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Research Papers

Computational Aspects of Elasto-Plastic Deformation in Polycrystalline Solids

[+] Author and Article Information
Ronaldo I. Borja

Department of Civil and Environmental Engineering,  Stanford University, Stanford, CA 94305borja@stanford.edu

Helia Rahmani

Department of Civil and Environmental Engineering,  Stanford University, Stanford, CA 94305

J. Appl. Mech 79(3), 031024 (Apr 06, 2012) (9 pages) doi:10.1115/1.4005898 History: Received July 26, 2011; Revised November 10, 2011; Posted February 13, 2012; Published April 06, 2012; Online April 06, 2012

The overall elasto-plastic behavior of single crystals is governed by individual slips on crystallographic planes, which occur when the resolved shear stress on a critical slip system reaches a certain maximum value. The challenge lies in identifying the activated slip systems for a given load increment since the process involves selection from a pool of linearly dependent slip systems. In this paper, we use an “ultimate algorithm” for the numerical integration of the elasto-plastic constitutive equation for single crystals. The term ultimate indicates exact integration of the elasto-plastic constitutive equation and explicit tracking of the sequence of slip system activation. We implement the algorithm into a finite element code and report the performance for polycrystals subjected to complicated loading paths including non-proportional and reverse/cyclic loading at different crystal orientations. It is shown that the ultimate algorithm is comparable to the widely used radial return algorithm for J2 plasticity in terms of global numerical stability.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Kinematics of crystal slips

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Figure 2

Euler angles defining crystal axes (xc , yc , zc ) relative to the fixed system (x, y, z)

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Figure 3

Uniaxial loading of a cubical solid with a square cross section

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Figure 4

Lateral movement of top end (cross-section with a mesh) relative to bottom end (cross-section without a mesh) at different crystal orientations. Displacements magnified 80 ×.

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Figure 5

Deformation band forming in the cubical sample for crystal orientation 1. Color bar is y-displacement in cm.

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Figure 6

Cyclic twisting of a cylindrical solid with a circular cross section

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Figure 7

Cyclic torsion versus angular twist for cylindrical solid at crystal orientations O1, O2, and O3

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Figure 8

Lateral and vertical movement of top end (cross-section with a mesh) relative to bottom end (cross-section without a mesh) at different crystal orientations. Lateral displacements magnified 80 ×. Crystal orientation 3 produced pure twisting with no rocking.

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Figure 9

Finite element mesh for a hollow cylinder subjected to torsional twisting. The cylinder has a height of 4 m, outer diameter of 2 m, and thickness of 0.1 m. The mesh has 5148 nodes and 2560 eight-node hexahedral elements, all integrated with the B-bar option.

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Figure 10

Deformation bands forming in the hollow cylinder subjected to torsional twisting: (a) uniform crystal orientation 3 and (b) crystal orientation 3 with an imperfection in the form of crystal orientation 1 in four adjacent elements. Color bar is second invariant of deviatoric plastic strain in percent.

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Figure 11

Predictor phase for crystal plasticity calculations

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Figure 12

Plastic integrator based on the ultimate algorithm [9]

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