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    Journal of the Mechanics and Physics of Solids

    Journal of the Mechanics and Physics of Solids 63 (2014) 491–505

    0022-50 http://d

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    journal homepage:

    Modelling crystal plasticity by 3D dislocation dynamics and the finite element method: The Discrete-Continuous Model revisited

    A. Vattré a,b,1, B. Devincre b,n, F. Feyel a, R. Gatti b, S. Groh b,2, O. Jamond a,b, A. Roos a

    a ONERA – The French Aerospace Lab, F-92322 Châtillon, France b LEM, CNRS–ONERA, 29 Avenue de la Division Leclerc, BP 72, 92322 Châtillon Cedex, France

    a r t i c l e i n f o

    Article history: Received 13 April 2012 Received in revised form 22 March 2013 Accepted 9 July 2013 Available online 20 September 2013

    Keywords: Dislocation dynamics Finite element Dislocation theory Crystal plasticity Discrete-Continuous Model

    96/$ - see front matter & 2013 Elsevier Ltd.

    esponding author. Tel.: þ33 1 46734449; fa ail address: (B. Dev esent address: CEA, DAM, DIF, F-91297 Arpa esent address: Institute for Mechanics and , Germany.

    a b s t r a c t

    A unified model coupling 3D dislocation dynamics (DD) simulations with the finite element (FE) method is revisited. The so-called Discrete-Continuous Model (DCM) aims to predict plastic flow at the (sub-)micron length scale of materials with complex boundary conditions. The evolution of the dislocation microstructure and the short-range disloca- tion–dislocation interactions are calculated with a DD code. The long-range mechanical fields due to the dislocations are calculated by a FE code, taking into account the boundary conditions. The coupling procedure is based on eigenstrain theory, and the precise manner in which the plastic slip, i.e. the dislocation glide as calculated by the DD code, is transferred to the integration points of the FE mesh is described in full detail. Several test cases are presented, and the DCM is applied to plastic flow in a single-crystal Nickel- based superalloy.

    & 2013 Elsevier Ltd. All rights reserved.

    1. Introduction

    Plastic deformation of crystalline materials results from the collective movement of dislocations under the influence of the interactions between the dislocation cores and the crystal lattice, their mutual interactions and the externally applied loads. The resulting spatial distribution of dislocations is intrinsically heterogeneous and the manner in which this affects the mechanical response of material samples is a complex problem. Today, predicting these phenomena quantitatively with mathematical and numerical models remains one of the most challenging problems in materials science.

    Outside their core region of a few Angstroms, dislocations are mostly linear defects with long-range elastic fields. These fields can be calculated with a classical procedure proposed long ago by Volterra (1907). The elastic strain field of a dislocation loop is also expressed in the form of a line integral along the loop with a kernel involving the elastic Green functions in an infinite domain (Mura, 1987). These solutions and other derived expressions are the basis of many models of plastic deformation.

    All rights reserved.

    x: þ33 1 46734155. incre). jon, France. Fluid Dynamics, University of Technology Bergakademie Freiberg, Lampadiusstrasse 4, D-09599

  • A. Vattré et al. / J. Mech. Phys. Solids 63 (2014) 491–505492

    3D dislocations dynamics (DD) simulations are among the most advanced techniques to model the collective properties of large dislocation ensembles (Kubin et al., 1992; Verdier et al., 1998; Zbib et al., 1998; Schwarz, 1999; Ghoniem and Sun, 1999; Cai et al., 2004; Devincre et al., 2011). They are reasonably mature now and the computation of up to a few percent of plastic strain in a volume with several hundreds of dislocations has become routine. Nevertheless, in DD simulations boundary value problems are usually much simplified (Van der Giessen and Needleman, 1995; Devincre et al., 2003; Deng et al., 2008). For instance, it is often assumed that external loads induce a uniform stress field inside the simulation volume or in the vicinity of surfaces and interfaces. Subsequently, approximations are commonly used to calculate image forces and many of the existing DD simulations codes are limited to isotropic elasticity, where analytical expressions for the stress fields for dislocation segments exist. In some 3D DD codes solutions for anisotropic media are used (Han et al., 2003; Chen and Biner, 2007), but they are computationally expensive due to the lack of closed-form solution of Green's functions for that case. A significant step forward concerns the analytical expressions derived by Yin et al. (2010, 2012), based on the matrix formalism with the Willis–Steels–Lothe formula, but to the best of our knowledge these expressions have not yet been used in large-scale DD simulations.

