The role of heterogeneous deformation on damage nucleation at grain boundaries in single phase metals

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Abstract

The mechanical response of engineering materials evaluated through continuum fracture mechanics typically assumes that a crack or void initially exists, but it does not provide information about the nucleation of such flaws in an otherwise flawless microstructure. How such flaws originate, particularly at grain (or phase) boundaries is less clear. Experimentally, “good” vs. “bad” grain boundaries are often invoked as the reasons for critical damage nucleation, but without any quantification. The state of knowledge about deformation at or near grain boundaries, including slip transfer and heterogeneous deformation, is reviewed to show that little work has been done to examine how slip interactions can lead to damage nucleation. A fracture initiation parameter developed recently for a low ductility model material with limited slip systems provides a new definition of grain boundary character based upon operating slip and twin systems (rather than an interfacial energy based definition). This provides a way to predict damage nucleation density on a physical and local (rather than a statistical) basis. The parameter assesses the way that highly activated twin systems are aligned with principal stresses and slip system Burgers vectors. A crystal plasticity-finite element method (CP-FEM) based model of an extensively characterized microstructural region has been used to determine if the stress–strain history provides any additional insights about the relationship between shear and damage nucleation. This analysis shows that a combination of a CP-FEM model augmented with the fracture initiation parameter shows promise for becoming a predictive tool for identifying damage-prone boundaries.

Introduction

Continuum fracture mechanics has provided a wealth of methodologies for modeling the evolution of damage, but these methods all depend on knowing where the damage nucleated; hence a pre-existing void or crack is normally introduced arbitrarily. The process by which undamaged material develops damage (here defined as the generation of a new free surface where there was none before) is not very well understood. An understanding of this damage nucleation process in the context of microstructural evolution will allow properties that are of great importance to designers, such as toughness, ductility, and fatigue life, to become more predictable. Damage nucleation frequently develops in two stages, where nascent or pre-damage conditions develop during monotonic deformation resulting from forming operations, followed by growth to a critical size during service, e.g. growth of short cracks at a scale smaller than the grain size, to one larger than the microstructural scale during subsequent loading. In this case nucleation and growth of fatigue cracks depends strongly on microstructure evolution during prior forming history. Thus, a paradigm is needed to understand how the process of plastic deformation interacting with microstructural features leads to the development of subcritical cracks or voids.

From both experimental and computational studies, it is commonly held that damage nucleation occurs in locations of large strain concentrations (from the continuum perspective, as in Fig. 1a), or microstructurally, where substantial heterogeneous deformation occurs. If large local strains are effective in accommodating required geometry changes they may prevent damage nucleation, whereas it is conceivable that damage may nucleate where insufficient strain or shape accommodation occurs, as illustrated schematically in Fig. 1b. Such variability in shape accommodation is connected to crystal orientations and crystallographic deformation mechanisms. Experimentally, heterogeneous deformation is often assessed using slip trace analysis, which can be accomplished with both optical and electron microcopy, and can be greatly enhanced and made more quantitative by using tools such as orientation imaging microscopy and strain mapping. However studies that fully analyze the operating deformation mechanisms in the context of the stress–strain history and observed microstructure evolution are rare. More qualitative experiments commonly show cracks and voids developing preferentially in some boundaries but less in others, indicating the significance of heterogeneity in local deformation history.

