formulation for modeling
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
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.