Anisotropic yield function of hexagonal materials taking into account texture development and anisotropic hardening B. Plunkett a , R.A. Lebensohn b , O. Cazacu a, * , F. Barlat c a Department of Mechanical and Aerospace Engineering, University of Florida/REEF, 1350 N Poquito Road, Shalimar, FL 32579-1163, USA b Los Alamos National Laboratory, MST8, MS G755, Los Alamos, NM 87545, USA c Materials Science Division, Alcoa Inc., Alcoa Technical Center, 100 Technical Drive, Alcoa Center, PA 15069-0001, USA Received 13 October 2005; received in revised form 4 May 2006; accepted 4 May 2006 Available online 1 August 2006 Abstract Because of twinning and texture evolution, the yield surface for hexagonal close-packed (hcp) metals significantly changes its shape with accumulated plastic deformation. Traditional hardening laws cannot accurately model these phenomena. In this paper, an aniso- tropic model that captures the influence of evolving texture on the plastic response of hcp metals is proposed. Initial yielding is described using a recently developed analytical yield function that accounts for both anisotropy and strength differential effects. To describe the change of the shape of the yield surface during monotonic loading, the evolution of the anisotropic coefficients involved in the expression of the yield function is considered. The evolution laws for the anisotropic coefficients are obtained based on experimental data and crystal plasticity theory, together with a macroscopic-scale interpolation technique. This approach is further applied to the description of the mechanical behavior of high-purity zirconium at room temperature. Validation of the proposed model is provided by applying it to the simulation of the three-dimensional deformation of a beam subjected to four-point bending along different directions with respect to the hard-to-deform Æcæ-axis predominant orientation of the material. Comparison between predicted and measured macroscopic strain fields and beam sections shows that the proposed model describes very well the difference in response between the tensile and compressive fibers and the shift of the neutral axis. Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Twinning; Texture evolution; Yielding asymmetry; Anisotropic yielding; Zirconium 1. Introduction Characterization of the plastic response in metals requires the specification of a yield function and a flow rule by which subsequent inelastic deformation can be calcu- lated for specified loadings and displacements. Tradition- ally, the evolution of the yield surface is described by a combination of isotropic and kinematic hardening laws. Isotropic hardening implies a proportional expansion of the surface, without any changes in shape or position. An isotropic hardening model is only truly valid for monotonic loading along a given strain path assuming that every strain path hardens at the same rate. For simulation of sheet forming operations of cubic metals (both face- centered cubic (fcc) and body-centered cubic (bcc)), such an assumption is reasonably adequate [1]. Pure translation of the initial yield surface could be described by the classic linear kinematic hardening laws [2,3]. To model more accu- rately the smooth elastic–plastic transition upon reverse loading, multi-surface models as well as nonlinear kine- matic hardening models have been proposed. Reviews of such models may be found in Refs. [4–6]. Because of non-negligible twinning activity accompa- nied by grain reorientation and highly directional grain interactions, the influence of the texture evolution on hard- ening of hexagonal close-packed (hcp) materials cannot be 1359-6454/$30.00 Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.05.009 * Corresponding author. Tel.: +1 850 833 9350; fax: 1 850 833 9366. E-mail address: cazacu@gerc.eng.ufl.edu (O. Cazacu). www.actamat-journals.com Acta Materialia 54 (2006) 4159–4169