A numerical study of crack tip constraint in ductile single crystals

Abstract

In this work, the effect of crack tip constraint on near-tip stress and deformation fields in a ductile FCC single crystal is studied under mode I, plane strain conditions. To this end, modified boundary layer simulations within crystal plasticity framework are performed, neglecting elastic anisotropy. The first and second terms of the isotropic elastic crack tip field, which are governed by the stress intensity factor K and T-stress, are prescribed as remote boundary conditions and solutions pertaining to different levels of T-stress are generated. It is found that the near-tip deformation field, especially, the development of kink or slip shear bands, is sensitive to the constraint level. The stress distribution and the size and shape of the plastic zone near the crack tip are also strongly influenced by the level of T-stress, with progressive loss of crack tip constraint occurring as T-stress becomes more negative. A family of near-tip fields is obtained which are characterized by two terms (such as K and T or J and a constraint parameter Q) as in isotropic plastic solids.

Introduction

There have been rapid strides in modeling capabilities in recent years, which have enabled simulation of the behavior of individual grains in a polycrystalline aggregate through finite element computations employing crystal plasticity theory. These computations can provide an understanding of the role of texture and hardening considering dislocation interaction on fracture and formability of engineering alloys. As a first step in this direction, it is important to investigate plastic deformation at a crack tip in a ductile single crystal. In this context, it must be noted that when the crack opening displacement is much less than the grain size, the crack tip fields are contained in a single grain. The need to understand ductile–brittle transition also requires characterization of the slip patterns arising in the vicinity of crack tips in ductile single crystals. Further, some key structural components are being fabricated in single crystal form. For example, blades in high pressure turbines of jet engines are made of single crystal, nickel-based superalloys. Therefore, it is necessary to understand and characterize the stress and deformation fields present at the tip of a crack in a single crystal.

Rice (1987) proposed an asymptotic solution for the crack tip stress field in ductile single crystals under mode I plane strain conditions within small strain, ideal plasticity framework. His analysis considered the cases of a crack on the (0 1 0) plane with crack front along $[10\overline{1}]$ direction, and a crack on the (1 0 1) plane with crack front along $[10\overline{1}]$ direction for FCC and BCC crystals, respectively. The motivation for considering these crack orientations is because they have been frequently observed to occur in experimental studies on fracture of ductile single crystals (see, Neumann, 1974a, Neumann, 1974b; Garrett and Knott, 1975). The solution of Rice (1987) consists of sectors of constant stress, with concentrated kink and slip shear deformation on the sector boundaries. Subsequently, Saeedvafa and Rice (1989) extended these results by assuming power law hardening and proposed HRR-type asymptotic solutions for crack tip singular fields. A similar solution was also derived by Cuitino and Ortiz (1996) for FCC crystals obeying both diagonal and isotropic power law hardening. Drugan (2001) proposed asymptotic solutions near a stationary crack tip in an elastic-ideally plastic ductile single crystal that do not contain kink-type plastic shearing bands.

Rice et al. (1990) conducted preliminary finite element simulations under 2D plane strain, small scale yielding (SSY) conditions in ductile single crystals within small strain plasticity framework. They found that the near-tip fields are consistent with the asymptotic solution of Rice (1987). They also reported some preliminary computational results for the center cracked panel (CCP) configuration considering a planar double-slip model. Mohan et al. (1992) subsequently performed 2D plane strain finite element analysis of a stationary crack tip in FCC and BCC crystals subjected to mode I loading under SSY conditions accounting for finite deformation and lattice rotations. Their calculations were based on a saturation-type hardening rule. Their observations are in partial agreement with earlier analytical and numerical solutions (Rice, 1987, Rice et al., 1990). Cuitino and Ortiz (1996) conducted 3D finite element analysis of the four point bending configuration using a dislocation hardening model. Their results showed notable differences in slip activity at the free surface and in the interior of the specimen. In a recent study, Flouriot et al. (2003) carried out 3D finite element simulations of a compact tension (CT) specimen made from a single crystal of nickel-based superalloy. Their computations were performed for three different orientations by assuming elastic-ideally plastic behavior. A good agreement was found between numerical results and experimental data.

Motivated by the predictions of the above noted analytical and finite element solutions, some researchers have experimentally investigated the near-tip deformation fields in a variety of single crystals. Shield and Kim (1994), Shield (1996) and Crone and Shield (2001) studied these fields in a four point bend single crystal specimen using Moiré interferometry. Their experimental results follow the general structure of the analytical solution (Rice, 1987), namely existence of constant stress sectors with sharp boundaries. Kysar and Briant (2002) performed electron back-scattered diffraction (EBSD) analysis of an aluminum single crystal specimen subjected to mode I loading. Their results showed existence of the kink shear sector boundary consistent with Rice's (1987) solution. Flouriot et al. (2003) also demonstrated the presence of kink bands near the crack tip at the free surface in CT specimens of nickel-based superalloy (FCC) single crystals through EBSD analysis.

Although several studies on crack tip fields in ductile single crystals have been undertaken in the literature as discussed above, some important issues still need to be addressed. The influence of higher order terms in the asymptotic solution which have been shown to play an important role in isotropic plastic solids (see, for example, Sharma and Aravas, 1991, O’Dowd and Shih, 1991) has not been examined in the context of single crystals. In particular, O’Dowd and Shih (1991) have demonstrated that a two-parameter characterization of crack tip fields involving J and a triaxiality (or constraint) parameter Q is necessary to satisfactorily describe the configuration dependence of fracture response in isotropic plastic solids, especially under large scale yielding conditions. They found that in tension dominated geometries, such as CCP and single edge notch under tension (SENT), Q can attain a significantly negative value leading to loss of crack tip constraint or stress triaxiality. Rice (1987) noted that his asymptotic solution may not apply under large scale yielding conditions in low constraint (single crystal) fracture geometries such as those mentioned above wherein the stresses around the crack tip may be much lower. He also pointed out the possible strong configuration dependence of the near-tip deformation fields. Thus, considerable work is needed to understand these issues pertaining to fracture of ductile single crystals.

The objective of the present work is to examine the effect of crack tip constraint on near-tip fields in a FCC single crystal having the orientation chosen by Rice (1987) under mode I plane strain condition. In order to simplify the analysis and interpretation, elastic anisotropy is ignored. A family of crack tip fields is generated by a two-parameter ($K–T$ based) modified boundary layer approach. The finite element simulations are carried out within a continuum crystal plasticity framework and isotropic hardening response characterized by the Pierce–Asaro–Needleman (PAN) model (Peirce et al., 1983) is assumed. The results show that the near-tip deformation field, especially, the development of kink or slip shear bands, is sensitive to the constraint level. The stress distribution and the size and shape of plastic zone near the crack tip are also strongly influenced by the level of T-stress. The value of the constraint parameter Q becomes highly negative with increase in T-stress in the negative direction, which is akin to isotropic plastic solids. This will have profound implications on crack initiation by cleavage cracking as well as by micro-void growth and coalescence in ductile single crystals.