Computationally, two approaches to modeling evolution of microstructure have developed, statistical methods based upon Taylor theory, and finite element polycrystal plasticity approaches (atomistic or discrete dislocation density models can typically model volumes much smaller than a cubic micron (e.g. Farkas, 2005, Arsenlis et al., 2004), making them most useful for modeling nanocrystals). Statistical models developed on the foundation of Taylor theory (e.g. Chen and Gray, 1996, Nemat-Nasser et al., 1998, Nemat-Nasser and Guo, 2000) homogenize deformation characteristics, which is useful for modeling deformation phenomena at the scale of forming operations. This kind of analysis motivates models for yield surface evolution, e.g. Barlat et al. (2003). Homogenization is not helpful for investigating damage nucleation, however, which is a statistically rare event that reflects deviations from homogeneous behavior. This shortcoming can be partially overcome using viscoplastic self-consistent polycrystal plasticity codes that allow strains and stresses to vary in different crystal orientations (e.g. Lebensohn and Tome, 1993, Lebensohn, 2001, Karaman et al., 2000). Nevertheless, self-consistent codes are still based upon a statistical representation of a microstructure. Hence, damage that originates from strain incompatibilities in specific sites cannot be meaningfully predicted with statistical models such as the large body of literature based upon continuum damage mechanics (e.g. review of Lin et al., 2005), because the specific strain history depends on both the local strain behavior near an interface, as well as the strain history in adjacent grains or even within regions of the same grain (non-local strain). Self consistent models homogenize the grain neighborhood and, therefore, cannot provide detailed information at the local scale.

Modeling of site-specific stress–strain histories can be accomplished with crystal plasticity finite element modeling of representative microstructural volumes (oligocrystals or microstructure patches). Several approaches have recently been developed and compared with experimental observations (e.g. Hao et al., 2003, Hao et al., 2004, Heripre et al., 2007, Querin et al., 2007, Dunne et al., 2007, Clayton and McDowell, 2004, Bhattacharyya et al., 2001, Raabe et al., 2001, Ma and Roters, 2004, Ma et al., 2006a, Ma et al., 2006b, Zaafarani et al., 2006, Cheong and Busso, 2004, Cheong and Busso, 2006, Dawson et al., 2002, Kalidindi and Anand, 1993). To date, most modeling attempts of this kind have simulated high ductility damage resistant metals such as steel, copper, or aluminum. Characterizing damage nucleation events microscopically in such high ductility metals is challenging due to the large strains and high dislocation densities that precede damage nucleation.

The ability to predict damage nucleation and evaluate whether it will lead to the fatal flaw is one of the major goals of computational plasticity. Such predictions require multiscale modeling approaches that are under development in a number of groups and laboratories (Hao et al., 2003, Hao et al., 2004, Clayton and McDowell, 2004, Voyiadjis et al., 2004, Buchheit et al., 2005, Dunne et al., 2007, Cheong et al., 2007). While heterogeneous deformation is understood to be a precursor to damage nucleation, the actual initiation step between heterogeneous deformation and damage nucleation is not clearly understood. This connection is crucially important, because if the locations of damage are not properly predicted, then any simulations of microstructural evolution that evolve thereafter will be unreliable (merely fiction). A comprehensive review of multi-scale modeling of plastic deformation shows that solutions to practical problems often have the nanoscale effectively interacting with microscale, which cannot be handled by atomistic methods (Liu et al., 2004, Hao et al., 2003, Hao et al., 2004). Currently, there are no effective handoff methods between atomistic and microstructure scales. Hence, there is an opportunity for bridging across length scales if damage nucleation (intrinsically a nano-scale phenomenon) can be predicted reliably on the basis of heterogeneous microscale deformation.

Interfaces represent a profound challenge to modeling heterogeneous deformation and damage nucleation. Damage in particle-free materials normally nucleates at discontinuous interfaces such as grain or phase boundaries.1 At interfaces, strain must be somehow transferred from one grain to another through the boundary. In this process, damage may nucleate at a specific (rather than a generic) interface, due to both local and non-local effects. Rules for predicting which interfaces become damage nucleation sites are not known, though some have used slip transfer criteria as a means to identify suspicious locations (e.g. Ashmawi and Zikry, 2003a, Ashmawi and Zikry, 2003b). From the review that will follow, it will become clear that damage nucleation at interfaces depends on

  • i.

    the orientations of crystals on either side of the interface,

  • ii.

    the boundary orientation and structure (energy),

  • iii.

    the activated deformation systems on either side of the boundary, and

  • iv.

    the stress–strain gradient history in the grains on either side of an interface.

