A Self-Consistent Approach for Necking Correction in Tensile Specimens With Rectangular Cross-Section Using a Novel Mirr
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THE
necking behavior, observed during tensile testing of metals, has been studied extensively.[1–12] Bridgman[1] analyzed the necking behavior for specimens with circular cross-section. In Bridgman’s correction, the necking geometry is quantified using the curvature of the necked region, R, and the radius of the specimen, a. Then the equivalent true stress, req , is computed by multiplying the experimentally calculated axial stress, rZZ ; by a correction factor which is a function of the neck geometry: r ZZ req ¼ ½1 2R a 1 þ a In 1 þ 2R The correction factor can be represented simply by f, req ¼ rzz f
½2
Where,
K.E. YAZZIE, formerly Graduate Research Associate, with Materials Science and Engineering, Arizona State University, Tempe, AZ 85287-6106, and is now Senior Engineer with Assembly Test and Technology Development, Intel Corporation, Chandler, AZ. H. FEI, formerly Graduate Research Associate, with Mechanical and Aerospace Engineering, Arizona State University, and is now Senior Engineer with Assembly Test and Technology Development, Intel Corporation. H. JIANG, Associate Professor, is with Mechanical and Aerospace Engineering, Arizona State University. N. CHAWLA, Fulton Professor of Materials Science and Engineering, is with Materials Science and Engineering, Arizona State University. Contact e-mail: [email protected] Manuscript submitted April 1, 2012. Article published online August 7, 2012 5058—VOLUME 43A, DECEMBER 2012
f¼
1 1
ð
ÞInð
1þ2R a
a 1þ2R
Þ
prior to necking after necking
½3
The Bridgman correction shows that correcting for necking requires a two-step process: First the neck geometry must be quantified, and then a correction factor can be calculated to obtain the equivalent true stress from the axial stress. Though the Bridgman correction has been widely used, it can only be directly applied to samples with circular cross-section because it assumes an axisymmetric stress distribution.[1–3] Specimens with rectangular cross-section are frequently used due to ease of machining or limitations imposed by the starting material. However, efforts have been made to develop a necking correction for rectangular specimens. Aronofsky[4] showed that the stress distribution is not uniform in a necked rectangular specimen. Tvergaard[5] showed that the necking geometry is complex for rectangular tensile specimens and varies with aspect ratio. Some researchers have foregone the quantification of neck geometry by substituting a/R in Eq. [3] with the reduced strain, e–en where e is the true plastic strain and en is the true plastic strain at the onset of necking, multiplied by a constant.[6,7] While it has been shown that this method is within the error inherent to the Bridgman correction, it involves rigorous experimental work to determine the correct constant for each geometry and material.[8] Zhang et al.[9] developed an approximation to determine the necking geometry, but they relied on the Bridgman correction for determining true stress. Ling[10] developed a weighted average method to calculate
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