Eur. J. Mech. A/Solids 18 (1999) 805–818  1999 Èditions scientifiques et médicales Elsevier SAS. All rights reserved Necking in rectangular tensile bars approximated by a 2-D gradient dependent plasticity model Lars Pilgaard Mikkelsen 1 Cambridge University Engineering Department, Trumpington street, Cambridge CB21PZ, England (Received 13 July 1998; received and accepted 4 February 1999) Abstract – Necking and post-necking of a uniaxially stretched tensile test specimen with a rectangular cross-section is investigated. The analysis is based on an enhanced 2-D plane stress finite element model, where the third dimension (the thickness effect) is taken into account by an incorporated length scale. The length scale, fixed with respect to the current deformed state, is incorporated in the continuum description by a finite strain version of a gradient dependent J 2 -flow theory. A convincing agreement with the true 3-D post-necking behaviour is obtained by this approximative 2-D plane stress model. The model gives a realistic prediction of the development of the post-necking zone in the tensile bar. Furthermore, the model is found to capture the transition from a dominating diffuse necking mode for a tensile bar with a square cross section to a dominating oblique localised necking mode for a wide strip; effects which can not be captured by an analysis based on a conventional (local) plane stress model.  1999 Èditions scientifiques et médicales Elsevier SAS tensile test specimen / post-necking behaviour / internal length scale 1. Introduction Necking in elastic-plastic bodies under tensile loading has been studied by a large number of authors. In sheet metal forming, the necking behaviour is an important precursor to the final failure, Keeler (1968). Because the post-necking behaviour is determined by 3-D effects, most of these analyses are performed for specimens where some approximations regarding the deformation is possible. For a round bar, an axisymmetric solution describes the neck development. Hutchinson and Miles (1973) found that bifurcation into a diffuse necking mode occurs slightly after the maximum loading point with an increased delay for a higher thickness to length ratio. For a long thin bar, this agrees with the classical Considére (1885) criterion. In the case of uniaxial tension of a thin wide sheet, Hill (1952) shows that a localised neck will occur along a line inclined to the normal of the tensile direction. This localised necking mode will be initiated at a critical strain equal to twice the critical strain based on the Considére criterion. Alternatively, based on a conventional plasticity theory, the post-necking behaviour of a tensile test specimen with a rectangular cross-section requires a full 3-D finite element analysis. This analysis has been performed by Tvergaard (1993). It is an example for which the resulting post-necking behaviour is due to a competition between the two necking modes described above. In the present paper, a re-analysis of Tvergaard (1993)’s post-necking analysis is performed. Contrary to the 3-D finite element model used by Tvergaard (1993), the analysis herein is based on an enhanced 2-D plane stress model in which an incorporated length scale accounts for the third dimension (the thickness effects), see Benallal and Tvergaard (1995) and Mikkelsen (1997a). Note that a conventional 2-D plane stress model only predict the pre-necking behaviour accurately; post-necking analysis would results in a physically unrealistic 1 Present address: Dept. of Building Technology and Structural Engineering, Aalborg University, DK-9000 Aalborg, Denmark