An Eulerian multiplicative constitutive model of finite elastoplasticity Mahdi Heidari a , Abolhassan Vafai a, * , Chandra Desai b a Department of Civil Engineering, Sharif University of Technology, P.O. Box 11365-8639, Tehran, Iran b Department of Civil Engineering and Engineering Mechanics, University of Arizona, AZ 85721, Tucson, USA article info Article history: Received 28 July 2008 Accepted 7 May 2009 Available online 21 May 2009 Keywords: Finite strain Elastic-plastic material Constitutive behavior Corotational rate abstract An Eulerian rate-independent constitutive model for isotropic materials undergoing finite elastoplastic deformation is formulated. Entirely fulfilling the multiplicative decomposition of the deformation gradient, a constitutive equation and the coupled elastoplastic spin of the objective corotational rate therein are explicitly derived. For the purely elastic deformation, the model degenerates into a hypoe- lastic-type equation with the Green–Naghdi rate. For the small elastic- and rigid-plastic deformations, the model converges to the widely-used additive model where the Jaumann rate is used. Finally, as an illustration, using a combined exponential isotropic-nonlinear kinematic hardening pattern, the finite simple shear deformation is analyzed and a comparison is made with the experimental findings in the literature. Ó 2009 Elsevier Masson SAS. All rights reserved. 1. Introduction For many engineering applications, a comprehensive analysis of finite elastoplastic deformation is necessary to ensure that accurate results are obtained from simulations in the process of designing or analyzing structures. Progress in the computational capabilities of computers has made it feasible to implement more complex and realistic models. Hence, constitutive modeling of finite elastoplas- ticity has invoked considerable attention. To formulate a plasticity theory accommodated for finite deformation, the classical laws well established for infinitesimal deformation should appropriately be extended. There are several serious barriers in the process of this generalization, which have resulted in conflicting propositions affecting the way formulations and computations are made. Among them is the ambiguity in the decomposition of the total deformation into the elastic and plastic parts (Naghdi, 1990). A variety of decompositions, retaining the additive split for infinitesimal deformation, have already been proposed and debated in the literature. Developing a plasticity theory based on the continuum thermodynamics, Green and Naghdi (1965) additively decomposed the Green strain tensor where the plastic part was introduced as a primitive variable. Nemat-Nasser (1982) elaborated the additive split of the strain rate tensor where the elastic part was prescribed by a hypoelastic relation. Lee (1969) adopted the multiplicative decomposition of the deformation gradient, which became appealing and found increasing applications due to its rich physical content and simple mathematical form. It is not derived merely from the direct extension of the small deformation case, but motivated by physical considerations. The central idea is the notion of local intermediate stress-free configuration at each particle defined by an imaginary destressing process. Thus far, various aspects and formats of the decomposition have been scrutinized in the literature. Nemat- Nasser (1990) showed that the multiplicative decomposition of the deformation gradient could always be recast in the form F[V e QU p , where V e and U p were stretch tensors and Q a rotation tensor. For the small elastic distortion, he established a constitutive model, where the corotational rate was associated with the spin, _ QQ T . Schieck and Stumpf (1995) proposed a multiplicative decomposi- tion as F ¼ Q b U e U p , where U p was a Lagrangean plastic stretch, b U e a back-rotated Lagrangean-type elastic stretch and Q a rotation tensor. Meanwhile, some authors have made efforts to unify the strain rate additive and deformation gradient multiplicative decompositions so that the main discrepancies would disappear (Agah-Tehrani et al., 1987; Xiao et al., 2000; Lubarda, 2004). Ther- modynamical consideration of the multiplicative elastoplasticity has been the subject of numerous publications, where the second law of thermodynamics is recast as the dissipation inequality (Tsakmakis, 2004; Gurtin and Anand, 2005; Lin et al., 2006). Employing the principle of maximum plastic dissipation, Itskov (2004) deduced that contrary to some Lagrangean additive models, only the multiplicative model delivers a non-spurious shear * Corresponding author. Tel.: þ98 21 66005419; fax: þ98 21 66012983. E-mail address: vafai@sharif.edu (A. Vafai). 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.2009.05.002 European Journal of Mechanics A/Solids 28 (2009) 1088–1097