Combined hardening and softening constitutive model of plasticity: precursor to shear slip line failure A. Ibrahimbegovic, D. Brancherie Abstract In this work, we present a ﬁnite element model capable of describing both the plastic deformation which accumulates during the hardening phase as the precursor to failure and the failure process leading to softening phenomena induced by shear slip lines. This is achieved by activating subsequently hardening and softening mecha- nisms with the localization condition which separates them. The chosen model problem of von Mises plasticity is addressed in detail, along with particular combination of mixed and enhanced ﬁnite element approximations which are selected to control the locking phenomena and guar- antee mesh-invariant computation of plastic dissipation. Several numerical simulations are presented in order to illustrate the ability of the presented model to predict the ﬁnal orientation of the shear slip lines for the case of non- proportional loading. Keywords Strain localization, Shear bands, B-bar method, Incompatible modes method 1 Introduction Ever increasing demand to achieve a more economical structural design requires that a better understanding be obtained of the non-linear behavior of a particular struc- tural system and a reliable estimate be furnished of its limit load. The limit state of a complex system Crisﬁeld 1997a and Crisﬁeld 1997b often implies the presence of particular components whose peak resistance has been already defeated forcing them at that stage to function in post-peak or softening regime. It is well known by now that the strain-softening phe- nomena can not reliably be represented by the classical continuum mechanics model, and different remedies leading to a non-standard interpretation of different model ingredients have been proposed: non-local continuum (Bazant et al. 1984), higher order gradient plasticity (Coleman and Hodgdon 1985), viscoplasticity regulariza- tion (Needleman 1988) or Cosserat continuum (De Borst and Sluys 1991), among others. All the modiﬁcations of this kind, usually referred to as localization limiters (see Belytschko and Lasry 1989) lead to a signiﬁcant increase of complexity in formulating the strain localization problems and even more in solving it numerically. For that reason, they seem to be nowadays often replaced by a particular modiﬁcation of the classical continuum which allows that either displacement or strain discontinuities enter the formulation (Ortiz et al. 1987; Simo et al. 1993; Belytschko et al. 1988; Oliver 1995; Armero and Garikipati 1995; Runesson et al. 1991; Jirasek and Zimmermann 2001 or Qiu et al. 2001 among others). The main advantage of the modiﬁed continuum models of this kind is to provide the adequate measure of the total inelastic dissi- pation of the strain softening component regardless of the chosen ﬁnite element mesh, which is the principal information of interest for the limit load computation. However, the vast majority of previous works are typically developed in combination with elastic response, so they can consider only either elastic or strain softening states and thus completely ignore the possible inelastic defor- mation which can spread outside the strain-softening zone (i.e. the components acting in strain hardening regime). In this work, we develop a model capable of taking into account both the contribution of a strain hardening as well as a strain softening model component. Such a model is considered to be a more realistic representation of the limit state of a massive structural system where both kind of inelastic dissipations (either strain-hardening or strain- softening) are considered to be non negligible. The model problem considered in detail is the metal plasticity with the von Mises yield criterion governing the strain hard- ening behavior and shear slip line representing the strain softening behavior. In this case, taking into account the hardening phase will lead in general to stress redistribu- tion and better estimate of shear slip line orientation. It is considered that these two types of behavior are separated by the localization condition, with the strain-hardening behavior appearing as the precursor to shear slip line creation. The crucial condition which allows to connect two states, the states of pre- and post-localization, pertains to imposing the equivalence of the corresponding dissi- pations, which can be cast as the stress orthogonality with respect to localization induced enhanced strain ﬁeld. The standard ﬁnite element implementation ought to be modiﬁed in order to account for this orthogonality con- dition and addition of corresponding displacement modes Computational Mechanics 31 (2003) 88–100 Ó Springer-Verlag 2003 DOI 10.1007/s00466-002-0396-x 88 A. Ibrahimbegovic (&), D. Brancherie E ´ cole Normale Supe ´rieure de Cachan, Laboratoire de Me ´canique et Technologie de Cachan 61, av. du pre ´sident Wilson 94235 Cachan, France e-mail: ai@lmt.ens-cachan.fr Dedicated to the memory of Prof. Mike Crisﬁeld, for his cheerfulness and cooperation as a colleague and friend over many years. This work was supported by the French Ministry of Research and ACI research program. This support is gratefully acknowledged.