Research that considers all four of these factors is rare. For example, the grain boundary engineering paradigm focuses on grain boundary energy (item ii) as a metric for “good” or “bad” grain boundaries, but little has been done to examine how slip processes affect the character of “good” Σ boundaries differently from their “bad” random boundary counterparts. Item iii has rarely been examined experimentally or computationally, and when it has, it has not been done with fine detail. Studies of deformation transfer have led to identification of some rules by which a dislocation in one grain can penetrate into a neighboring grain (Clark et al., 1992, de Koning et al., 2002, de Koning et al., 2003). However, it is not clear how deformation transfer and damage nucleation are related, and this open question provides the primary motivation for this paper. Clearly, knowledge of a boundary’s propensity to generate damage could provide an effective bridge between atomistic and continuum scale models.

To assess the role of slip processes at interfaces on damage nucleation, it is important to have a reliable representation of heterogeneous deformation, the character of the grain boundary, and slip transfer mechanisms. These three topics and current approaches to integrate them are reviewed in some detail in order to provide motivation and a foundation for a new approach that identifies a deformation system based definition of grain boundary character. This new definition of grain boundary character was developed on the basis of experimental observations, and it may be able to determine which kind of deformation system interactions at the boundary will lead to damage nucleation. One example of a deeply characterized microstructure from this experimental work is examined using a current polycrystal plasticity finite element model to identify how mesoscale computational modeling may be used in combination with this new definition of grain boundary character to predict locations of damage nucleation.

Analysis of heterogeneous strain near boundaries can be traced back to Livingston and Chalmers (1957), who observed that more slip systems are active near bicrystal grain boundaries than in the grain interior. However, bicrystals with arbitrarily oriented grains generally activate only one slip system in the grain interior (unless orientations are chosen that have the same Schmid factor for multiple slip systems), unlike polycrystals that generally require activation of two or more slip systems due to compatibility constraints. Thus, bicrystals will not typically generate deformation states that correspond to those found in polycrystal interfacial boundaries, and consequently cannot provide a reliable basis for predicting deformation conditions in polycrystals (though they can and have provided much insight and understanding about processes of deformation transfer).

Poly or multicrystal aluminum alloys and pure copper have been used for focused studies to characterize and model heterogeneous slip near grain boundaries (e.g. Yao and Wagoner, 1993, Delaire et al., 2000). Within a given grain, slip traces of deformation systems with high Schmid factors extended across grains, while planes with moderate Schmid factors had slip traces that extended part way from a boundary into the grain interior. Experimentally measured surface strain maps on high purity copper polycrystals show that heterogeneous strains extend 20–100 microns into the grain interior (Thorning et al., 2005, Clayton and McDowell, 2004). Local rotations have been measured using orientation imaging microscopy, which has allowed direct comparisons with polycrystal plasticity-finite element method (CP-FEM) models (Bhattacharyya et al., 2001, Raabe et al., 2001, Sachtleber et al., 2002, Tatschl and Kolednik, 2003, Prasannavenkatesan et al., 2005, Cheong and Busso, 2004, Cheong and Busso, 2006, Cheong et al., 2007). Raabe et al. (2001) used a local micromechanical Taylor factor to better predict local crystal rotations and strains that were measured using high resolution strain mapping techniques. Simulations of HCP metals and alloys have also been done (e.g. Barton and Dawson, 2001, Diard et al., 2005), though there is a lesser degree of direct comparison with experiment than with investigations on cubic metals. The FEM models generally assume no specific grain boundary properties such that the discontinuity of plastic properties accounts for the observed heterogeneous strain.

While boundaries clearly cause heterogeneous strain in adjoining crystals, it is not clear how this strain affects the cohesive properties of the grain boundary, because a strongly cohesive boundary may force heterogeneous strain in an adjacent grain in order to maintain compatibility. On the other hand, large heterogeneous strains may cause large tensile tractions in the boundary that could nucleate damage (e.g. Bieler et al., 2005a, Bieler et al., 2005b, Querin et al., 2007). Thus, the evolution of local stress–strain history and boundary properties will affect damage nucleation.

