Effects of heterogeneous elasticity coupled to plasticity on stresses and lattice rotations in bicrystals: A Field Dislocation Mechanics viewpoint

Abstract

In the present paper, the static Field Dislocation Mechanics (FDM) theory is first recalled and next formulated for heterogeneous elasticity and plasticity with respect to a reference homogeneous medium. The issue of an infinite bicrystal with planar boundary undergoing piecewise uniform plastic distortions and elastic properties is then considered. The static FDM theory is accordingly used to derive explicit closed-form solutions of stress and lattice misorientation fields in the general context of heterogeneous anisotropic elasticity. These expressions are validated by Crystal Plasticity Finite Element simulations performed on a bicrystal with periodic boundary conditions. As a result of analytical expressions, it is possible to quantify the contribution of the different incompatibility sources, namely the contribution arising from elastic incompatibilities alone, that due to plastic incompatibilities alone and the one due to couplings between elastic and plastic incompatibilities. Besides, residual stresses and lattice misorientations are accordingly calculated in elastic/plastic bicrystals for different orientations of the elastic crystal. Results are compared with the corresponding homogeneous isotropic elasticity approximation. It is shown that elastic–plastic coupling incompatibilities may have consequent effects on residual stresses and lattice misorientations depending on the degree of elastic anisotropy of the material and the relative orientation of both crystals. As a conclusion, the consideration of coupling incompatibilities in multiscale plasticity models when predicting internal stresses and texture developments is discussed.

Introduction

Many studies coupling experiments and theoretical investigations (e.g., Gemperlova et al., 1989; Hirth, 1972; Liang and Dunne, 2009; Mayama et al., 2009; Ohashi et al., 2009; Rey and Zaoui, 1980, Rey and Zaoui, 1982; Saada, 2006; Sun et al., 1998; Sutton and Balluffi, 1995; Vehoff et al., 1987, Vehoff et al., 2004) have been carried out on bicrystals in order to get a better understanding of the specific role of grain boundaries during the deformation of polycrystals. At a material surface of discontinuity (grain, twin or phase boundary), incompatibility stresses can develop due to material elastic anisotropy. Much theoretical work (e.g., Gemperlova et al., 1989; Qamar and Husain, 1989; Tewary et al., 1989), including Finite Element simulations (e.g., Schick et al., 2000; Vehoff et al., 2004), has been devoted to estimate these elastic incompatibility stresses in order to predict crack initiation and propagation path. When plasticity enters into play, plastic strain incompatibilities add to the picture and yield additional stresses and lattice rotations. Among these fields, it may actually be distinguished between those uniquely due to a difference in plastic distortions and those arising from a coupling between elastic and plastic heterogeneities. Many models make an assumption of homogeneous isotropic elasticity or of pure rigid viscoplastic behaviour (no elasticity) and thus overlook the contribution of both elastic and elastic–plastic coupling incompatibilities. The present paper aims notably at underlying the potentially significant role of coupling incompatibilities. For this purpose, explicit closed-form solutions of stress and lattice misorientation fields are derived for two semi-infinite crystals undergoing uniform plastic distortions in the general context of heterogeneous anisotropic elasticity. The derivation is performed by means of the concepts of the elastic theory of continuously distributed dislocations (ECDD) initiated by Kröner, 1958, Kröner, 1981 and Willis (1967). It deals with the Nye's tensor α (Nye, 1953) which reflects only the so-called geometrically necessary dislocations (GND) densities following Ashby (1970), also referred to as “excess” or “polar” dislocation densities in the literature. The Nye's tensor α is a continuous rendition of the collective arrangements of dislocations and the associated lattice incompatibility. This theory was rethought some years ago by Acharya (2001) (see also Acharya, 2004; Acharya and Roy, 2006) in considering the permanent deformation arising due to dislocation motion. In the Field Dislocation Mechanics (FDM) theory of Acharya (2001), the elastic distortions are uniquely decomposed into compatible and incompatible parts whereas the evolution equation for the GNDs (Mura, 1963) is added within an efficient numerical set-up.

Incompatibility problems constitute a long-standing issue of materials mechanics (see the pioneering works of Bilby, 1955; Frank, 1950; Indenbom, 1966; Kröner, 1958; Willis, 1967). Different techniques have been envisaged to address these issues and determine stresses and lattice rotations: Green functions (Berveiller and Zaoui, 1980; Berveiller et al., 1987; Kröner, 1989), stress potential functions for plane problems (Kröner, 1981; Berveiller, 1978) or Fourier Transforms (Berbenni et al., 2008; Rey and Saada, 1976; Saada, 1979). In the case of bicrystals, Gemperlova et al. (1989) imagine the total stresses as a sum of the applied and compatibility stresses and the total strains as a sum of elastic strains in relation with applied stresses, elastic strains in relation with compatibility stresses and plastic strains. Then, they consider boundary conditions for stresses and compatibility conditions for strains at the interface to find out the stresses. For three-dimensional heterogeneous elasticity problems, Michelitsch and Wunderlin (1996) have extended the Kröner stress function method. It is also noteworthy that some homogenization schemes for laminates microstructures were derived to calculate the stresses using elastic stress concentration and influence tensors (e.g., Franciosi and Berbenni, 2007, Franciosi and Berbenni, 2008; Stupkiewicz and Petryk, 2002). Nevertheless, to the authors' knowledge, no explicit closed-form formulas of stresses have been given for bicrystals with completely anisotropic elasticity and heterogeneous plasticity. In addition, the context of the present paper is different from the preceding approaches since it aims at directly applying the FDM theory (Acharya, 2001) in its static form.

The paper outline is as follows. In Section 2, notation conventions are settled. The static FDM theory is recalled in Section 3 and formulated for heterogeneous elasticity and plasticity with respect to a reference homogeneous medium. Conventional continuity conditions at surfaces of discontinuity are also recalled in this section, in addition to the expression of the surface-dislocation density tensor and the jump condition ensuring the conservation of the Burgers vector. Section 4 deals with the issue of an infinite bicrystal with planar boundary undergoing piecewise uniform plastic distortions and elastic properties. The static FDM theory is accordingly used to derive analytical expressions of stress and lattice rotation fields in the general case of heterogeneous anisotropic elasticity. The validations of these explicit formulae are checked in Section 5 from Crystal Plasticity Finite Element simulations on a bicrystal with periodic boundary conditions assuming each crystal has a cubic symmetry (fcc metals). The contribution of the different sources of incompatibilities is further analysed. In Section 5, analytical expressions of residual stresses and lattice misorientations are also used to perform comparisons with results derived from the isotropic approximation. Finally, concluding remarks follow.