Effects of lattice incompatibility-induced hardening on the fracture behavior of ductile single crystals
Introduction
Many experimental studies (Reimanis et al., 1991; Wang and Anderson, 1991; Elssner et al., 1994; Bagchi and Evans, 1996) have shown that cracks can grow slowly via cleavage in the presence of substantial plastic flow. For rate-independent materials modeled by conventional J2 plasticity theory, the maximum normal stress ahead of the crack tip is at most five times as much as the initial yielding stress, σY (Wei and Hutchinson, 1997). Such a prediction falls short of stress levels of the order 10 times σY or more that are required for producing atomic decohesion of a lattice or a strong interface. For rate-sensitive materials, modeling of high strain rate and inertial effects arising from rapid crack advance lead to attaining the cleavage stress (Freund and Hutchinson, 1985). Modeling the quasi-static propagation of a crack by atomic decohesion in ductile crystals, however, poses a challenge requiring the development of a high opening stress for the process.
A series of studies (Xia and Hutchinson, 1996; Jiang et al., 2001) have been carried out to investigate the effects of strain gradients on crack-tip fields in isotropic materials, with particular emphasis on elevation of tractions ahead of the crack tip. The work of Wei and Hutchinson (1997) employed a phenomenological strain gradient theory (Fleck and Hutchinson, 1997) that includes the effects of both stretching and rotation gradients within the small-strain framework, along with the embedded fracture process zone (EPZ) (Tvergaard and Hutchinson 1992, Tvergaard and Hutchinson 1993) model. This resulted in significant elevation of the cohesive surface tractions. The numerical results demonstrated that the elevation occurs only for a steadily propagating crack tip. For a stationary crack tip, however, a physically unacceptable compressive traction ahead of a mode-I crack tip appears (Chen et al., 1999).
Crack-tip fields in ductile single crystals based on conventional crystal plasticity theory have been studied by Rice (1987), Saeedvafa and Rice (1989), Rice et al. (1990), Cuitiño and Ortiz (1992), Saeedvafa (1992), Cuitiño and Ortiz (1996), Crone and Shield (2003), and Crone et al. (2004). As shown in Rice (1987) and the aforementioned works, the deformation fields around the crack tip of a ductile single crystal are vastly different from those in an isotropic material.
Motivated by the work of Wei and Hutchinson (1997), this paper focuses on gradient effects in fracture of single crystals modeled by finite deformation crystal plasticity theory. It has been demonstrated (Acharya and Beaudoin, 2000; Beaudoin et al., 2000; Kok et al., 2002) that a modification of a widely used conventional hardening law based on a physically appropriate accounting of the effects of geometrically necessary dislocations (GNDs) via lattice incompatibility can help predict a higher hardening rate and stage IV hardening. Such a factor can be expected to play a vital role on the normal opening traction elevation ahead of the crack tip in the presence of substantial plastic flow. Here, we explore this idea and the effects of lattice incompatibility on crack-tip fields and the fracture process, predicated on the hardening model developed in Acharya and Beaudoin (2000) within the framework of the ‘Simple’ gradient theory of crystal plasticity proposed in Acharya and Bassani (2000). In that model, the ‘slip plane lattice incompatibility’ reflects the presence of forest dislocations threading corresponding slip planes, and the hardening produced by such dislocations is modeled by incorporating the incompatibility into the instantaneous hardening moduli. No higher-order stresses are introduced, and this is in contrast to the gradient theories of Fleck and Hutchinson (1993), Gurtin (2000) and Huang et al. (2000).
We believe this to be the first study to investigate fracture behavior in ductile single crystals incorporating the effects of distortion gradients, crystalline anisotropy, and finite deformations. In particular, we strive to understand cleavage in a ductile single crystal with a planar crack whose tip is far-removed from external surfaces of the crystal except the crack flanks. Consequently, the study conducted here is based on a ‘plane strain’ analysis in the following sense. A 3D analysis with out-of-plane displacements constrained to vanish is considered. Correspondingly, all three equilibrium equations are solved, with shear tractions on external faces perpendicular to the out-of-plane direction imposed to vanish. While it is hard to justify the shear traction boundary condition just mentioned in the deep interior of a nonlinear solid where the crack tip being considered is located in reality, we assume this device to be an acceptable idealization due to the symmetric nature of the slip system geometry. At any rate, for the considered slip system geometry, our modeling assumption is at least as good or better than conventional plane strain analysis for single crystals where one equilibrium equation is ignored along with traction boundary conditions on the faces perpendicular to the out-of-plane direction.