A growing body of research has followed the paradigm that strong and weak grain boundaries can be identified based upon the structure of the boundary, i.e. grain boundary character (Watanabe, 1984). However, the fundamental meaning of strong and weak boundaries is not well defined. For example, a strong boundary may simply have a high cohesive strength, but such a boundary may not always be a strong barrier to dislocation motion (e.g. twins can either transmit dislocation slip easily or be a strong barrier, depending on the actual the slip system interaction). Thus, the idea of grain boundary strength can be approached from several perspectives.

Many studies have correlated properties of boundaries with their interfacial structure using the coincident site lattice (CSL) model, where a low Σ value indicates a high degree of common lattice points in adjacent grains. Beneficial properties of low Σ boundaries have been identified: Low Σ boundaries have low solubility for alloy or impurity elements, which makes them less susceptible to corrosion or nucleation of second phase particles (Palumbo et al., 1991, Watanabe, 2000, Lejcek and Hofmann, 2002, Lejcek et al., 2003, McMahon, 2004). Low Σ boundaries maintain their low energy configurations with a few degrees of deviation, accommodated by grain boundary dislocations (Brandon, 1966, Frary and Schuh, 2003, Bollmann, 1982 Materials with large numbers of low Σ boundaries (Palumbo et al., 1998, Kim et al., 2003, Watanabe and Tsurekawa, 2004, Randle, 2004) that are well connected in networks (Schuh et al., 2003, McGarrity et al., 2005) exhibit higher flow stress and ductility than materials with few low Σ boundaries. Because low Σ boundaries are less able to absorb lattice dislocations than random boundaries (Kokawa et al., 1981), many researchers have attributed material strength, and/or resistance to damage nucleation, to the presence of low Σ boundaries (Watanabe, 1984, Watanabe and Tsurekawa, 1999, Watanabe and Tsurekawa, 2005, Tsurekawa et al., 1999).

Another class of boundaries referred to as special boundaries have interfaces with low energy surfaces and repeating polyhedral structural units (generally a subset of low Σ boundaries). Of the five parameters that describe a boundary, three for the misorientation, and two for the boundary normal, low Σ boundaries are sufficiently defined by the misorientation. Many studies show that lowest energy configurations result when the boundary normal is a low index crystal plane, or has a common crystal direction about which there is a tilt or twist (Rohrer et al., 2004, Rohrer et al., 2006, Tschopp and McDowell, 2007, Wolf, 1990). Hence, low angle boundaries (referred to as Σ1) are special. In analysis of beneficial grain boundary character, the most beneficial boundaries are found to those with low surface energy, which also have structural repeating polyhedral units in the boundary plane (Randle, 2001, Rohrer et al., 2006). Recent studies by Tschopp et al., 2007, Tschopp et al., 2008 have shown that dislocation emission is correlated with the presence of particular kinds of polyhedral structural units.

However, low Σ or special boundary attributes are neither a necessary nor sufficient definition of a strong or beneficial boundary. First, the beneficial effect of these boundaries cannot be exclusively ascribed to lower solute content, because solute atoms can also strengthen grain boundaries, e.g. B doping in aluminides. Second, even though the benefit of such boundaries is statistically convincing, some low Σ boundaries do develop damage, while many more random boundaries do not (e.g. Lehockey and Palumbo, 1997), suggesting that additional criteria for identifying strong and weak boundaries exist, such as the influence of active deformation systems. There is little study of effects of deformation systems on boundary character (Davies and Randle, 2001); only a few special cases have been examined e.g. (Su et al., 2003, Pyo and Kim, 2005). Third, some general boundaries (or very high Σ boundaries) have special properties based upon the rotation axis (Lejcek and Paidar, 2005), or “plane-matching boundaries”, which are statistically more common than low Σ boundaries (Kawahara et al., 2005). Fourth, the benefit of low Σ or special boundaries has rarely been examined in non-cubic materials, even though the structure of low Σ boundaries is known (e.g. HCP – Wu et al., 2004, L10 – Singh and King, 1993). Much of the grain boundary engineering literature is more focused on creating low Σ boundaries with heat treatments than examining why they are effective.

