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Research Papers

# Lattice Rotation Patterns and Strain Gradient Effects in Face-Centered-Cubic Single Crystals Under Spherical Indentation

[+] Author and Article Information
Y. F. Gao

Department of Materials Science and Engineering,
University of Tennessee,
Knoxville, TN 37996
Materials Science and Technology Division,
Oak Ridge National Laboratory,
Oak Ridge, TN 37831
e-mail: ygao7@utk.edu

B. C. Larson

Materials Science and Technology Division,
Oak Ridge National Laboratory,
Oak Ridge, TN 37831
e-mail: lasonbc@ornl.gov

J. H. Lee

Research Reactor Mechanical Structure
Design Division,
Korea Atomic Energy Research Institute,
Daejeon 305-353, South Korea

L. Nicola

Department of Materials Science and Engineering,
Delft University of Technology,
Delft 2628 CD, The Netherlands

J. Z. Tischler

X-Ray Sciences Division,
Argonne National Laboratory,
Argonne, IL 60439

G. M. Pharr

Department of Materials Science and Engineering,
University of Tennessee,
Knoxville, TN 37996
Materials Science and Technology Division,
Oak Ridge National Laboratory,
Oak Ridge, TN 37831

1Corresponding authors.

Manuscript received March 16, 2015; final manuscript received April 17, 2015; published online April 30, 2015. Editor: Yonggang Huang.

The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States government purposes.

J. Appl. Mech 82(6), 061007 (Jun 01, 2015) (10 pages) Paper No: JAM-15-1143; doi: 10.1115/1.4030403 History: Received March 16, 2015; Revised April 17, 2015; Online April 30, 2015

## Abstract

Strain gradient effects are commonly modeled as the origin of the size dependence of material strength, such as the dependence of indentation hardness on contact depth and spherical indenter radius. However, studies on the microstructural comparisons of experiments and theories are limited. First, we have extended a strain gradient Mises-plasticity model to its crystal plasticity version and implemented a finite element method to simulate the load–displacement response and the lattice rotation field of Cu single crystals under spherical indentation. The strain gradient simulations demonstrate that the forming of distinct sectors of positive and negative angles in the lattice rotation field is governed primarily by the slip geometry and crystallographic orientations, depending only weakly on strain gradient effects, although hardness depends strongly on strain gradients. Second, the lattice rotation simulations are compared quantitatively with micron resolution, three-dimensional X-ray microscopy (3DXM) measurements of the lattice rotation fields under 100 mN force, 100 μm radius spherical indentations in $〈111〉$, $〈110〉$, and $〈001〉$ oriented Cu single crystals. Third, noting the limitation of continuum strain gradient crystal plasticity models, two-dimensional discrete dislocation simulation results suggest that the hardness in the nanocontact regime is governed synergistically by a combination of strain gradients and source-limited plasticity. However, the lattice rotation field in the discrete dislocation simulations is found to be insensitive to these two factors but to depend critically on dislocation obstacle densities and strengths.

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## Figures

Fig. 1

(a) Indentation hardness H as a function of indentation depth h for the Berkovich indentation on iridium single crystal. (Reproduced with permission from Swadener et al. [14] and Huang et al. [36]. Copyright 2002 and 2006 by Elsevier Science Ltd.) (b) The size dependence is often interpreted by the extra hardening due to the GNDs, i.e., the Nix–Gao model for pyramidal indentation. (c) The generalized Nix–Gao model for spherical indentation.

Fig. 2

The total deformation gradient can be decomposed into a plastic part due to crystalline slip and an elastic part that considers stretching and rotation

Fig. 3

(a) Local coordinates ξi used in the abaqus C3D10M element with ten nodes. (b) Isoparametric mapping of the four Gaussian integration points, which form a dashed tetrahedron in (a), to a tetrahedron in the local coordinates ζi.

Fig. 4

Geometric setup of the finite element simulations by a spherical indenter. Only one-quarter of the substrate is shown for illustrative clarity. Contour plot of the(111)[01¯1] slip strain is plotted at peak load of 100 mN for the indenter radius of 100 μm.

Fig. 5

Lattice rotations about the 〈110〉 direction on the cross section (as denoted by the white box in Fig. 4) for 〈111〉 indentations with three different indenter radii: (a) R = 100 μm, (b) R = 30 μm, and (c) R = 15 μm. The strain gradients have weak effects on these rotation fields.

Fig. 6

Finite element simulations of the strain gradient crystal plasticity predictions. (a) A negligible dependence of rotation angle (given in degrees) on the indenter radius. (b) A strong dependence of indentation hardness on the indenter radius. (c) The hardness relates to the indenter radius by the relationship of H/H0=1+R*/R.

Fig. 7

Two sets of load–displacement (P–h) curves for 〈111〉 indentation of Cu using a 100 μm radius spherical indenter and 100 mN loads; the P–h curves are separated laterally for clarity. The P–h curves corrected for finite machine stiffness were fit with finite element modeling simulations yielding constitutive parameters g0 = 2 MPa, gs = 35 MPa, h0 = 550 MPa, and q = 1.3.

Fig. 8

Indentation geometry and lattice rotation fields produced by P = 100 mN load and R = 100 μm radius spherical indentations along the 〈111〉, 〈001〉, and 〈110〉 directions in Cu. (a) Schematic illustration of the indentation geometry for the 〈111〉 direction case; the blue (clockwise) and red (counterclockwise) rotations depicted by the curved arrows are about a 〈110〉 type direction, as is the case for each of the three indentation directions in part (b) below. (b) 3DXM experimental measurements of indenter induced lattice rotations about a 〈110〉 axis performed on beamline 34ID-E at the APS (left) and by finite element simulations (right). The respective 〈111〉, 〈001〉, and 〈110〉 indentation directions are indicated in the 20 μm by 40 μm panels (see 10 μm scale bars). (See online figure for color.)

Fig. 9

Two-dimensional discrete dislocation simulations of wedge contact indentation considering only the horizontal slip system. (a) Simulations with no obstacles with schematic illustrations of dislocation patterns and lattice rotations overlaid on the plot. (b) Simulations with an obstacle density of 400 μm−2.

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