Computational Physics Group

Karel Matous









Finite element formulation for modeling

nonlinear viscoelastic elastomers

P. Areias1 and  K. Matous1,2

1Computational Science and Engineering
2Department of Aerospace Engineering
University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA.


Nonlinear viscoelastic response of reinforced elastomers is modeled using a three-dimensional mixed finite element method with a nonlocal pressure field. A general second-order unconditionally stable exponential integrator based on a diagonal Pade approximation is developed and the Bergstrom-Boyce nonlinear viscoelastic law is employed as a prototype model. An implicit finite element scheme with consistent linearization is used and the novel integrator is successfully implemented. Finally, several viscoelastic examples, including a study of the unit cell for a solid propellant, are solved to demonstrate the computational algorithm and relevant underlying physics.


We have formulated a novel integration algorithm and implemented it into a three-dimensional computational framework to simulate the viscoelastic response of reinforced elastomers. Both material and geometric nonlinearities are treated and the Bergstrom-Boyce viscoelastic model is employed. The finite element framework used in our work is based on a mixed Galerkin method with a nonlocal pressure field and a stabilization bubble, but a different numerical scheme can be used to solve the underling PDE.

The highly nonlinear viscous constitutive law is integrated by a new second-order, unconditionally stable exponential integrator based on a diagonal Pade approximation.  Exact preservation of a unit determinant of a traceless second-order tensor in 2D and the supremum and infimum of determinant in 3D are obtained. A consistent linearization of the resulting system of nonlinear equations has been derived and leads to an efficient solution of the complex, highly nonlinear problem.

Various viscoelastic examples were solved. Large magnitude stretches 75% can be instantaneously applied with the proposed numerical scheme.  To illustrate the ability of the numerical scheme to capture the effect of nonuniform particle spacing and size on viscous flow, we have analyzed a twenty-seven-particle composite system (an idealized solid propellant). The method was shown to capture the viscous flow due to stress concentrations in the vicinity of the particles.

The emphasis of this work has been on the development of a three-dimensional computational framework for the simulation of highly nonlinear viscoelastic reinforced elastomers. For many materials, such as solid propellants, it should also incorporate particle-matrix decohesion and matrix tearing. These two requirements will increase the computational costs associated with the analysis, therefore requiring an efficient parallel implementation of the computational scheme.


The authors gratefully acknowledge support from Alliant Techsystems (ATK-21316), with J. Thompson and Dr. I.L. Davis serving as program monitors, and from the Center for Simulation of Advanced Rockets (CSAR) under contract number B523819 by the U.S. Department of Energy as a part of its Advanced Simulation and Computing program (ASC). The authors also thank Prof. Michael Heath for numerous suggestions that improved the presentation of this paper.

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© 2009 Notre Dame and Dr. Karel Matous