7 PVP-Vol. 369, Recent Advances in Solids/Structures and Application of Metallic Materials ASME 1997 STRAIN LOCALIZATION IN SINGLE CRYSTALS: THERMODYNAMIC C ONSIDERATIOI\ Khanh Chau Le Lehrstuhl fiir Allgemeine Mechanik Ruhr-Universitdt Bochum 44780 Bochum, Germany ABSTRACT This paper presents a model describing strain localization in singie cr-"*stals. Constitutive relations for slip systems a.re proposed within the framework of thermodynamics. A generalized Schmid law is formulated in terms of the Eshelby stress tensor. The strain localization is found to be possible when the hardening rate for active slip systems has fallen to a certain critical positive value h".. The latter is determined for double active slip systems. A comparison with corresponding results available in the lirerature is provided. Introduction The strain locaiization in form of coarse slip bands or macroscopic shear bands is one of the most remarkable feature of ductiie crystaline materials (Price & Kelly. 1964; Chang & Asa.ro, 1981; Spitzig, 1981; Peirce et aI., 1982. Dao & Asa.ro, 1996). This strain localization ofben occurs short before the formation of voids and microcracks. On the other hand, it is not clear up to now. how this irreversibie Drocess can be explained properly within the framework of thermodynamics. Asaro & Rice (1977) analized the localization criteria for materials that essentialiy obey Schmid's lar,r' but dispiay modest departure from it. They showed clearly that Schmid's law (Schmid. 1935) in it,s classicai form is not appropriate to expiain the strain localization in single crystais. Horvever, the relationship between rhe constitutive equations proposed therein. which include so-called non-Schmid lerms. and the entrop)' inequality is not clear. The presenr paper aims at deriving a geueralized Schmid law from the entropy inequality'. which enabies one !o describe the strain localization. W-e srarr from the assumption that the free energy ciensitv per unit volume of the referetrce crystal depends only on the lattice (elastic) deformation (Le & Stumpf. i993.1994). The free energ]'density per unit voiume of the initial configuration will depend :hen on the Cauch-u-'-Green deformation tensor and on the plastic distortion. Substituting this free energy deasitv into the entropy inequality and requiring the latter to be fulfilled for arbitrary processes, one can obtain an "elastic part" of constitutive equations describing the lattice responce, as well as a flow ruie for the slip systems. It is shown that the dissipation is equai to the power of the shear components 81