Local solution of the stress and strain fields in the necking section of cylindrical bars under uniaxial tension Sharif Shahbeyk a, * , Davood Rahiminejad a , Nik Petrinic b a Department of Civil Engineering, Tarbiat Modares University, Jalal Ale Ahmad Highway, P.O. Box 14115,143 Tehran, Iran b Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK article info Article history: Received 17 June 2009 Accepted 15 October 2009 Available online 23 October 2009 Keywords: Necking Tensile test Stress–strain curve Plasticity FDM abstract The problem of finding the stress and strain fields over the minimum cross section of necked cylindrical bars under uniaxial tensile load has been solved locally using a new fast numerical method. The scheme delivers both the accuracy of the finite element analysis and the applicability of simple closed-form analytical solutions. The required inputs are the distributions of curvature radii for both isostatic and material lines. It is numerically observed that the mathematical formulas available in the literature fail to adequately predict these distributions. Introducing the stress normalized strain-hardening rate as the most decisive parameter affecting the curvature radii, a database and interpolation technique have been developed in order to estimate the necessary information based on the results of the previously FE analyzed samples. Finally, a practical case has been solved and compared with the FE results. Ó 2009 Elsevier Masson SAS. All rights reserved. 1. Introduction The determination of complete effective stress-effective plastic strain curve is a key element in successful constitutive modeling of metals. Among all applicable procedures to extract such data, uniaxial monotonic and cyclic tensile tests on cylindrical intact and notched bars are the most employed in practice. However, the initiation and evolution of the necking region in simple bars or the variation of root radius in notched samples lead to a complex problem in terms of both stresses and strains inside the minimum cross section. Therefore, an accurate solution of the neck problem is inevitable if a representative material stress–strain curve has been sought. This demand has been led to a considerable volume of studies which can be categorized into two main groups: analytical and FEA. The first clever analytical solution for the distribution of stresses in the minimum cross section of a necked cylindrical bar is attrib- uted to Bridgman (1944, 1956). The most important assumption in his approach is the uniformity of the radial strain over the necking section. This small assumption, supported by the early experi- mental observations of Davidenkov and Spiridonova (1946), dramatically simplifies the problem and results in a closed form prediction of the axial stress as a function of the radial coordinate. In addition, the uniformity of the radial strain will lead to the equality of the radial and tangential strains, the uniformity of the axial strain, and consequently a constant level of the equivalent plastic strain over the section. The solution has been modified by Davidenkov and Spiridonova (1946) for some of the quantities such as the curvature radius of the isostatic lines. Using a quadratic form of the axial strain distribution over the section, Yamashita (1966) has refined the solution to get more accurate results. Kaplan (1973) and Jones et al. (1979) have extended the Bridgman’s solution beyond the minimum cross section to estimate the neck profile. Hutchinson and Neale (1977) have proposed an analytical three- dimensional solution for an incipient neck. The stress state on the revolution axis away from the minimum cross section has been determined in a closed form by Argon et al. (1975). Eisenberg and Yen (1983) and Eisenberg (1985) have extended the Bridgman’s solution to anisotropic bars. Recently, in the absence of the shear stress, Valiente (2001) has introduced an important kinematic relation between the radial strain rate, the variation of curvature radius, and the partial derivative of the axial strain rate in respect to the radial coordinate at the free edge of an axisymmetric notched tensile bar. This rela- tion is used to complete Bridgman’s formula in order to predict the entire load-minimum diameter curve of an axisymmetric blunt notched bar under tension. Mirone (2004) introduces two inno- vative material independent MLR functions based on the compre- hensive experimental results on various metals. In addition, he assumes that the distribution of axial stress over the minimum cross section has a quadratic form. Comparing with the FE results, it * Corresponding author. Tel.: þ98 21 82884359; fax: þ98 21 82884914. E-mail address: shahbeyk@modares.ac.ir (S. Shahbeyk). Contents lists available at ScienceDirect European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol 0997-7538/$ – see front matter Ó 2009 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2009.10.003 European Journal of Mechanics A/Solids 29 (2010) 230–241