7. CPFE approaches to local damage analysis

7.1. Continuum approaches to modeling damage

where D is a scalar quantity ranging from 0 (no damage) to 1 (fully damaged) (the plastic response follows as a consequence). Many authors have recognized that damage represented by the parameter D is not isotropic, and have introduced vectorial and tensorial modifications to this idea to simulate failure processes associated with crystallographic planes, intergranular fracture or distortion and growth of voids, e.g. Luccioni and Oller [332], Menzel et al. [333], or Voyiadjis and Dorgan [334].

As continuum approaches are valuable for design and modeling at the component scale, it is desirable to develop a method by which the continuum anisotropic damage formulations can be informed by physically modeled plastic processes. Thus, if models of microstructures using CPFE methods can identify how dislocation-based deformation processes cause damage nucleation and evolution, then these can be expressed in forms that are ready for use in continuum scale models.

7.2. Microstructurally induced damage

Micromechanics studies imply that shear localization at the microscale occurs as a result of microstructure characteristics such as inclusion morphology and distribution, grain boundary character, texture, grain shape, and the operation of slip systems coupled with damage site locations. Figure 18 illustrates how two cavities that are close together may not coalesce preferentially if the slip systems are not favorably oriented. Thus, shear localization, and hence the toughness, will depend upon anisotropic microstructural details. This anisotropic effect is observed at the macroscale by 25 % variations in KIc with respect to direction in rolled sheet material [335]. However, there has been limited study of how the crystallographic processes lead to failure mechanisms that depend on local grain and grain boundary orientations, making damage modeling using CPFE approaches an important area for future study. In particular, CPFE allows the direct modeling of experimentally characterized microstructures where damage has been observed, in order to evaluate theories of damage nucleation and early growth stages.

Grain boundaries are often sources of critical damage nucleation, even when pre-existing cracks may be present within a grain. In Al alloys, the primary mechanisms for grain boundary crack nucleation are void coalescence between grain boundary precipitates [336, 337]. In fatigue conditions, sub-critical short cracks either pre-exist due to cracked inclusions (such as constituent particles, see figure 18) or from cracking early during fatigue cycling, but do not propagate past a limiting grain boundary or past a triple line [338, 339]. When short cracks are able to penetrate a grain boundary, they make the transition to longer cracks that can then be modeled with established continuum fracture mechanics. This penetration event often takes place late in the cycling process, indicating that crack penetration of the grain boundary may control fatigue life.

7.2.1. Heterogeneous plastic deformation

Both experimental and computational studies suggest that damage nucleation occurs in locations of large strain concentrations, which develop in locations of substantial heterogeneous deformation near microstructural features such as grain or phase boundaries. However, if a large local strain is effective in accommodating a required local geometry change (due to boundary conditions imposed by differential deformation amounts in the local neighborhood) a locally large strain may prevent damage nucleation. In contrast, damage may nucleate where an insufficient amount of strain occurs to accommodate a locally required shape change, such that opening a free surface may require less energy than further intragranular deformation. Clearly, not all large strain sites are damage sites, and damage may develop where strains are modest.

It is well known that strain varies from grain to grain due to the effects of differing deformation processes in neighboring grains, figure 3. The spread of deformation within a grain does not only depend on the orientations of the neighboring grains, but also on the constraints provided by neighboring grains that diminish but are still significant several grains away [106, 340, 197]. Within a given grain, slip traces of deformation systems with high Schmid factors may extend across grains, while planes with moderate Schmid factors may reveal slip traces that extend 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 [341, 322]. Local lattice rotations have been measured using orientation imaging microscopy, which has allowed direct comparisons between experiment and CPFE models [90, 38, 14, 342, 343]. Simulations of local rotations measured using high resolution strain mapping and local strain accumulation effects were better understood at the grain scale when a local micromechanical Taylor factor was used to identify the activated slip systems [38].

Because damage originates from strain incompatibilities in specific sites, it is most appropriate to investigate conditions that lead to damage nucleation using CPFE methods that model experimentally realistic microstructures

(e.g. Hao et al. [344], Clayton and McDowell [322], Bhattacharyya et al. [90], Raabe et al. [38], Ma and Roters [23], Ma et al. [24, 25], Zaafarani et al. [171], Cheong and Busso [22], Dawson et al. [345], Kalidindi and Anand [346]). As damage events reflect interactions between the microstructure scale and the atomic scale, they are intrinsically nanoscopic. Thus, multiscale modeling approaches that include atomistic scale computations are under development in a number of groups [347, 348, 322, 349, 350].

