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Title: An Eulerian projection method for quasi-static elastoplasticity

Abstract

A well-established numerical approach to solve the Navier-Stokes equations for incompressible fluids is Chorin's projection method, whereby the fluid velocity is explicitly updated, and then an elliptic problem for the pressure is solved, which is used to orthogonally project the velocity field to maintain the incompressibility constraint. In this paper, we develop a mathematical correspondence between Newtonian fluids in the incompressible limit and hypo-elastoplastic solids in the slow, quasi-static limit. Using this correspondence, we formulate a new fixed-grid, Eulerian numerical method for simulating quasi-static hypo-elastoplastic solids, whereby the stress is explicitly updated, and then an elliptic problem for the velocity is solved, which is used to orthogonally project the stress to maintain the quasi-staticity constraint. We develop a finite-difference implementation of the method and apply it to an elasto-viscoplastic model of a bulk metallic glass based on the shear transformation zone theory. We show that in a two-dimensional plane strain simple shear simulation, the method is in quantitative agreement with an explicit method. Like the fluid projection method, it is efficient and numerically robust, making it practical for a wide variety of applications. We also demonstrate that the method can be extended to simulate objects with evolving boundaries. Here, wemore » highlight a number of correspondences between incompressible fluid mechanics and quasi-static elastoplasticity, creating possibilities for translating other numerical methods between the two classes of physical problems.« less

Authors:
 [1];  [2];  [3]
+ Show Author Affiliations
  1. Harvard Univ., Cambridge, MA (United States). Paulson School of Engineering and Applied Sciences; Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Dept. of Mathematics
  2. Simon Fraser Univ., Burnaby, BC (Canada). Dept. of Mathematics
  3. Weizmann Inst. of Science, Rehovot (Israel). Chemical Physics Dept.
Publication Date:
Research Org.:
Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1525116
Alternate Identifier(s):
OSTI ID: 1250020
Grant/Contract Number:  
AC02-05CH11231
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 300; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Fluid mechanics; Chorin-type projection method; Plasticity; Elastoplasticity

Citation Formats

Rycroft, Chris H., Sui, Yi, and Bouchbinder, Eran. An Eulerian projection method for quasi-static elastoplasticity. United States: N. p., 2015. Web. doi:10.1016/j.jcp.2015.06.046.
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Rycroft, Chris H., Sui, Yi, & Bouchbinder, Eran. An Eulerian projection method for quasi-static elastoplasticity. United States. https://doi.org/10.1016/j.jcp.2015.06.046
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Rycroft, Chris H., Sui, Yi, and Bouchbinder, Eran. Sun . "An Eulerian projection method for quasi-static elastoplasticity". United States. https://doi.org/10.1016/j.jcp.2015.06.046. https://www.osti.gov/servlets/purl/1525116.
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@article{osti_1525116,
title = {An Eulerian projection method for quasi-static elastoplasticity},
author = {Rycroft, Chris H. and Sui, Yi and Bouchbinder, Eran},
abstractNote = {A well-established numerical approach to solve the Navier-Stokes equations for incompressible fluids is Chorin's projection method, whereby the fluid velocity is explicitly updated, and then an elliptic problem for the pressure is solved, which is used to orthogonally project the velocity field to maintain the incompressibility constraint. In this paper, we develop a mathematical correspondence between Newtonian fluids in the incompressible limit and hypo-elastoplastic solids in the slow, quasi-static limit. Using this correspondence, we formulate a new fixed-grid, Eulerian numerical method for simulating quasi-static hypo-elastoplastic solids, whereby the stress is explicitly updated, and then an elliptic problem for the velocity is solved, which is used to orthogonally project the stress to maintain the quasi-staticity constraint. We develop a finite-difference implementation of the method and apply it to an elasto-viscoplastic model of a bulk metallic glass based on the shear transformation zone theory. We show that in a two-dimensional plane strain simple shear simulation, the method is in quantitative agreement with an explicit method. Like the fluid projection method, it is efficient and numerically robust, making it practical for a wide variety of applications. We also demonstrate that the method can be extended to simulate objects with evolving boundaries. Here, we highlight a number of correspondences between incompressible fluid mechanics and quasi-static elastoplasticity, creating possibilities for translating other numerical methods between the two classes of physical problems.},
doi = {10.1016/j.jcp.2015.06.046},
journal = {Journal of Computational Physics},
number = C,
volume = 300,
place = {United States},
year = {Sun Nov 01 00:00:00 EDT 2015},
month = {Sun Nov 01 00:00:00 EDT 2015}
}
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Journal Article:
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Publisher's Version of Record
https://doi.org/10.1016/j.jcp.2015.06.046
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Cited by: 13 works
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Figures / Tables:

Fig. 1 Fig. 1: A schematic representation of the timestep in (a) the projection method for the incompressible Navier–Stokes equations and (b) the projection method for quasi-static elastoplasticity.
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