**Multiscale modeling of solid propellants: From
particle packing to failure**

K. Matous^{1,2}, H.M Inglis^{1}, X. Gu^{1},
D. Rypl^{3}, T.L Jackson^{1} and P.H.
Geubelle^{1,2}

^{1}Center for Simulation of Advanced Rockets

^{2}Department of Aerospace Engineering

University of Illinois at Urbana-Champaign

Urbana, IL 61801, USA.

^{3}Department of Structural Mechanics

Czech Technical University in Prague

Prague, 160 00 P6, Czech Republic
#### Abstract

We present a theoretical and computational framework for
modeling the multiscale constitutive behavior of highly
filled elastomers, such as solid propellants and other
energetic materials. Special emphasis is placed on the
effect of the particle debonding or dewetting process
taking place at the microscale and on the macroscopic
constitutive response. The microscale is characterized by
a periodic unit cell, which contains a set of hard
particles (such as ammonium perchlorate for AP-based
propellants) dispersed in an elastomeric binder. The unit
cell is created using a packing algorithm that treats the
particles as spheres or discs, enabling us to generate
packs which match the size distribution and volume
fraction of actual propellants. A novel technique is
introduced to characterize the pack geometry in a way
suitable for meshing, allowing for the creation of
high-quality periodic meshes with refinement zones in the
regions of interest. The proposed numerical multiscale
framework, based on the mathematical theory of
homogenization, is capable of predicting the complex,
heterogeneous stress and strain fields associated, at the
microscale, with the nucleation and propagation of damage
along the particle–matrix interface, as well as the
macroscopic response and mechanical properties of the
damaged continuum. Examples involving simple unit cells
are presented to illustrate the multiscale algorithm and
demonstrate the complexity of the underlying physical
processes.

#### Conclusions

A fully automated mathematical/numerical framework for
multiscale modeling of heterogeneous propellants from
particle packing to failure has been presented. The
microscale description is based on a periodic unit cell
consisting of particles dispersed in a blend and
incorporates the local non-homogeneous stress and
deformation fields present in the unit cell during the
failure of the particle/matrix interfaces. A packing
algorithm, treating the embedded particles as spheres or
discs, is used to generate packs which match the size
distribution and volume fraction of actual propellants.
Moreover, a sophisticated pre-processing tool has been
developed to generate a geometric model based on Bezier
curves and/or surfaces. This geometric model is then used
in a general meshing tool, T3d, to create high-quality
periodic meshes. Since the identical meshing of the
periodic entities using the advancing front technique is
not usually viable, a different approach based on
mirroring has been adopted. Next, the mathematical theory
of homogenization based on the asymptotic expansion of the
displacement, strain and stress fields has been derived
and used in modeling debonding (or dewetting) damage
evolution in reinforced elastomers.

Various examples involving 2D unit cells and macroscopic
deformation histories of an idealized solid propellant
have been considered to study the link between the failure
process taking place at the particle size scale and its
effect on the macroscopic stress–strain curves and the
evolution of void volume. The emphasis of this work has
been to develop a damage analysis tool at multiple scales
from particle packing to failure. Further research will
involve the inclusion of large deformations, a more
complex, rate-dependent description of the binder and a
matrix tearing model needed to capture the initiation and
propagation of cracks in the solid propellant during void
coalescence. Moreover, the size of the representative
volume element in the presence of damage needs to be
investigated.

#### Acknowledgment

The work of K. Matous, H.M. Inglis, X. Gu, T.L. Jackson
and P.H. Geubelle was supported by the Center for
Simulation of Advanced Rockets (CSAR) under contract
number B341494 by the U.S. Department of Energy as a part
of its Advanced Simulation and Computing program (ASC). K.
Matous and P.H. Geubelle also acknowledge support from
ATK/Thiokol, with J. Thompson and Dr. I.L. Davis as a
program monitors. The work of Dr. Rypl was supported by
the Grant Agency of the Czech Republic under contract
number GACR 103/05/2315.