Int. Journ. of Fracture 21 (1983) 261-278. 0376-9429/83/04/0261-1850.20 © 1983 Martinus Nijhoff Publishers, The Hague. Printed in The Netherlands Localization of deformation in rate sensitive porous plastic solids J. PAN*, M. SAJE** and A. NEEDLEMAN Division of Engineering, Brown University, Providence, RI 02912, USA (Received October 14, 1981;in revised form April 3, 1982) ABSTRACT The effect of material rate sensitivity on the localization of deformation in a porous visco-plastic solid is examined under plane strain tension and axisymmetric tension conditions. The plastic flow rule proposed by Gurson [3], modified to account for material rate sensitivity, is adopted to model the plastic softening behavior that arises due to void nucleation and growth. An initial imperfection in the form of a planar band is assumed and a material instability is sought as the deformation proceeds. Comparisons are made with the results of a rate-independent analysis I-10].The present rate-dependent results show that the retardation effecton flowlocalization is larger when the material is more rate-sensitive, and that, with a given rate sensitivity, the retardation effecton flow localization is greater in plane strain tension than in axisymmetric tension. Results are also obtained by employing parameter values representative of spheroidized carbon steels studied by Fisher [21], and the predictions of the model are in good agreement with experimental observations. 1. Introduction The localization of deformation into a shear band is a commonly observed phenomenon in experiments when ductile solids deform into the plastic range. Rice [1] has given a general review of the theoretical framework for analyzing the localization of plastic deformation in rate-independent solids, including both the bifurcation theory and the analysis of the growth of initial imperfections. One mechanism which may induce localization is, as suggested by Berg [2], the nucleation and growth of micro-voids. Gurson [3] has developed a yield function for a voided material, which has subsequently been employed in several analyses [4- 10, 28]. The work carried out here is parallel to the work of [ 10], but material rate sensitivity is incorporated into the present analysis. We consider an element of solid, constrained so as to rule out geometric instabilities, subject to loading conditions that could give rise to a homogeneous deformation state outside the planar imperfection band. We then seek the condition under which the macroscopic constitutive relation permits the deformation rate within the band to become infinite, with the deformation rate outside the band remaining finite. As in the work by Yamamoto [5], we assume a slightly higher initial porosity within the band, motivated by the fact that weakly- bonded inclusions (such as MnS in steel) or second-phase particles (such as carbides in steel) can crack or debond from the matrix during the earliest stages of plastic deformation. Furthermore, the nucleation of voids throughout the deformation history is taken into account. Two kinds of nucleation criteria have been proposed within the context of Gurson's * Present Address: Stress Analysis and Fracture Section, Battelle Columbus Laboratories, Columbus, OH 43221, USA. ** Permanent Address: Faculty for Architecture, Civil Engineering and Survey, Edvard Kardelj University, 61000 Ljubljana, Jamova 2, Yugoslavia.