An analysis of the nonisothermal tensile test
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I.
INTRODUCTION
THE formability of metals has
received considerable experimental and theoretical attention over the past several decades. ~': This work has focused on failure in both primary, or bulk, forming operations conducted under largely compressive states of loading as well as secondary forming processes such as sheet forming. The latter, involving tensile deformation, include such operations as deep drawing and stretching. 3'4'5 For deep drawing, formability is usually quantified by the limiting drawing ratio (LDR), D/d, where D is the maximum diameter of a circular blank that can be successfully drawn into a right cylindrical cup of diameter d. Relatively simple analyses exist for predicting the LDR of metals as a function of such properties as the normal plastic anisotropy. 6'7'8 In stretching, a number of material, geometric, and lubrication factors have been found to influence formability, and various test techniques have been devised to evaluate them. These include punch Cout-of-plane") stretching and "in-plane" stretching. 9-~2 Forming limits for punch stretching are difficult to predict, often requiring sophisticated finite element methods. 13 Those for in-plane stretching, on the other hand, are more amenable to simple analyses such as that first proposed by Marciniak and Kuczynski. H.~4Often these simple techniques postulate the presence of a material imperfection as the source of the localized thinning which controls failure. The material imperfection is usually cited as consisting of an inhomogeneity in thickness, strength, or other material property, ts The rate of growth of the imperfection to form a localized neck is then determined by the imposed deformation path and the magnitude of the strain and strain-rate hardening capacities of the material. At present, relatively little effort has been put into determining the effect of heat generation and heat transfer on forming limits of sheet metals. 16 That temperature effects can play an important role in deep drawing is now S. L. SEMIATIN is Principal Research Scientist, Metalworking Section, Battelle's Columbus Laboratories, Columbus, OH 43201. ROBERT A. AYRES is Staff Development Engineer, CPC Headquarters Building, General Motors Corporation, Warren, MI 48090. J.J. JONAS is Professor, Department of Metallurgical Engineering, McGill University, Montreal, ~PQ, H3A 2A7, Canada. Manuscript submitted March 29, 1985. METALLURGICALTRANSACTIONSA
recognized. 17In contrast, such phenomena in stretching are largely unexplored from either an experimental or a theoretical viewpoint. The present work was undertaken as a first step in the development of a model for the stretching process which includes thermal effects. To this end, an analysis of the simplest form of tensile deformation, namely, that of the uniaxial tensile test, was conducted. The results of the simplified one-dimensional approach taken here demonstrate the importance of temperature gradients in limiting ductility in tension and other modes of forming. II.
ANALYSIS
A. Problem Formula
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