While the CSL approach considers how well the lattices on either side of a boundary are aligned, another approach for assessing grain boundary character is based upon understanding the geometry of grain boundary dislocations (Brandon, 1966, Frary and Schuh, 2003, Bollmann, 1982 developed the O-lattice approach to identify grain boundary dislocations, which has been used to explain diffraction contrast features in grain boundaries, (e.g. Solenthaler and Bollmann, 1986). Grain boundary dislocation Burgers vectors may or may not reside in the boundary plane, making them mobile or sessile, respectively. Even if boundary dislocations are mobile, they will face barriers at triple lines (where three boundaries meet), where they may or may not be able to continue to propagate, depending on whether the triple line is hard or soft (Fedorov et al., 2003). Triple lines are often described as I- or U-lines (Bollmann, 1984, Bollmann, 1988, Bollmann, 1991), where I-lines are typically intersections of Σ boundaries. I-lines do not have dislocations entering the boundary from adjacent grains, whereas U-lines do, resulting in disclinations, where crystal dislocations terminate along the triple lines. Dislocation transmission is possible through I-lines without development of dislocation debris, so they allow slip transfer. From this paradigm, triple line characteristics affect properties (Bollmann, 1991, Randle, 1995), including the likelihood of triple junction cracking (Wu, 1997, Wu and He, 1999). U-lines have higher energy (due to unbalanced dislocation content), providing sources or sinks for lattice dislocations during deformation, and are more susceptible to cavitation damage than I-lines.

There is a possible disconnect between the slip transparency of I-lines (junctions of low Σ boundaries), and the sense of boundaries being strengthening elements that resist dislocation motion (Tsurekawa et al., 1999, Kobayashi et al., 2005, Lim and Raj, 1985). Clearly, the influence of low Σ or special boundaries and associated I-lines on damage nucleation mechanisms is only partially understood. More importantly, the random boundaries that are more likely to develop damage nucleation need focused and systematic attention in order to identify how damage develops, because there will normally be a significant number of random boundaries in polycrystals.

From the prior section it is clear that a good understanding of how deformation systems interact with boundaries is necessary before deformation transfer mechanisms can be distinguished from damage nucleation mechanisms. Three types of deformation transfer can be imagined near boundaries:

  • (1)

    the grain boundary acts as an impenetrable boundary that forces operation of additional intragranular (self accommodating) slip systems that generate localized rotations (Zaefferer et al., 2003) in order to maintain boundary continuity;

  • (2)

    the boundary is not impenetrable, and slip in one grain can progress into the next grain with some degree of continuity (leaving residual boundary dislocations, and perhaps only partial ability to accommodate a shape change, as suggested in Fig. 1b);

  • (3)

    the boundary is transparent to dislocations, and (near) perfect transmission can occur, i.e. no deformation resistance (e.g. I-lines, low Σ boundaries (Lim and Raj, 1985) or low angle boundaries (Zaefferer et al., 2003, Kobayashi et al., 2005)). This type is most typically modeled in an FEM mesh grain boundary.

Additionally, boundaries impose a threshold stress effect, such that strain bursts through a boundary occur with increasing stress/strain due to achieving a stress sufficient to activate a grain boundary source (Wang and Ngan, 2004, Hasnaoui et al., 2004, Kobayashi et al., 2005)). Also, the characteristics of boundaries change with strain (Sun et al., 2000); e.g. the localized rotation gradients that reflect the geometrically necessary dislocations can change with increasing strain due to dislocation absorption or emission from the boundary that alters the boundary character. Furthermore, solute atom content can affect the activation energy for a dislocation source at a boundary (Floreen and Westbrook, 1969).

A number of studies have focused on boundary type (2), i.e. conditions where slip transfer is likely, but geometrically imperfect. Livingston and Chalmers (1957), who examined the geometry of slip transfer in bicrystals, concluded that deformation near boundaries required at least four activated slip systems within the two grains. Clark et al. (1992) and Lagow et al. (2001) determined that the slip transmission process often leaves residual dislocations in the boundary and requires a change in direction of the Burgers vector, along with a change in the plane orientation that results in two intersecting lines in the grain boundary plane. This geometry is illustrated in Fig. 2, and the three ‘rules’ that summarize conditions for slip transfer are:

  • 1.

