Computational Physics Group

Karel Matous









Boundary condition effects on multiscale analysis of

damage localization

H.M. Inglis1, P.H. Geubelle2 and  K. Matous2,3

1Department of Mechanical Science and Engineering
2Department of Aerospace Engineering
3Computational Science and Engineering
University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA.


The choice of boundary conditions used in multiscale analysis of heterogeneous materials affects the numerical results, including the macroscopic constitutive response, the type and extent of damage taking place at the microscale and the required size of the Representative Volume Element (RVE). We compare the performance of periodic boundary conditions and minimal kinematic boundary conditions [1] applied to the unit cell of a particulate composite material, both in the absence and presence of damage at the particle-matrix interfaces. In particular, we investigate the response of the RVE under inherently non-periodic loading conditions, and the ability of both boundary conditions to capture localization events that are not aligned with the RVE boundaries. We observe that, although there are some variations in the evolution of the microscale damage between the two methods, there is no significant difference in homogenized responses even when localization is not aligned with the cell boundaries.


We have compared the behavior of a particulate composite system under periodic boundary conditions and under the minimal kinematic boundary conditions introduced by Mesarovic and Padbidri [1]. For an undamaged material system, the computed shear stiffness is 6-7% higher with periodic boundary conditions
than with minimal kinematic boundary conditions, consistent with the results obtained by Mesarovic and Padbidri. In a system where interfacial damage is modeled, periodic boundary conditions successfully capture weak localization associated with the particle debonding process even when that weak localization is not aligned with the domain axes. For some pack geometries and some loading cases, the additional constraint of periodicity is satisfied by the formation of more than one band of partial localization. Characteristic features of the homogenized solution, including the initial slope, the initial peak, and evolution of damage and failure, are similar for the two boundary conditions across multiple packs.
    The results suggest that the multiscale scheme based on periodic boundary conditions, which is supported by a wealth of theoretical development and is attractive because of its mathematical tractability, can be used even in the case of off-axis damage localization. The multiscale scheme based on MKBC presents the key advantage of not requiring periodic RVE’s, and can therefore be applied to a wider range of microstructures, especially those extracted directly from actual micrographs.
    The multiscale scheme based on minimal kinematic boundary conditions does not perform well when particles are too close to the boundaries. The integral constraint is then satisfied by excessive straining of a narrow ligament, rather than by deformation of the entire domain. The introduction of a penalty term in the integral boundary condition may improve the performance of this method.


This work was supported by the Center for Simulation of Advanced Rockets (CSAR) under contract number B523819 by the U.S. Department of Energy. Karel Matous would also like to acknowledge support from ATK/Thiokol (ATK-21316), with J. Thompson and Dr. I. L. Davis serving as program monitors.

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© 2009 Notre Dame and Dr. Karel Matous