7.2.2. Interfaces

Interfaces represent a profound challenge to modeling heterogeneous deformation and damage nucleation. The cohesive strength of the boundary in real polycrystals varies according to the atomic scale arrangement; some boundaries have more disorder than others [351], leading to lower interfacial cohesive strength. Many studies have correlated properties of boundaries with their interfacial structure through coincident site lattice (CSL, or low Σ) boundaries. Because low Σ boundaries have less free volume due to better packing efficiency, these boundaries are assumed to be strong. Materials with large numbers of low Σ boundaries [352, 353, 354] that are well connected as networks [355] exhibit higher flow stress and ductility than materials with few low Σ boundaries (a weak boundary percolation can occur if there are less than 78 % low Σ boundaries [356]). Because low Σ boundaries are less able to absorb lattice dislocations than random boundaries [357], many researchers have attributed material strength, and/or resistance to damage nucleation, to the presence of low Σ boundaries [358]. This characterization of the boundary state is useful in computational modeling, as the grain boundary energy used in a Griffith criterion provides a criterion for nucleating a crack.

7.2.3. Cohesive zone boundary modeling

The energy-based definition of the grain boundary character has been modeled in CPFE modeling using the cohesive force model first presented by Needleman [359] [360], who described the cohesive energy as an empirical scalar function that relates displacement to normal and shear traction evolution in the boundary plane figure 19). Such formulations have been adopted in damage nucleation models [361, 344]. Clayton and McDowell [322] used non-local models to accurately predict local stress–strain history, and hence, tractions on the boundary. From this analysis, they identified a parameter which 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 (molecular dynamics modeling can overcome this, but at a much smaller scale). Such models have been evaluated to identify how nucleation and growth of voids affects subsequent deformation processes (e.g. figure 20). 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.

Low Σ boundary attributes are not a sufficient definition of a strong or weak boundary. First, the beneficial effect of low Σ 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 low Σ boundaries is statistically convincing, some low Σ boundaries do develop damage, while many more random boundaries do not, e.g. Lehockey and Palumbo [325], figure 21, suggest that additional criteria for identifying strong and weak boundaries exist, such as the influence of active deformation systems. Third, some general boundaries have special properties based upon the rotation axis [362], or ’plane-matching boundaries’, which are statistically more common than low Σ boundaries [363]. Fourth, the benefit of low Σ boundaries has rarely been examined in noncubic materials, even though the structure of low Σ boundaries is known

(e.g. hcp -[364], L10 -[365]). Finally, grain boundary dislocations are also important, as they are interrelated with the structure of the boundary, and they affect how slip can be transferred across a boundary, as discussed in the next section. Much of the grain boundary engineering literature is more focused on creating networks of low angle boundaries with heat treatments than examining why they are effective.

7.2.4. Grain boundary slip transfer

The analysis of heterogeneous strain near boundaries was initiated by Livingston and Chalmers [366], who observed that more slip systems are active near bicrystal grain boundaries than in the grain interiors. 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. While bicrystal deformation provides insights about mechanisms of deformation transfer, the results cannot be directly transferred to general grain boundaries in polycrystals.

Studies of deformation transfer have led to identification of some rules by which a dislocation in one grain can penetrate into a neighboring grain [249, 367]. These rules have been confirmed with atomistic scale simulations by de Koning et al. [368]. 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, resulting in two intersecting lines in the grain boundary plane. This geometry is illustrated in figure 22, and the three ’rules’ that summarize conditions for slip transfer are:

• The angle between the lines of intersection between the grain boundary and each slip system (Θ) must be a minimum.
• The magnitude of the Burgers vector of the dislocation left in the grain boundary (correlated to the magnitude of κ) must be a minimum.
• The resolved shear stress on the outgoing slip system must be a maximum.

Semi-quantitative geometrical expressions describing the likelihood of a slip transmission event have been developed. Luster and Morris [369] noted that large values of cos ψ cos κ, were correlated with observed instances of slip transmission. Slip transmission criteria depend strongly on the degree of coplanarity of slip systems engaged in deformation transfer (Θ will be small if ψ is small). Other studies of deformation transfer have focused more on the misalignment of the Burgers vector colinearity (cos κ in figure 18), such as Gibson and Forwood [370], who found that twin impingement at boundaries in TiAl is accommodated by a/2<110] ordinary dislocation slip on a variety of planes on both sides of the boundary, with residual dislocations left in the boundary.

The process of slip transfer is also dependent on grain boundary dislocations. 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 face barriers at triple lines, where they may or may not be able to continue to propagate. Triple lines are often described as I-or U-lines [371], where I-lines are typically intersections of Σ boundaries. Dislocation transmission without development of dislocation debris is possible through I-lines, so that they permit slip transfer, whereas U-lines providing sources or sinks for lattice dislocations during deformation. Thus, triple line characteristics affect properties [371, 372], e.g. cavitation and cracking are more likely at U-lines [373].