    The angle between the lines of intersection between the grain boundary and each slip system (θ) must be a minimum.

  • 2.

    The magnitude of the Burgers vector of the dislocation left in the grain boundary must be a minimum.

  • 3.

    The resolved shear stress on the outgoing slip system must be a maximum.

Based on these qualitative observations, quantitative geometrical expressions describing the likelihood of a slip transmission event have been developed. For example, Luster and Morris (1995) noted that large values of cos ψ cos κ, were correlated with observed instances of slip transmission, where ψ and κ are the angles between the slip plane normals or slip directions respectively of two slip systems on either side of a boundary. Ashmawi and Zikry (2003a) considered a slip transmission criteria based on the degree of coplanarity of slip systems engaged in deformation transfer that included both θ and ψ. Other instances of deformation transfer have focused more on Burgers vector colinearity (cos κ in Fig. 2), such as Gibson and Forwood (2002), who found that twin impingement at boundaries in TiAl is accommodated by á/2<110) ordinary dislocation slip on a variety of planes on both sides of the boundary, with residual dislocations left in the boundary. In a subsequent examination of equiaxed TiAl (Simkin et al., 2003a, Simkin et al., 2003b, Bieler et al., 2005c, Fallahi et al., 2006), slip transfer was examined in the context of the global stress state (discussed further in section II).

Computational modeling of dislocation-grain boundary interactions has been carried out at several length scales, from molecular dynamics (MD), to discrete dislocation dynamics (DD), to CP-FEM analyses. However, the experimental characterization necessary to validate modeling has not kept pace with the detail available with such simulations. While orientation imaging is valuable, it is only recently 3-dimensional (e.g. Spowart et al., 2003, Zaafarani et al., 2006, Konrad et al., 2006), and it lacks angular accuracy (∼1° uncertainty in orientation). Consequently, there is an increasing effort to non-destructively obtain experimentally measured 3-D data sets (Juul-Jensen, 2005, Liu et al., 2005, Ice et al., 2005, Barabash et al., 2005, Shan and Gokhale, 2004), from which detailed computational models can be built. Quantitative experimental descriptions of dislocation activity near grain boundaries with known boundary conditions is possible using electron channeling contrast imaging (ECCI) methods (Simkin and Crimp, 1999, Simkin et al., 2003b).

Modeling and simulation of dislocation emission and interactions at the atomic scale has provided mechanistic insights (Yoo et al., 1995, Saraev and Schmauder, 2003, Hirth et al., 2006). Processes of dislocation nucleation from selected boundaries have been identified and described, and correlated with presence of particular polyhedral structural units (Spearot et al., 2005, Spearot et al., 2007, Spearot et al., 2008a, Spearot, 2008b, Tschopp et al., 2007, Tschopp et al., 2008, Brown and Mishin, 2007). Simulations have been able to identify mechanisms for grain boundary dislocation induced grain boundary motion (Cahn et al., 2006, Shen and Anderson, 2006). Dislocation absorption (slip transfer) has been modeled in Σ11 boundaries (de Koning et al., 2002, de Koning et al., 2003), which have confirmed the TEM observations of Clark et al. (1992) (above).

While simulations based upon discrete dislocations (DD) have focused more on crack propagation (e.g. Noronha and Farkas, 2004) or intragranular dislocation motion at an indenter tip (Miller et al., 2004), there are a few studies with grain boundaries using simple models (Espinosa et al., 2006, Balint et al., 2005, Cheong and Busso, 2004). It is possible to computationally combine atomistic simulations with dislocation dynamics modeling in a multi-scale approach (e.g. Noronha and Farkas, 2004, de Koning et al., 2003). While these simulations are insightful, for these methods to be reliable, they must be properly correlated with experimental observations.