Plane strain deformation fields are expected to be dominant in the near-tip region far from the surfaces of a finite single crystal (cf. Cuitiño and Ortiz, 1996). These fields play significant roles on both crack initiation and growth. We first address the effects of lattice incompatibility on crack-tip fields. The objective is to show that cleavage in the presence of substantial plastic flow can be explained on the basis of the gradient theory considered herein. In the context of a gradient modification of phenomenological J2 plasticity theory along the lines of Acharya and Bassani (1996), Niordson (2002) has shown predictions of higher normal stresses ahead of a mode-I crack as well as reduction in steady state fracture toughness. In the second part of the study, the effects of lattice incompatibility on crack growth resistance behavior are studied. A cohesive zone methodology (Xu and Needleman, 1994; Rahulkumar et al., 2000) that employs a phenomenological interface constitutive relation is used to simulate the fracture process. The relation between the work of separation and the macroscopic fracture work, in the context of the gradient theory considered here, is studied. Such a relationship has been considered earlier in the studies of Rice and Wang (1989), Tvergaard and Hutchinson (1992), and Elssner et al. (1994). Here, the dependence of the fracture resistance on fracture process parameters—the work of separation and peak separation stress—is studied in the presence of gradient effects.
Section snippets
Lattice incompatibility and hardening law
The elastic deformation of the crystal lattice is usually not compatible with a single-valued displacement field. A measure of the slip plane lattice incompatibility (Acharya and Beaudoin, 2000) associated with the slip system α is defined aswhere nα is the unit normal of the slip plane associated with the slip system α and curl refers to the appropriate operation on the current configuration. Denoting the magnitude of the Burgers vector by b, the term λα/b may be
Problem definition and dimensional analysis
The asymptotic analysis of Rice (1987) revealed the existence of discontinuities of stresses and shear displacements around a plane strain, stationary, tensile crack tip in elastic-ideally plastic single crystals, with intense shear deformation along the rays of discontinuities emanating from the crack tip. Rice showed that the rays lie either parallel (glide shear) or perpendicular (kink shear) to the slip plane traces. Although the analysis used small strain theory and elastic-ideally plastic
Implementation
The finite element analysis uses a rate-dependent formulation within a finite deformation framework along with a forward gradient method (Willam, 1978; Peirce 1983, Peirce 1984; Becker, 1992) for the integration of constitutive response. Details of the numerical procedure followed here are given in Peirce et al. (1983). In this section, we briefly outline the implementation of the hardening law with lattice incompatibility.
The increment in slip at time t is estimated as
Results and discussion
Unless otherwise specified, the values of various material parameters employed in the analysis are given in Table 1. These values are suitable for a ductile single crystal with moderate hardening rate. Here the introduced material constant, k0, due to gradient effects takes a base value of 20.2. This value is a reasonable fit obtained in the study of polycrystal grain size effects in Ni (Acharya and Beaudoin, 2000). In the study conducted here, this base value is varied over a factor of 25 to
Summary
The present study shows that cleavage in the presence of plastic flow in ductile single crystals can be predicted by a hardening model accounting for the effects of lattice incompatibility. To our knowledge, this work is the first effort in modeling the physical phenomenon accounting for all the essential nonlinearities of the plasticity of single crystals. The effects of lattice incompatibility on traction elevation ahead of the crack tip depend strongly on a non-dimensional parameter k0b/R0,
Acknowledgements
Financial support from the NSF-Sandia Life Cycle Engineering Initiative under contract No. SNL-25530 is gratefully acknowledged. Helpful comments from anonymous referees are also acknowledeged.
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