There is an interesting disconnect between the slip transparency of I-lines, and the fact that networks of such boundaries may even increase the strength [374, 375]. Clearly, the influence of low Σ boundaries and associated I-lines on damage nucleation mechanisms is only partially understood. However, it appears important to focus attention on damage nucleation mechanisms in high Σ and random boundaries that are more likely to develop damage, because there will always be a significant number of random boundaries in polycrystals.

Consideration of the geometry of slip transfer suggests that there are three classes of boundaries with respect to their mechanical behavior:

• (1) The grain boundary acts as an impenetrable interface that forces operation of additional intragranular (self accommodating) slip systems that generate localized rotations [95] 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). 

• (3) The boundary is transparent to dislocations, and (near) perfect transmission can occur (e.g. low Σ boundaries related to I-lines, [376] or low angle boundaries [95, 375], this type of boundary is most naturally modeled with CPFE methods).

Further complications are suggested from experimental observations. From nanoindentation experiments it is known, that grain 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 [377]. The misorientations of boundaries, and hence, their properties, change with strain [378]. For example, a change in boundary character that affects dislocation absorption or emission from the boundary will affect the localized rotation gradients arising from geometrically necessary dislocations.

This discussion clearly shows that before damage nucleation can be predicted, deformation transfer mechanisms must be modeled in a reasonable manner. Further, if a relationship between deformation transfer and damage nucleation can be developed, this would provide an effective bridge between atomistic and continuum scale models.

To make computational modeling of damage nucleation possible in the CPFE paradigm, grain boundary elements that allow physically realistic deformation transfer are necessary. Two approaches of modeling grain boundary deformation have been proposed by Ma et al. [25] and Ashmawi and Zikry [379]. In both cases, grain boundary elements with finite thickness were used. Ashmawi and Zikry [379] used the grain boundary element to track the evolution of dislocation density in elements in an envelope fanning into the grain interior on either side of the boundary. The most active slip system in this envelope was evaluated, and then this density was tracked. Grain boundary elements accumulated the impinging dislocation density as a damage factor in a continuum element similar to D in equation 87 above; this is interpreted as a pileup that causes cavitation to develop, and hence the reduction in stress carrying capability. The density was reduced if slip transfer occurred, i.e. if Θ < 15°and κ< 35^° in figure 22 (based upon experimental observations of Werner and Prantl [367]), then slip transfer was permitted in proportion to the geometrical factor cos Θ cos κ to reduce the accumulated dislocation density in the grains on either side. This formulation used arbitrary square crystal plasticity elements in square grains with thinner grain boundary elements having the continuum based damage nucleation model, so it was not examined using practical microstructures.

In contrast, Ma et al. [25] developed a boundary element with crystal plasticity components with an increased resistance to flow stress based upon the fractional dislocation debris left in a boundary when slip transfer occurs (see details in section 4.3.2). This increase in flow resistance is expressed as an increase in the activation energy barrier for dislocation slip within the grain boundary element, and hence the deformation process in the boundary is kept crystallographic. However, in both cases, the process of what happens to dislocations that retain some sense of their identity as they penetrate into the neighboring grain is neglected for simplicity in the interest of capturing at least some of the physics of the process. There is clearly opportunity for further insightful development of a practical grain boundary element that can capture both the dislocation slip transfer and the damage nucleation processes in practical and realistic ways.

7.2.5. Experimental studies of fracture initiation criteria

While much research in CPFE has focused on ductile cubic metals, damage nucleation is much more critical in low ductility metals and intermetallics, at both ambient and high temperature conditions. Due to more limited slip, it is easier to experimentally identify relationships between slip, twinning, and damage nucleation in slip-limited materials. After making unsuccessful attempts to correlate damage with slip transfer using only geometric parameters, Simkin and Bieler [380] developed a fracture initiation parameter (fip) that is based upon the activity of slip and twinning systems in a stressed condition for adjacent grains in TiAl. The fip is analogous to a probability statement about how likely it is for a given grain boundary to crack when subjected to a stress field. A fip consists of several physical/geometrical factors that could enhance crack nucleation due to localized shear strain concentrated at the boundary. Variations of this idea are presented in equations as F_i, where i is a label [380, 381, 382, 383, 384]. For example, the fip parameter F₁ is the product of three terms:

