Research Papers

Quantitative Stress Analysis of Recrystallized OFHC Cu Subject to Deformation In Situ

[+] Author and Article Information
Joel V. Bernier1

Engineering Technologies Division, Lawrence Livermore National Laboratory, B141/R1114, L-229, Livermore, CA 94551bernier2@llnl.gov

Matthew P. Miller

Sibley School of Mechanical and Aerospace Engineering, Cornell University, 195 Rhodes Hall, Ithaca, NY 14853mpm4@cornell.edu

Jun-Sang Park

Sibley School of Mechanical and Aerospace Engineering, Cornell University, 195 Rhodes Hall, Ithaca, NY 14853jp118@cornell.edu

Ulrich Lienert

Advanced Photon Source, Argonne National Laboratory, B431/A007, Argonne, IL 60439lienert@aps.anl.gov

Contracted indexing is implied; e.g., for the second-rank symmetric tensor, ϵ:ϵ11ϵ1, ϵ22ϵ2, ϵ33ϵ3, ϵ23ϵ4, ϵ13ϵ5, and ϵ12ϵ6.


Corresponding author.

J. Eng. Mater. Technol 130(2), 021021 (Mar 28, 2008) (11 pages) doi:10.1115/1.2870234 History: Received August 08, 2007; Revised January 23, 2008; Published March 28, 2008

Quantitative strain analysis (QSA) provides a means for assessing the orientation-dependent micromechanical stress states in bulk polycrystalline materials. When combined with quantitative texture analysis, it facilitates tracking the evolution of micromechanical states associated with selected texture components for specimens deformed in situ. To demonstrate this ability, a sheet specimen of rolled and recrystallized oxygen-free high conductivity Cu was subject to tensile deformation at APS 1-ID-C. Strain pole figures (SPFs) were measured at a series of applied loads, both below and above the onset of macroscopic yielding. From these data, a lattice strain distribution function (LSDF) was calculated for each applied load. Due to the tensorial nature of the LSDF, the full orientation-dependent stress tensor fields can be calculated unambiguously from the single-crystal elastic moduli. The orientation distribution function (ODF) is used to calculate volume-weighted average stress states over tubular volumes centered on the 100[100], 311[100], and 111[100] fibers—accounting for 50% of the total volume—are shown as functions of the applied load along [100]. Corresponding weighted standard deviations are calculated as well. Different multiaxial stress states are observed to develop in the crystal subpopulations despite the uniaxial nature of the applied stress. The evolution of the orientation-dependent elastic strain energy density is also examined. The effects of applying stress bound constraints on the SPF inversion are discussed, as are extensions of QSA to examine defect nucleation and propagation.

Copyright © 2008 by American Society of Mechanical Engineers
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Figure 1

(a) shows a stress-strain curve obtained for LD∥RD. The test was run at room temperature in air at a constant engineering strain rate of 10−4s−1. The markers denote loads at which the sample was briefly held to measure a SPF: σa∊{133MPa,167MPa,200MPa,242MPa,262MPa,268MPa}. (b) shows an optical micrograph of the mechanically polished and etched Cu sheet, with the grain boundaries in high contrast.

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Figure 2

(a) shows level surfaces of the ODF at 2.5 and 25 MUD plotted over the cubic fundamental region. The lower surface is semi-transparent, while the higher is the opaque sphere near the center. (b) shows calculated IPDFs corresponding to LD, TD, and ND (left to right) plotted over the standard stereographic triangle for cubic symmetry as an equal-area projection (vertices are ⟨100⟩, ⟨110⟩, and ⟨111⟩ ccw from lower left). Units are MUD.

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Figure 3

The directional modulus ED{hkl} plotted as an IPF in an equal-area projection along the tensile axis [100]. Units are GPa.

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Figure 4

Schematic of the geometry for the in situ test showing the tensile specimen, image plate, and goniometer angle conventions (χ,ω) for relating the sample LD-TD-ND frame to the laboratory frame {X,Y,Z}. The X′ and Y′ axes correspond to the detector-based horizontal and vertical coordinates. The sample is shown at (−25deg,20deg), and the incident beam is along −Z. A diffracted beam along with its associated scattering vector s are shown for a particular 2θ and η. Note that s is coplanar with the incident and diffracted beams and cos−1(s⋅Z)=90deg−θ.

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Figure 5

The PF coverage pattern from the in situ experiment is shown as an equal-area projection along ND. The seven “rings” of points are shaded by the order of measurement (left to right in the table; black to white in the figure).

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Figure 6

Left: A diffraction pattern for the Cu specimen at normal incidence (χ,ω=0deg) and σa=268MPa. The direct beam position is marked with an X. The rings with homogeneous azimuthal intensities arise from the CeO2 standard on the downstream face. The sectors used for polar rebinning are defined by the dashed lines. Right: The rebinned image in polar coordinates (ρ,η) with Δρ=70μm and Δη=10deg. The strains are visible in the indicated Cu reflections as the η-dependent shifts in ρ, with minima at ηLD=0deg, 180deg corresponding to tensile strains and maximum at ηTD=90deg and 270deg corresponding to compressive strains. The corresponding SPFs are shown in Fig. 7.

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Figure 7

(a) shows the measured (top) and recalculated (bottom) SPFs for σa=268MPa. The glyph sizes are scaled by the associated PDF values; i.e., larger points⇒larger volume fraction of crystals along underlying fiber. The RP¯ errors from left to right are 11.8%, 5.3%, and 5.8%. (b) and (c) show the corresponding stress component distributions on the boundary and midplanes of the cubic fundamental region of the Rodrigues space. Note the sign reversals in the shears and the rough symmetry of the components.

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Figure 8

Inverse stress PFs for the indicated s and each of the six applied loads shown in Fig. 1 in an equal-area projection over the stereographic triangle (from left to right, top to bottom). Since LD-TD-ND is aligned with {ŝi}, these represent the weighted averages of the σ1, σ2, and σ3 components using Ñsiσ (see Eq. 9). Note that these are normal tensor components, not principal stresses.

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Figure 9

The shaded points represent the cubature points used to define volumes within 10deg of the indicated fibers. The volume fraction of materials associated with each tube centered on ⟨100⟩∥LD, ⟨311⟩∣LD, and ⟨111⟩∥LD are 34.6%, 11.6%, and 11.5%, respectively. The weighted mean stress tensors over each tube are listed in Table 2 for the six values of σa shown in Fig. 1.

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Figure 10

PFs showing elastic strain energy density for the indicated s and each of the six applied loads shown in Fig. 1 (from left to right, top to bottom). Each point represents an ODF-weighted mean, similar to that defined in Eq. 9 with s⋅σ(R)⋅s replaced with U(R)=12σ(R):ϵ(R). Units are kJ∕m3.




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