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De Souza Neto E.A.,University of Swansea | Blanco P.J.,Lncc Mcti Laboratorio Nacional Of Computacao Cientifica | Blanco P.J.,Inct Macc Instituto Nacional Of Ciencia E Tecnologia Em Medicina Assistida Por Computacao Cientifica | Sanchez P.J.,CONICET | And 2 more authors.
Mechanics of Materials | Year: 2015

A multiscale theory of solids based on the concept of representative volume element (RVE) and accounting for micro-scale inertia and body forces is proposed. A simple extension of the classical Hill-Mandel Principle together with suitable kinematical constraints on the micro-scale displacements provide the variational framework within which the theory is devised. In this context, the micro-scale equilibrium equation and the homogenisation relations among the relevant macro- and micro-scale quantities are rigorously derived by means of straightforward variational arguments. In particular, it is shown that only the fluctuations of micro-scale inertia and body forces about their RVE volume averages may affect the micro-scale equilibrium problem and the resulting homogenised stress. The volume average themselves are mechanically relevant only to the macro-scale. © 2014 Elsevier Ltd. All rights reserved.

Sanchez P.J.,CONICET | Blanco P.J.,Lncc Mcti Laboratorio Nacional Of Computacao Cientifica | Blanco P.J.,Inct Macc Instituto Nacional Of Ciencia E Tecnologia Em | Huespe A.E.,CONICET | And 2 more authors.
Computer Methods in Applied Mechanics and Engineering | Year: 2013

This contribution presents the theoretical foundations of a Failure-Oriented Multi-scale variational Formulation (FOMF) for modeling heterogeneous softening-based materials undergoing strain localization phenomena. The multi-scale model considers two coupled mechanical problems at different physical length scales, denoted as macro and micro scales, respectively. Every point, at the macro scale, is linked to a Representative Volume Element (RVE), and its constitutive response emerges from a consistent homogenization of the micro-mechanical problem. At the macroscopic level, the initially continuum medium admits the nucleation and evolution of cohesive cracks due to progressive strain localization phenomena taking place at the microscopic level and caused by shear bands, damage or any other possible failure mechanism. A cohesive crack is introduced in the macro model once a specific macroscopic failure criterion is fulfilled. The novelty of the present Failure-Oriented Multi-scale Formulation is based on a proper kinematical information transference from the macro-to-micro scales during the complete loading history, even in those points where macro cracks evolve. In fact, the proposed FOMF includes two multi-scale sub-models consistently coupled: (i) a Classical Multi-scale Model (ClaMM) valid for the stable macro-scale constitutive response. (ii) A novel Cohesive Multi-scale Model (CohMM) valid, once a macro-discontinuity surface is nucleated, for modeling the macro-crack evolution. When a macro-crack is activated, two important kinematical assumptions are introduced: (i) a change in the rule that defines how the increments of generalized macro-strains are inserted into the micro-scale and (ii) the Kinematical Admissibility concept, from where proper Strain Homogenization Procedures are obtained. Then, as a consequence of the Hill-Mandel Variational Principle and the proposed kinematical assumptions, the FOMF provides an adequate homogenization formula for the stresses in the continuum part of the body, as well as, for the traction acting on the macro-discontinuity surface. The assumed macro-to-micro mechanism of kinematical coupling defines a specific admissible RVE-displacement space, which is obtained by incorporating additional boundary conditions, Non-Standard Boundary Conditions (NSBC), in the new model. A consequence of introducing these Non-Standard Boundary Conditions is that they guarantee the existence of a physically admissible RVE-size, a concept that we call through the paper "objectivity" of the homogenized constitutive response. Several numerical examples are presented showing the objectivity of the formulation, as well as, the capabilities of the new multi-scale approach to model material failure problems. © 2012 Elsevier B.V. All rights reserved.

Toro S.,CONICET | Sanchez P.J.,CONICET | Huespe A.E.,CONICET | Giusti S.M.,CONICET | And 4 more authors.
International Journal for Numerical Methods in Engineering | Year: 2014

In the first part of this contribution, a brief theoretical revision of the mechanical and variational foundations of a Failure-Oriented Multiscale Formulation devised for modeling failure in heterogeneous materials is described. The proposed model considers two well separated physical length scales, namely: (i) the macroscale where nucleation and evolution of a cohesive surface is considered as a medium to characterize the degradation phenomenon occurring at the lower length scale, and (ii) the microscale where some mechanical processes that lead to the material failure are taking place, such as strain localization, damage, shear band formation, and so on. These processes are modeled using the concept of Representative Volume Element (RVE). On the macroscale, the traction separation response, characterizing the mechanical behavior of the cohesive interface, is a result of the failure processes simulated in the microscale. The traction separation response is obtained by a particular homogenization technique applied on specific RVE sub-domains. Standard, as well as, Non-Standard boundary conditions are consistently derived in order to preserve objectivity of the homogenized response with respect to the micro-cell size. In the second part of the paper, and as an original contribution, the detailed numerical implementation of the two-scale model based on the finite element method is presented. Special attention is devoted to the topics, which are distinctive of the Failure-Oriented Multiscale Formulation, such as: (i) the finite element technologies adopted in each scale along with their corresponding algorithmic expressions, (ii) the generalized treatment given to the kinematical boundary conditions in the RVE, and (iii) how these kinematical restrictions affect the capturing of macroscopic material instability modes and the posterior evolution of failure at the RVE level. Finally, a set of numerical simulations is performed in order to show the potentialities of the proposed methodology, as well as, to compare and validate the numerical solutions furnished by the two-scale model with respect to a direct numerical simulation approach. © 2013 John Wiley & Sons, Ltd.

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