Effects of work-hardening and rate sensitivity on the sheet tensile test
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I.
INTRODUCTION
SINCE the classical analysis by Consid~re ~of the relationship between the uniform elongation in a uniaxial tensile test of a rate-insensitive material and the strain-hardening exponent, n, efforts have been extended to rate-sensitive materials to clarify the effect of the rate-sensitivity, m, on instability and necking. Uniform elongations (i.e., maximum load points) and the total elongations have been investigated using analytical 2-s and experimental methods. 9-~2 The analytical works are mostly limited to one-dimensional, pre-necking, qualitative analysis, while experimental works are limited to attainable combinations of n and m values and may include variations in other material parameters. By applying the finite element method, for a wide range of n and m values, the effect of n and m can be examined separately from each other and from other properties. The finite element modeling (FEM) allows examination of both the uniform strain range and the post-uniform, or necking region. The two-dimensional capability of the FEM employed enables comparison of accuracy with onedimensional simplified analyses and the development of improved understanding of the strain localization process. An arbitrary failure criterion is required in conjunction with FEM results to obtain total elongation values because no failure occurs within the continuum framework imposed on the problem. This work is similar to Ghosh's numerical analysis. 4 The main difference is in the numerical method employed; a semi-two dimensional analysis was carried out using a finite difference method in Reference 4. Also, quantitative expressions of the failure elongation and uniform elongation in terms of a wide range of n and m, and the analysis of biaxial stress effect on the uniform elongations are additions in this work. It will be found that the qualitative conclusions are, however, in good agreement. K. CHUNG, formerly Postdoctoral Research Associate, Department of Metallurgical Engineering, The Ohio State University, is with Alcoa Laboratories, Alcoa Center, PA 15069. R.H. WAGONER is Professor, Department of Metallurgical Engineering, The Ohio State University, Columbus, OH 43210. Manuscript submitted October 28, 1986. METALLURGICALTRANSACTIONS A
II.
PROCEDURE
The uniaxial sheet tension test has been simulated using a two-dimensional finite element method for materials having various hardening coefficients and rate sensitivities, in order to examine the effect on elongations at the maximum tensile force (uniform elongation) and total elongation at the failure point. A detailed discussion follows.
A. Material Modeling In order to model the material, a rate-sensitive, Hollomontype material hardening law 13 -~ = g ~ - n ( - ~ / ~ o ) m
[1]
was used, where o-, 8 (= f d-~), K, n, m, and s0 refer to effective stress, effective strain, strength coefficient, workhardening coefficient, logarithmic strain rate sensitivity index, and the base strain rate, respectively. The dot in Eq. [ 1] denotes the material time derivative. Throughou
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