## 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.