The first term is the Schmid factor of the most highly stressed twinning system in a grain pair, mtw, which identifies twins that cause the largest shear discontinuity at a grain boundary. The second term is the scalar product of the unit vector of this twin’s Burgers vector direction, b^ˆ_tw, and the unit vector pointing in the direction of the maximum tensile stress t^ˆ, i.e. b^ˆ_tw · ^ˆt, which identifies the strength of a mode I opening component at the boundary. This term is the part of the Schmid factor related to the slip direction. The third term, ∑_ord | b^ˆ_tw · b^ˆ_ord| , is the sum of scalar products between the Burgers vector of a highly-stressed twin system in the initiating grain (with Schmid factor m_tw) and the Burgers vector of available ordinary slip systems in either the same grain or the neighboring (responding) grain. This term describes how well the local shear direction at the boundary can be accommodated by dislocation activity in the neighboring or initiating grain, i.e. the scalar product defined by the angle κ in figure 22. This quantitatively expresses one of the three requirements identified by Clark et al. [249] for slip transfer. The sum term is maximized when two or more slip systems have a modest value of κ, because when the scalar product is near 1 for one slip system, the scalar product is much smaller for the rest. Thus, the sum is large when the opportunity for imperfect slip transfer is large.

From experimental measurements, the fip is larger for cracked boundaries than intact boundaries, implying that imperfect slip transfer (which leaves residual dislocation content in the boundary) is strongly correlated with crack nucleation. This approach has been shown to be statistically significant in two different materials under different loading and temperature conditions, ambient temperature deformation in equiaxed (duplex) TiAl, and during creep in high stress creep of a cobalt-based superalloy [384]. While this suggests that the fip concept may be a robust predictor of damage nucleation, further examination of this concept in other material systems is needed.

7.2.6. Strain energy as a criterion for damage

Strain energy is a commonly used criterion for damage. A recent example that illustrates this approach in crystal plasticity studies is in the work of Dunne et al. [340], who used cumulative plastic slip as a means to predict damage sites in CPFE studies of Ni and Ti alloys in low cycle fatigue studies. In the maximum stress region of a continuum model of a 3-point bend specimen, they inserted a crystal plasticity section with the same grain configuration as a carefully analyzed experiment. Planes with highly active slip corresponded to planes with high Schmid factors and observed slip bands in the experiment. With a one dimensional damage model similar to equation 87, they were able to simulate the locations of persistent shear bands and crack positions in a 3-point fatigue bending specimen. However, the details of the crack nucleation differed between the experiment and simulation. This was in part due to simplifying assumptions regarding the stress state. This work showed that shear bands and cracks are very sensitive to the actual local geometry of grains.

The importance of local geometry was further emphasized in a systematic computational study of fatigue facet (crack) formation in hard orientations of titanium in polycrystals [340]. Particular orientations of adjacent crystals and particular grain boundary inclinations were found to be most likely to generate slip penetration from the adjacent soft grain into the hard grain such that tensile stresses developed normal to basal planes. Such conditions facilitate formation of facets that develop into fatigue cracks. This computational study was consistent with features in deeply characterized experiments of Sinha et al. [385] and Bieler et al. [386].

CPFE is particularly valuable for identifying microstructural conditions where strain incompatibility develops (this is exaggerated in slip-limited materials). These incompatibilities develop due to activation of slip systems that cause shears in very different direction in adjacent grains, leading to significant local triaxial stress states and load shedding to harder orientations. Self-consistent modeling of generic microstructural characteristics have been used to estimate plausible stress states, e.g. Bieler et al. [386], but as the prior examples show, the actual grain geometry leads to very significant variations around such estimates. It is clear that strain energy is an important metric for predicting locations where damage is possible, but clearly, there are additional criteria that must be considered (and identified) to account for the fact that not all sites with potentially dangerous characteristics actually develop damage.

7.3. Assessment of current knowledge about damage nucleation

CPFE modeling is an enabling tool for examining conditions that lead to damage nucleation. However, the physical understanding that is needed to develop computationally e&#64259;cient and robust criteria is relatively undeveloped. Rather than stating what only may be true on the basis of recent studies, relationships between heterogeneous strain and damage nucleation can be provided as a list of hypotheses that can be explored in combined experimental and CPFE computational modeling research programs.

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. [387].
7. Damage nucleation is highly correlated with severe local strain heterogeneity, e.g. lattice curvature.
8. Dislocation density-based (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 depends upon cohesive strength of the boundary, i.e. energy needed to separate an existing interface — Griffith criterion.
10. Damage nucleation probability is proportional to local hydrostatic tensile stress.
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. Twin boundaries resist damage because they repel dislocations from the boundary.
15. Twin boundaries resist damage because they allow efficient slip transfer.
16. Twin boundaries are schizophrenic (sometimes resistant, sometimes susceptible to damage nucleation).
17. Fatal flaws are located where there is the highest density of local damage sites.
18. Fatal flaws are located where the size of nucleated damage grows the fastest.