Boundary condition effects on multiscale analysis of
H.M. Inglis1, P.H.
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  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 . 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.
© 2009 Notre Dame and Dr.