    Hybrid approaches have been proposed to solve the mechanical boundary problem of a finite volume containing dislocations. The first of these approaches superposes the mechanical fields associated to dislocations in an infinite medium to the elastic solution of a modified boundary condition problem in a dislocation-free finite body. Initially applied to 2D DD simulations by Van der Giessen and Needleman (1995), the superposition principle is now routinely used in 3D DD simulations for different problems using finite element (FE) (Fivel and Canova, 1999; Weygand et al., 2001) or boundary element (BE) (Takahashi and Ghoniem, 2008; El-Awady et al., 2008) methods to solve the modified boundary problem. Recently, anisotropic elasticity has been taken into account in 2D DD (Shishvan and Mohammadi, 2011) to deal with the boundary value problem of a polycrystalline medium. The advantage of the superposition framework is the accurate description of the short-range dislocation stress field when dislocations are not too close to surfaces and interfaces. Unfortunately, numerical difficulties arise when the dislocation segments approach the boundaries (Weygand et al., 2001; Tang et al., 2006). Other methods include the extended finite element method (X-FEM) (Belytschko and Gracie, 2007; Gracie et al., 2007, 2008), and the level set method (Xiang et al., 2003). In the following we will refer to conventional codes when the boundary conditions are not accounted for, and to hybrid codes otherwise.

    Another hybrid approach, which addresses some of these issues, is revisited in the present paper. Here, dislocation loops are represented as thin plate-like inclusions causing an eigenstrain field in an isotropic or anisotropic elastic medium. The slipped areas calculated with a DD simulation are regularised as plastic strains at the integration points (IPs) of a FE mesh (Lemarchand et al., 2001; Devincre et al., 2003). The FE code can then calculate the associated equilibrium stress field inside a simulation volume of any shape, which can then be used to further drive the dislocation dynamics in a subsequent time increment, and so on. This coupling between a DD simulation and a FE calculation, the so-called Discrete-Continuous Model (DCM), is numerically straightforward: the DD simulation simply replaces the usual crystallographic constitutive law in the FE calculation. The main difference with the superposition method is that the FE code not only takes into account the boundary conditions of the finite volume, but also the long-range dislocation–dislocation interactions. Problems with periodic boundary conditions can also be treated without having to take into account explicitly the contributions of periodic replicas, so at almost no additional computational cost. It was first developed by Lemarchand et al. (1999a, 1999b, 2000, 2001) and subsequently by Groh et al. (2003a, 2003b). A comparable approach is sometimes used in phase-field simulations to model the influence of dislocations on phase transformations (Rodney and Finel, 2001; Wang et al., 2001; Rodney et al., 2003). Another approach has been recently used in the framework of finite deformation theory (Liu et al., 2009; Gao et al., 2010).

    The DCM has already been used to study plastic relaxation in anisotropic heteroepitaxial thin films (Groh et al., 2003b), plastic deformation in metal matrix composites (Groh et al., 2003a), compression tests in a micro-pillar of copper (Liu et al., 2009), orientation dependence of the yield strength (Vattré et al., 2010a) and channel size effects (Vattré et al., 2010b) in single crystal superalloys. However, the algorithmic details of the regularisation procedure from slipped areas to plastic strains, which constitutes the core of the DCM, have never been published, and they have also been improved in recent years in terms of precision and efficiency. The purpose of the present paper is thus to present these details (Section 2) together with validation tests and a more complex application to a single crystal Nickel-based superalloy (Section 3). The last section gives a brief discussion and conclusions.

    2. The Discrete-Continuous Model

    This section discusses t