## Finite element formulation for modeling 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 is modeled using a three-dimensional updated
Lagrangian finite element formulation based on the
perturbed Petrov-Galerkin method for the treatment of
nearly incompressible behavior. The progressive failure of
the particle-matrix interface is modeled by a cohesive law
accounting for mode mixity. The meso-scale is
characterized by a unit cell, which contains particles
dispersed in a homogenized blend. A new, fully implicit
and efficient finite element formulation, including
consistent linearization, is presented. The proposed
finite element model is capable of predicting the
non-homogeneous meso-fields and damage nucleation and
propagation along the particle-matrix interface. Simple
deformations involving an idealized solid rocket
propellant are considered to demonstrate the algorithm.

#### Conclusions

We have formulated and implemented a 3D computational
model to simulate dewetting evolution in reinforced
elastomers subject to finite strains. The particle-matrix
interface is modeled by a cohesive law that accounts for
irreversibility and mode mixity. The finite element
framework is based on a stabilized updated 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. Through a set of examples involving one- and
four-particle unit cells, we have shown the ability of the
numerical scheme to capture the non-homogeneous stress and
deformation fields present in the matrix and the damage
nucleation and propagation along the particle-matrix
interface. In particular, the scheme was shown to capture
effects associated with the interface strength and
nonuniform particle spacing and size. The existence of a
bifurcation in the solution path was also briefly
investigated. The present work is a first step toward
linking the macro-scale to the meso-scale through the
computational homogenization, where a meso-structure is
fully coupled with the deformation at a typical material
point of a macro-continuum. The formulation and
implementation of a truly multiscale model for the effect
of microstructural damage on the macroscopic constitutive
response of reinforced elastomers is the topic of our
future research.

#### 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.