proposed theory provides a clear, logically-structured framework within which existing formulations can be rationally justified and new, more general multiscale models can be rigorously derived in well-defined steps. Its generality allows the treatment of problems involving phenomena as diverse as dynamics, higher order strain effects, material failure with kinematical discontinuities, fluid mechanics and coupled multi-physics. This is illustrated in a number of examples where a range of models is systematically derived by following the same steps. Due to the variational basis of the theory, the format in which derived models are presented is naturally well suited for discretization by finite element-based or related methods of numerical approximation. Numerical examples illustrate the use of resulting models, including a non-conventional failure-oriented model with discontinuous kinematics, in practical computations.

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“…The general framework proposed in [33] encompasses a wide spectrum of multi-scale modeling scenarios, ranging from conventional constitutive-inspired multi-scale theories (see [36,37,38,39]) to more sophisticated and novel approaches, as for example: multi-scale techniques linking high order macro-scale description with standard (first order) micro-scale kinematics [40], thermo-mechanical models (see [41]), multi-scale formulations accounting for inertia and body forces effects (see [42]), etc.…”

“…The full model formulation has been described in Toro et al [44]. Thus, the readers are addressed to that work for obtaining further details of the formulation, as well as to [33] for consulting specific aspects of the MMVP theory.…”

“…These homogenization procedures are rationally justified within the framework of the MMVP theory ( [33] and [44]). Each of them is typified as follows:…”

“…The scale bridging equations are established from the following two axioms and the consequences derived from them, see further details in [33]: H1) Kinematical admissibility requirements connecting the kinematics at macro and microscales. This hypothesis imposes the following constraints on the micro-scale strains:…”

“…H2) Identity, in a variational sense, of internal virtual power densities between both scales. This hypothesis is the Principle of Multi-scale Virtual Power, according with [33], and is a generalized version of the Hill-Mandel Principle. Hypothesis (H2) has been specified in [44] to include the presence of strong discontinuities at both scales of analysis.…”