Simple and Effective Approach for Polar Decomposition of the Deformation Gradient Tensor
Publication: Journal of Engineering Mechanics
Volume 140, Issue 5
Abstract
A precise iterative strategy to compute the polar decomposition of the three-dimensional deformation gradient tensor is presented for nonlinear solid mechanics analysis software. By exploiting relationships between various stretch tensors, polar decomposition is transformed into the solution of six nonlinear (quadratic) simultaneous equations, via Newton-Raphson (N-R), and a subsequent matrix inversion and multiplication. The approach is easy to program and is versatile in its applicability to statics or dynamics. With only modest computational increases, the approximations and potential numerical drift of incremental decomposition schemes is avoided. Convergence can be accelerated using stretch histories, which, coupled with a modified N-R approach, can reduce computer times. Using an explicit central difference time integration scheme, convergence is shown to be attained in only a few iterations with very tight tolerances for severe deformations. Numerical examples compare different method variations that are demonstrated to be robust and efficient. The results also demonstrate its effectiveness for large time increments (the entire analysis), which can be especially useful for static and implicit solvers.
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Acknowledgments
Permission to publish was granted by the Director of the Geotechnical and Structures Laboratory. The work was supported in part by a grant of computer time from the Department of Defense (DoD) High Performance Computing Modernization Program at the U.S. Army Engineer Research and Development Center (ERDC) DoD Supercomputing Resource Center (DSRC). The author thanks Professor Rebecca Brannon of the University of Utah for testing the present method on her large set of example deformation gradient tensors.
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Published In
Journal of Engineering Mechanics
Volume 140 • Issue 5 • May 2014
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Mar 4, 2013
Accepted: Oct 15, 2013
Published online: Oct 17, 2013
Published in print: May 1, 2014
Discussion open until: Jun 14, 2014
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