Computational Physics Group

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.
© 2009 Notre Dame and Dr. Karel Matous