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

  • Pablo J. Blanco
  • Pablo J. Sánchez
  • Eduardo A. de Souza Neto
  • Raúl A. Feijóo
Original Paper

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 well-known 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.

Notes

Acknowledgments

This work was partially supported by the Brazilian agencies CNPq and FAPERJ. The support of these agencies is gratefully acknowledged. P.J. Sánchez acknowledges the financial support from CONICET (grant PIP 2013-2015 631) and from the European Research Council under the European Unions Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement N. 320815 (ERC Advanced Grant Project Advanced tools for computational design of engineering materials COMP-DES-MAT).

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Copyright information

© CIMNE, Barcelona, Spain 2014

Authors and Affiliations

  • Pablo J. Blanco
    • 1
    • 2
  • Pablo J. Sánchez
    • 3
    • 4
  • Eduardo A. de Souza Neto
    • 5
  • Raúl A. Feijóo
    • 1
    • 2
  1. 1.National Laboratory for Scientific ComputingLNCC/MCTIPetrópolisBrazil
  2. 2.National Institute of Science and Technology in Medicine Assisted by Scientific Computing (INCT-MACC)PetrópolisBrazil
  3. 3.CIMEC-UNL-CONICETSanta FeArgentina
  4. 4.GIMNI-UTN-FRSFSanta FeArgentina
  5. 5.Zienkiewicz Centre for Computational EngineeringSwansea UniversitySwanseaUK

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