The role of macroscopic hardening and individual length-scales on crack tip stress elevation from phenomenological strain gradient plasticity

Abstract

This paper quantifies the effect of strain gradient plasticity (SGP) on crack tip stress elevation for a broad range of applied loading conditions and constitutive model parameters, including both macroscopic hardening parameters and individual material length-scales controlling gradient effects. Finite element simulations incorporating the Fleck–Hutchinson SGP theory are presented for an asymptotically sharp stationary crack. Results identify fundamental scaling relationships describing (i) the physical length-scales over which strain gradients are prominent, and (ii) the degree of stress elevation over conventional Hutchinson–Rice–Rosengren (HRR) fields. Results illustrate that the three length-scale theory predicts much larger SGP effects than the single length-scale theory. Critically, the first length-scale parameter dominates SGP stress elevation: this suggests that SGP effects in fracture can be predicted using the length-scales extracted from nanoindentation, which exhibits similar behavior. Transitional loading/material parameters are identified that establish regimes of SGP relevance: this provides the foundation for the rational application of SGP when developing new micromechanical models of crack tip damage mechanisms and associated subcritical crack propagation behavior in structural alloys.

Introduction

The observation of brittle fracture in the presence of significant plastic deformation (Elssner et al., 1994) has sparked significant interest in the role of gradient-enhanced plasticity in modulating crack tip stresses (Chen et al., 1999; Jiang et al., 2001; Hwang et al., 2003; Qu et al., 2004). The central concept in this approach is that increased dislocation density associated with large gradients in plastic strain near the crack tip promotes strain hardening and leads to crack tip stresses that are much larger than those predicted using conventional plasticity. Previous simulations incorporating strain gradient plasticity (SGP) descriptions and a cohesive law for process zone fracture have shown that this stress elevation may be sufficient to drive atomic decohesion and rationalize the relationship between intrinsic and steady-state toughness (Wei and Hutchinson, 1997; Wei et al., 2004).

The fact that gradient-enhanced hardening can significantly elevate stresses has important implications for a number of fracture problems including interface debonding in thin films (Dauskardt et al., 1998), layered structure embrittlement (Broedling et al., 2008), microvoid damage (Srinivasan et al., 2008), fatigue (Brinckmann and Siegmund, 2008) and environmentally assisted fracture by both crack tip film rupture (Ford, 1990) and embrittlement by atomic hydrogen (H) (Gangloff, 2007). Crack tip hydrogen damage is of particular and growing interest (Woodtli and Kieselbach, 2000; Serebrinsky et al., 2004) because of the increasing focus on structural material integrity in hydrogen-based energy systems. Crack tip stresses play a central role in hydrogen-assisted cracking, both with regards to H localization and interface decohesion. Given the exponential dependence on H concentration with hydrostatic stresses at the crack tip, even moderate increases in stress over previous estimates dramatically increase the levels of crack tip hydrogen in the fracture process zone. Indeed, it was observed decades ago that hydrogen concentrations based on stresses from conventional plasticity are too low to rationalize the reduction in toughness due to hydrogen. This motivated alternative dislocation-based approaches to predict stresses larger than conventional plasticity (Kameda, 1986; Gerberich et al., 1991, Gerberich et al., 1993). Similar considerations are relevant to environment-enhanced fatigue crack propagation driven by highly localized and gradated cyclic plasticity, stress and H (Gangloff, 2002; Sunder, 2005, Sunder, 2007; Stoychev and Kujawski, 2008).

Clearly, any improvement over conventional predictions of crack tip stresses requires consideration of local material microstructure such as grain orientations, second phase particles, and of course, dislocations that arise in conjunction with steep gradients in plastic strains. Here, the focus is on the latter. The notion of exactly how enhanced dislocation hardening should be included in a constitutive description has spurred significant debate in the mechanics community. This debate will likely continue for some time because: (i) relatively few experimental methods for quantifying length-scale effects have emerged (e.g. nanoindentation (Nix and Gao, 1998), and more recently, bulge testing of thin films (Nicola et al., 2006)), (ii) competing approaches predict similar behavior and invariably require material-dependent parameters that must be determined from limited experimental results, and (iii) most rigorous “first-principles” modeling approaches are either limited in scope (e.g. two-dimensional simulations of edge dislocation interactions in face-centered cubic materials, Deshpande et al., 2005) or too computationally expensive to serve as the basis for prognosis of structural performance (Larsen et al., 2005).

With these considerations in mind, we adopt a phenomenological continuum strain gradient description to quantify dislocation-related changes in crack tip behavior (Fleck and Hutchinson, 2001). We utilize the reformulated Fleck/Hutchinson theory that, first and foremost, has been demonstrated to capture the effects of strain gradient-enhanced hardening in terms of established macroscopic yielding parameters and a limited number of parameters that dictate length-scale effects. Secondly, it is computationally efficient, which enables the present broad parameter study and increases the possibility of implementation with emerging prognosis tools (Larsen et al., 2005). Furthermore, the predictions from this approach are directly relevant to micromechanical models of the threshold and kinetics of important cracking processes (Gangloff, 2007).

The focus here is on identifying the relationships between macroscopic plasticity properties, applied loads and gradient effects, with two principle goals. First, we wish to identify the relationship between physical inputs and the length-scale over which gradient effects prominently enhance crack tip stresses. This facilitates the rational application of strain gradient theories, by clearly identifying regimes where length-scale effects are important over physical lengths that are appropriately modeled using continuum theories. Put in another way, we identify and graphically map combinations of material properties and loads that undoubtedly require explicit reference to material microstructure. Secondly, we seek to quantify gradient-induced changes in crack tip variables for scenarios where non-local continuum plasticity predictions are appropriate. The focus is on the magnitude of stress elevation and the physical length over which it occurs: both play important roles in evaluating the relative importance of competing crack advance mechanisms. The central question is simply: “at what level of applied stress intensity does one expect stresses significantly different from the HRR field ((Hutchinson, 1968; Rice and Rosengren, 1968), i.e. conventional plasticity), over a physical size scale where SGP is believable and for experimentally informed macroscopic hardening parameters, length-scale parameters, and applied stress intensity?” The answer to this question depends on not only given microstructure-dependent material length scales, l_i, but also the specific implementation of SGP: i.e., the details of how length-scale parameters and strain gradients are combined and inserted into the hardening law.

While previous simulations addressed some of these issues (Jiang et al., 2001; Wei and Hutchinson, 1997), they did not cover a broad range of material properties and applied loads. Moreover, they did not quantify the role of individual length-scale parameters, or establish connections between multiple-parameter theories and related single-parameter theories (such as those outlined for SGP (Fleck and Hutchinson, 2001) and in a different context by Aifantis (1984)). The comparison of single and multiple-parameter theories is of significant practical interest, considering that nanoindentation—the de facto choice for experimental calibration of SGP theories—is typically used to extract a single parameter. A meaningful length-scale can only be extracted when the same SGP theory is applied to interpret both depth-dependent hardness and a given application (e.g. fracture). The present study, which spans an unprecedented range of variables, enables one to infer the relevance of SGP to a specific material, loading scenario and crack growth damage mechanism based on published properties.