At the microstructural scale, it is difficult to model deformation that takes place by dislocation absorption or transmission in boundaries (Ashmawi and Zikry, 2003a, Ashmawi and Zikry, 2003b, Clayton and McDowell, 2004, Ma et al., 2006a), as atomistic details that stimulate dislocation sources such as ledges (Fischer et al., 2003, Edwards et al., 2005) cannot be discretely modeled. Recent comparisons with characterized microstructures have shown that non-local formulations based on dislocation density evolution simulate experimentally measured local strains much better than formulations based upon phenomenological representations of strain hardening (indirect representation of evolving dislocation density) (Cheong and Busso, 2004, Cheong and Busso, 2006, Rezvanian et al., 2006, Horstemeyer et al., 1999, McGinty and McDowell, 1999, Clayton and McDowell, 2003, Busso and Cailletaud, 2005, Ma et al., 2006a, Ma et al., 2006b, Bardella, 2007). Special elements to handle dislocation accumulation and/or transmission through boundaries have been developed, but for practical reasons, they have finite thickness in the FEM mesh (Ma et al., 2006a, Ma et al., 2006b, Ashmawi and Zikry, 2003a). However, there are few direct correlations between models and experiments with analysis sufficient to assess the model’s ability to capture dislocation scale mechanistic interactions.

Of the studies of heterogeneous deformation just described, the studies of Clayton and McDowell, 2004, Ashmawi and Zikry, 2003a, Ashmawi and Zikry, 2003b, Dunne et al., 2007, Cheong et al., 2007 sought ways to predict damage nucleation. Other recent studies have focused more on damage evolution, once it exists (e.g. Orsini and Zikry, 2001, Bennett and McDowell, 2002, Bennett and McDowell, 2003, Bjerken and Melin, 2004, Cao et al., 2005, Bieler et al., 2005a, Bieler et al., 2005b). Such studies show that the microstructure geometry has a large influence on void or crack growth once damage exists, but they offer no direct understanding about damage nucleation mechanisms.

An important and frequently used approach to model for void nucleation based upon cohesive interface energy was first presented by Needleman, 1987, Xu and Needleman, 1994, who described the cohesive energy as an empirical scalar function that relates displacement with normal and shear traction evolution in the boundary. Such formulations have been adapted in damage nucleation models (Arata et al., 2002, Hao et al., 2003, Hao et al., 2004, Clayton and McDowell, 2004, Sfantos and Aliabadi, 2007, Kabir et al., 2007). Clayton and McDowell (2004) used non-local models to accurately predict local stress–strain history, and hence, tractions on the boundary. From this analysis, they identified a parameter that could be used to predict damage nucleation locations, based upon how much accommodation by void damage is required by the material to deform to a given strain level. This model assumed isotropic interfacial energy for all boundaries (MD modeling can overcome this, but at a much smaller scale (Spearot et al., 2004, Spearot et al., 2005, Spearot et al., 2007, Spearot et al., 2008a, Spearot, 2008b). Cohesive interface energy models are appealing in that they are two dimensional, but they do not use the available information regarding operating slip systems to examine or analyze damage evolution.

Using a different approach, Ashmawi and Zikry, 2003a, Ashmawi and Zikry, 2003b used the Clark et al. (1992) slip transfer criteria (quantified as cos ψ cos θ) to correlate dislocation density with damage, based upon the assumptions that (1) damage develops with dislocation accumulation, and (2) that slip transfer mitigated dislocation accumulation. Because there is no reason to correlate concentrated dislocation activity with damage (concentrated dislocation activity could either mitigate or cause damage), this approach may not be effective. Ma et al., 2006a, Ma et al., 2006b also developed grain boundary elements that increase deformation resistance in boundaries and allow some slip transfer, but this has not been used to examine damage nucleation. There is clearly a need for an effective but simple way to identify how damage nucleates that takes into account the active slip system history in the vicinity of the boundary.

A summary of the state of knowledge about heterogeneous strain and damage nucleation can be provided as a list of hypotheses, starting with strain and slip system considerations, and continuing toward boundary structure considerations. Some of these are incompatible with each other, but they reflect this state of understanding/misunderstanding of damage nucleation, so they can guide experimental analysis and theoretical model development:

  • 1.

