Computational Physics Group

Karel Matous









Stabilized four-node tetrahedron with nonlocal pressure for

modeling hyperelastic materials

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 hyperelastic response of reinforced elastomers is modeled using a novel three-dimensional mixed finite element method with a nonlocal pressure field. The element is unconditionally convergent and free of spurious pressure modes. Nonlocal pressure is obtained by an implicit gradient technique and obeys the Helmholtz equation. Physical motivation for this nonlocality is shown. An implicit finite element scheme with consistent linearization is presented. Finally, several hyperelastic examples are solved to demonstrate the computational algorithm including the inf-sup and verifications tests.


We have developed a novel three-dimensional finite element scheme for nearly incompressible solids. The finite element framework is based on a mixed Galerkin method with a nonlocal pressure field and a stabilization bubble. The pressure spreading effect is governed by the Helmholtz equation and it is motivated by the physical nonlocal response of the reinforced elastomers. 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 hyperelastic examples were solved including the verification example to test the implementation. The element performance in the nearly incompressible limit was assessed by the inf-sup optimality and stability conditions. The emphasis of this work has been on the development of a three-dimensional computational framework for the simulation of highly nonlinear hyperelastic elastomers. For many materials, such as solid propellants, it should also incorporate particle-matrix decohesion, matrix tearing and nonlinear viscoelastic behavior of a binder. These 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