Computational Physics Group

Generalized Finite Element Method for Modeling Nearly Incompressible Bimaterial Hyperelastic Solids

K.R. Srinivasan¹, K. Matous^1,2 and P.H. Geubelle<sup>1

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

Abstract

An extension of the generalized finite element method to the class of mixed finite element methods is presented to tackle heterogeneous systems with nearly-incompressible nonlinear hyperelastic material behavior. In particular, heterogeneous systems with large modulus mismatch across the material interface undergoing large strains are investigated using two formulations, one based on a continuous deformation map, the other on a discontinuous one. A bimaterial patch test is formulated to assess the ability of the two formulations to reproduce constant stress fields, while a mesh convergence study is used to examine the consistency of the formulations. Finally, compression of a model heterogeneous propellant pack is simulated to demonstrate the robustness of the discontinous deformation map formulation.

Conclusions

The present work provides a numerical framework that combines the generalized finite element method with the classical mixed finite element method. Two formulations, based on a continuous and discontinuous deformation map, were derived and discretized for the motion of a bimaterial nearly-incompressible hyperelastic solid. The two formulations were assessed numerically on the low-order Q₁/P₀ element, which is very popular in engineering practice, using a bimaterial patch test and mesh convergence studies were carried out to evaluate the consistencies of the formulations. It was observed that both the continuous and discontinuous deformation maps yield convergent schemes for moderate modulus mismatches, while the continuous deformation map appears to be non-convergent for large mismatches. Finally, an idealized heterogeneous solid propellant pack is chosen as an example to demonstrate the capability and robustness of the discontinuous deformation map formulation.

Acknowledgment

 
This work was supported by the Center for Simulation of Advanced Rockets (CSAR) under contract number B341494 by the U.S. Department of Energy.  K. Matous also acknowledges support from ATK/Thiokol, ATK-21316 (Program Managers, J. Thompson and Dr. I. L. Davis). The authors would also like to thank Prof. C. A. Duarte for helpful discussions and comments.

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