    Damage nucleation always occurs at locations of maximum strain energy density (maximum area under local stress–strain curve).

  • 2.

    Large local strains can provide geometric accommodation that can prevent damage nucleation.

  • 3.

    Damage nucleation arises from slip interactions resulting from imperfect slip transfer through a boundary, which leaves residual dislocation content in the boundary plane.

  • 4.

    Damage nucleation occurs in particular boundaries where unfavorable slip interactions take place at the boundary to weaken the boundary.

  • 5.

    Slip interactions at the boundary are more (or less?) important than the magnitude of local strain for predicting damage nucleation.

  • 6.

    Damage nucleation occurs in locations where there is maximum geometric incompatibility arising from highly activated slip systems that cause dominant shears in very different directions (e.g. Bieler et al., 2005b).

  • 7.

    Damage nucleation is highly correlated with severe local strain heterogeneity, e.g. lattice rotations.

  • 8.

    Dislocation density (non-local) formulations of crystal plasticity models are necessary to adequately predict the local strains, and hence the slip system activity needed to predict damage nucleation.

  • 9.

    Damage nucleation probability is proportional to local hydrostatic tensile stress.

  • 10.

    Damage nucleation depends upon cohesive strength of the boundary, i.e. Griffith criterion – energy needed to separate an existing interface.

  • 11.

    Damage nucleation is more likely at triple lines than along boundaries, especially along U-lines.

  • 12.

    Slip directions are more influential on damage nucleation than slip planes.

  • 13.

    Low-Σ boundaries are less likely to accumulate damage than random boundaries.

  • 14.

    Boundaries with low index crystal normals (facets) are more (or less?) likely to resist damage.

  • 15.

    Twin boundaries resist damage because they repel dislocations from the boundary.

  • 16.

    Twin boundaries resist damage because they allow efficient slip transfer.

  • 17.

    Twin boundaries are schizophrenic (sometimes resistant, sometimes susceptible to damage nucleation).

  • 18.

    Fatal flaws are located where there is the greatest density of local damage sites.

  • 19.

    Fatal flaws are located where the size of nucleated damage grows the fastest.

Section snippets

New development of a slip system based definition of grain boundary strength

In contrast to high ductility metals such as pure copper, aluminum, and steel alloys damage nucleation is of crucial importance in low ductility metals and intermetallics, many of which are non-cubic. Such materials offer easier opportunities for analysis of active deformation systems, because they are fewer, and thus easier to analyze on the basis of microscopic evidence. Furthermore, an understanding of damage nucleation mechanisms is crucially important for technological applications in

Crystal plasticity-finite element modeling of microstructures with damage nucleation

Because the variables used for evaluating the fip are naturally computed in CP-FEM modeling of microstructures, it is possible to evaluate the fip in the CP-FEM setting. Furthermore, a CP-FEM analysis provides local stresses and strains that are not easily measured experimentally (strains and rotations can be partially assessed using differential image correlation and OIM data to track rotations and orientation gradients).

Conclusions

Based on this state of knowledge and the above experimental and CP-FEM modeling of TiAl polycrystal microstructures, we can evaluate four of the 19 hypotheses described at the end of section I affirmatively, specifically numbers 2, 3, 4, and 12, which are restated and justified with results from this analysis in italics:

2: Large local strains can provide geometric accommodation that can prevent damage nucleation. Damage nucleation was not observed at locations having the highest local strain

Acknowledgements

The experimental part of this research was supported by the Air Force Office of Scientific Research contract number under Grant # F49620-01-1-0116, monitored by Dr. Craig Hartley and by the Michigan State University Composite Materials and Structures Center. T.R.B. acknowledges sabbatical support from the Max-Planck-Institut für Eisenforschung and Michigan State University, and practical help with FEM calculations from Nader Zaafarani, Claudio Zambaldi, and Luc Hantcherli at MPIE. D.E.M.

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