The Effect of Inertia on Tensile Ductility XIAOYU HU, ROBERT H. WAGONER, GLENN S. DAEHN, and SOMNATH GHOSH One-dimensional numerical simulations of dynamic tensile tests have been carded out over a wide range of test velocities for materials having a Hollornon-type constitutive law with power- law strain-rate sensitivity. A variety of values of the strain-hardening exponent and strain-rate- sensitivity index have been used to analyze the effect of inertia on tensile ductility. Results show that the total elongation of the specimen is enhanced by inertia at high test velocities. This inertial effect varies with the strain-hardening exponent and strain-rate-sensitivity index and can be scaled with the normalized material density and the test velocity. Based on these results, the critical test velocity for the onset of the inertial effect as a function of material parameters has been numerically determined. To account for the effect of inertia on the enhancement of tensile ductility, a simple phenomenological explanation has been proposed. I. INTRODUCTION THE estimation of tensile ductility has been studied from a variety of theoretical perspectives, including those involving plastic-instability and flow-localization (necking) concepts. The earliest analysis describing the onset of instability in uniaxial tension was conducted by ConsidEre, ~j according to whom instability occurs at the load maximum, where the load increment caused by strain hardening is equal to the load decrement caused by geometrical softening. However, Consid6re's crite- rion is only valid for a rate-independent material. For this material, necking and failure follow rapidly after the maximum load is attained. For materials that exhibit rate-dependent behavior, on the other hand, the extensive analytical and experimental investigations of Hart t2j and others t3-26j indicate that strain-rate sensitivity delays the onset of instability per- taining to rate-independent materials and induces post- uniform elongation, leading to an enhancement of tensile ductility. For such materials, several criteria for pre- dicting the onset of instability have been proposed, and relationships between the failure strain, er, and material parameters (the strain-hardening exponent, n, and strain- rate-sensitivity index, m) have been established. Al- though it has been commonly accepted that m is unimportant prior to the maximum load and n is un- important after the maximum load, 1271 a detailed, two- dimensional finite element analysis of the sheet tensile test t22j has revealed that the total elongation, er, is strongly affected by both n and m. Furthermore, it has been of interest to note that for the given n and m values, the strain distribution and necking behavior of rate- dependent materials, which obey power-law strain-rate sensitivity, are independent of the strain rate under quasi-static and isothermal c o n d i t i o n s , t2~,28'291 Although the influence of strain rate on tensile in- stability and necking has been extensively studied, the XIAOYU HU, Postdoctoral Researcher, ROBERT H. WAGONER, Professor, and GLENN S. DAEHN, Associate Professor, Department of Materials Science and Engineering, and SOMNATH GHOSH, Assistant Professor, Department of Engineering Mechanics, are with The Ohio State University, Columbus, OH 43210. Manuscript submitted June 21, 1993. strain rates employed in these investigations were typi- cally limited to low values. When materials deform at low rates, the variation of the material velocity with time is small. In these cases, the material acceleration and the propagation of plastic waves are negligible, so that quasi-static equilibrium can be assumed. The develop- ment of strain gradients during such tests relies only on the static axial-force equilibrium at any cross section of the gage length. A tensile test at high strain rates must be distinguished from a test at low strain rates by the fact that inertia becomes significant. In this case, a dynamic-equilibrium condition should be satisfied. Experimentally, an improvement of tensile ductility at high strain rates has been observed in high-strength steel, ~3~ stainless steel, t311 oxygen-free, high- conductivity copper (OFHC copper), t32,331 tantalum, t34j and Ti alloys, t351 Compared with the results obtained at rates ranging from 10 -4 to 10 -2 S-l, the total elongation of the specimen for these materials increases by 20 to 50 pct at a strain rate around 3 • 103 s -l. Recently, it also has been shown that the plane-strain formability of interstitial-free iron sheet can be dramatically enhanced (to about three times the quasi-static, plane-strain form- ing limit) at a strain rate of 103 s -l. These experiments were performed with an electrohydraulic forming techniqueJ 361 In spite of these interesting observations, however, it is difficult to distinguish the effect of inertia on tensile ductility from strain-rate sensitivity experimentally, be- cause rate sensitivity can change with strain rates, t~sj This difficulty motivates the theoretical modeling of ten- sile ductility at high strain rates. Taylor et a l . 1371 presented a one-dimensional hydro- dynamic formulation to analyze the effect of inertia on the growth rate of an assumed perturbation in a thin, stainless steel sheet stretching dynamically. They found that many perturbations will be harmless at strain rates of 104 s -~ or greater. Within the framework of a one- dimensional theoretical model for uniaxial tension, Fressengeas and Molinari t3s,391 have examined the influ- ence of inertia and thermal softening together on the duc- tility of materials with both the linear perturbation and nonlinear analyses. The calculations revealed a more than 40 pct increase of the total elongation at 500 ~ and around 103 S-1 for tantalum, which was consistent with METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 25A, DECEMBER 1994--2723