## Multiscale Modeling of Particle Debonding in Reinforced

## Elastomers Subjected to Finite Deformations

K. Matous and P.H. Geubelle

Center for Simulation of Advanced Rockets

Department of Aerospace Engineering

University of Illinois at Urbana-Champaign

Urbana, IL 61801, USA.

#### Abstract

Interfacial damage nucleation and evolution in reinforced elastomers
subjected to finite strains is modeled using the mathematical theory
of homogenization based on the asymptotic expansion of unknown
variables. The microscale is characterized by a periodic unit cell,
which contains particles dispersed in a blend and the particle matrix
interface is characterized by a cohesive law. A novel numerical
framework based on the perturbed Petrov-Galerkin method for the
treatment of nearly incompressible behavior is employed to solve the
resulting boundary value problem on the microscale and the deformation
path of a macroscale particle is predefined as in the micro-history
recovery procedure. A fully implicit and efficient finite element
formulation, including consistent linearization, is presented. The
proposed multiscale framework is capable of predicting the
non-homogeneous micro-fields and damage nucleation and propagation
along the particle matrix interface, as well as the macroscopic
response and mechanical properties of the damaged continuum. Examples
are considered involving simple unit cells in order to illustrate the
multiscale algorithm and demonstrate the complexity of the underlying
physical processes.

#### Conclusions

The mathematical theory of homogenization based on the asymptotic
expansion of the displacement, deformation gradient and stress fields
has been derived and used in modeling debonding (or dewetting) damage
evolution in reinforced elastomers subject to finite strains. The
micro-scale 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 interface. A novel numerical
procedure is based on a stabilized Lagrangian formulation and adopts a
decomposition of the pressure and displacement fields to eliminate the
volumetric locking due to the nearly incompressible behavior of a
matrix. The consistent linearization of the resulting system of
nonlinear equations has been derived and leads to an efficient
solution of the complex highly nonlinear problem.
The hyperelastic behavior of an individual constituents is defined by
hyperelastic potentials and the particle matrix interface is
characterized by a cohesive law. A fully implicit nonlinear solver,
based on the arc-length procedure is applied allowing for large
loading steps.
Various examples involving simple 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 scale and its effect on the macroscopic stress-strain
curves and the evolution of the void volume. One of these examples has
illustrated the appearance of a bifurcation phenomenon associated with
the progressive or sudden debonding of particles. The emphasis of this
work has been on the development of the 3D
multiscale computational framework for the simulation of damage
evolution in reinforced elastomers. To provide reliable predictive
results, this multiscale model must allow for the simulation of a
larger more representative assembly of particles, possibly of different
sizes. For many materials, it should also incorporate a more complex,
rate dependent description of the matrix or blend response. These two
requirements will increase the computational costs associated with the
multiscale analysis, therefore requiring an efficient parallel
implementation of the multiscale scheme. On the modeling side, the
next step also involves the incorporation of a matrix tearing model
needed to capture the initiation and propagation of matrix cracks
between the voids.

#### Acknowledgment

The authors gratefully acknowledge support from the Center for
Simulation of Advanced Rockets (CSAR) at the University of Illinois,
Urbana-Champaign. Research at CSAR is funded by the U.S. Department of
Energy as a part of its Advanced Simulation and Computing (ASC)
program under contract number B341494.