Finite element simulation of large-strain single-crystal viscoplasticity: An investigation of various hardening relations T.M. Povall ⇑ , A.T. McBride, B.D. Reddy Centre for Research in Computational and Applied Mechanics, University of Cape Town, Private Bag, 7701 Rondebosch, South Africa article info Article history: Received 6 May 2013 Received in revised form 31 July 2013 Accepted 21 August 2013 Available online 18 September 2013 Keywords: Single crystals Large strain Plasticity Viscoplasticity Hardening Finite element method abstract Hardening relations describe the increase in resistance to deformation during plastic flow. Three harden- ing relations are compared here in the context of conventional large-strain single-crystal viscoplasticity. The first is an isotropic hardening relation. The second is a hardening relation that is expressed as an ordinary differential equation in the slip resistance. The third is a new relation, originally developed in the context of gradient crystal plasticity, in which the slip resistance is expressed explicitly in terms of the accumulated slip on each slip system. The numerical solution of the governing equations is found using the finite element method coupled with a predictor–corrector type algorithm. The features of the hardening relations are elucidated using a series of numerical benchmark problems. The parameters for the hardening relations are calibrated using a model problem. Various crystal structures are investi- gated, including single- and double slip, and face-centred cubic crystals. The hardening relations are com- pared and their relative features discussed. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction The description of plastic behaviour is characterised by (a) a yield criterion; (b) a hardening relation; and (c) a flow rule for plas- tic strain rates or equivalent quantities. A variety of hardening relations have been proposed for single- crystal plasticity. Popular models include a hardening relation that is expressed as an ordinary differential equation in the flow resis- tance [15,9]. A new hardening relation has recently been proposed by Gurtin and Reddy [8] in the context of strain-gradient single-crystal plasticity. This relation takes into account the interactions between slip systems. The purpose of this paper is to compare the fully-interactive hardening relation [8] with an implicit relation and an isotropic rela- tion. The fully-interactive relation was originally proposed in the context of small-strain rate-independent single-crystal gradient plasticity. The relation is extended here to the conventional large- strain viscoplastic setting. The isotropic relation is that defined by Steinmann and Stein [21] and is dependent on the sum of the accu- mulated slips, while the implicit relation used is that given in [9]. The finite element method (FEM) is used to solve the system of governing equations. The FEM implementation for conventional single-crystal plasticity is a well-established field with contribu- tions from Schröder and Miehe [18], Miehe et al. [13], Steinmann and Stein [21], and Anand and Kothari [1], to mention some impor- tant examples. A predictor–corrector algorithm, similar to that de- scribed by Steinmann and Stein [21], is used in conjunction with a consistent algorithmic tangent to accurately and efficiently de- scribe the plastic evolution. Two model problems are used to compare the hardening rela- tions. The first involves the indentation of an elastoplastic material by a rigid spherical indenter. The second problem is the shear of a bar subjected to cyclic deformations. The various hardening relations are examined in the indentation problem for three types of crystal structures: a single slip system, a double slip system and a face-cen- tred cubic (FCC) crystal. The shear problem examines the hardening relations using a double slip-system crystal and a FCC crystal. This paper is arranged as follows. Section 2 gives an overview of the notation, kinematics and the governing equations. Section 3 pro- vides a short derivation of the reduced dissipation inequality. A com- parison of rate-independent and rate-dependent formulations is given in Section 4. Section 5 details the three hardening relations. An explanation of the calibration process is given in Section 6.1. The results of the numerical simulation of the indentation and shear problems are discussed in Sections 6.2 and 6.3, respectively. A sum- mary of the results and the conclusion are given in Section 7. 2. Kinematics and governing equations 2.1. Kinematics A motion of a body B is a smooth function v that assigns to each material point X at time t a point in Euclidean space, x = v(X, t). The point x is referred to as the spatial (or current) point occupied by X 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.08.043 ⇑ Corresponding author. Tel.: +27 (0)21 650 3817. E-mail addresses: pvltim001@myuct.ac.za (T.M. Povall), andrew.mcbride@uc- t.ac.za (A.T. McBride), daya.reddy@uct.ac.za (B.D. Reddy). Computational Materials Science 81 (2014) 386–396 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci