## 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.