Royal Society Publishing

A systematic study of hcp crystal orientation and morphology effects in polycrystal deformation and fatigue

F.P.E Dunne, A Walker, D Rugg

Abstract

Elastically anisotropic, physically based, length-scale- and rate-dependent crystal plasticity finite element investigations of a model hcp polycrystal are presented and a systematic study was carried out on the effects of combinations of crystallographic orientations on local, grain-level stresses and accumulated slip in cycles containing cold dwell. It is shown that the most damaging combination is the one comprising a primary hard grain with c-axis near-parallel to the loading direction and an adjacent soft grain having c-axis near-normal to the load and a prismatic slip plane at approximately 70° to the normal to the load. We term such a combination a rogue grain combination. In passing, we compare results with the Stroh model and show that even under conditions of plasticity in the hcp polycrystal, the Stroh model qualitatively predicts some of the observed behaviours. It is shown that under very particular circumstances, a morphologicalcrystallographic interaction occurs which leads to particularly localized accumulated slip in the soft grain and the penetration of the slip into the adjacent hard grain. The interaction effect occurs only when the (morphological) orientation of the grain boundary in the rogue grain combination coincides (within approximately ±5°) with the (crystallographic) orientation of an active slip system in the soft grain. It is argued that the rogue grain combination and the morphological–crystallographic interaction are responsible for fatigue facet formation in Ti alloys with cold dwell, and a possible mechanism for facet formation is presented. The experimental observations of fatigue facet formation have been reviewed and they provide considerable support for the conclusions from the crystal plasticity modelling. In particular, faceting was found to occur at precisely those locations predicted by the model, i.e. at a rogue grain combination. Some experimental evidence for the need for a crystallographic–morphological interaction in faceting is also presented.

Keywords:

1. Introduction

The phenomenon of faceting—the development of microcracking at about the length of the grain size, almost always associated with an hcp basal plane—is of importance in a wide range of titanium alloy components operating in gas turbines in both the aerospace and power generation industries (Evans & Bache 1994; Evans 1998; Bache 2003; Bache & Evans 2003). The phenomenon is not yet fully understood, but its importance cannot really be understated: if faceting occurs at a particularly large grain (here considered to be a region of approximately uniform crystallographic orientation, rather than anything determined optically), which has certainly been observed to occur in practice, then the time or cycles necessary to propagate the crack to a length considered unacceptable is likely to be short. In such circumstances, component lifetime is largely determined by the nucleation—or faceting—process.

A feature of faceting in titanium alloys is that it seems often to be associated, or at least exacerbated, by cold dwell (used here to indicate a material rate dependence even at 20°C). There is now substantial experimental evidence that rate effects, including creep and stress relaxation, take place at 20°C in a range of titanium alloys (Evans & Bache 1994; Bache 2003; Bache & Evans 2003), and that the inclusion of load-hold periods at this temperature has a pronounced effect in reducing cycles to failure. A further complication is that while cold dwell effects, and the consequent reduction in lifetime, are observed in laboratory testing (Bache 2003), faceting is rarely observed in such circumstances, but is occasionally observed in components after loading in tests. This has led to the development of the view that material volume is important, but a definitive explanation has not yet been provided. The purpose of this paper is to present the results of systematic studies of crystallographic and morphological analyses using finite element crystal plasticity modelling and provide a possible explanation for faceting, i.e. to demonstrate which combination of crystallography and morphology is necessary in order to generate the conditions necessary for faceting and explain its dependence on the material volume.

Important information on the nature of faceting has been obtained from previous experimental work. In particular, it has been observed that facets are often associated with hcp crystal basal planes when oriented approximately normal to the loading direction. For example, in a textured titanium alloy sample extracted from a component, scanning electron microscopy (SEM) with electron backscatter diffraction (EBSD) was used to show that faceting was primarily associated with basals normal to the loading. In earlier work, Evans (1998) used this information to suggest an explanation for faceting, based on the elastic analysis carried out by Stroh (1954). The Evans model comprises a hard grain (i.e. unfavourably oriented to the loading) adjacent to the one that is soft (oriented appropriately for easy slip), which, with Stroh's analysis, shows that very large grain boundary stresses (or strictly, within the assumptions of the approach, those within the hard grain) are developed, depending on the orientation of the active slip system in the soft grain. In effect, it was postulated that faceting may result from a particularly unfortunate combination of crystallographic orientations in adjacent grains, with respect to the loading direction. However, if the assumptions in the Stroh model are assessed, it becomes clear that the model such as that of Evans cannot be used in a quantitative sense. Stroh (1954) presented the elastic solution for the stress field generated from the termination of a single line containing dislocations (the slip plane in Evan's model). The orientation of the slip plane relative to the loading direction gives the angle between the slip plane and the adjacent hard grain, and Stroh showed that the highest stress, in a direction parallel to the loading in the hard grain, occurred for a very particular slip plane orientation. At the grain boundary (i.e. the line termination in Stroh's model), the stresses, however, become singular because it was, after all, an elastic analysis. Additionally, in the context of hcp crystals that enable slip, in principle, on basal, prismatic and pyramidal planes, the assumption of elasticity alone and the presence of only one ‘slip plane’, while laudable in terms of physical insight, are unlikely to be (and indeed is not, as we show later) quantitatively predictive.

The role of rate effects in cold dwell and faceting has been investigated by Hasija et al. (2003), Venkataramani et al. (2006) and Dunne et al. (in press). In both studies, hcp rate-dependent crystal plasticity was used, in particular, to study stress redistribution local to a hard (badly oriented) grain in a matrix of softer material. An important result in the former study was that cold dwell, under conditions of applied load control, resulted in stress relaxation in the softer material (due to continued creep straining) which, by virtue of force equilibrium, resulted in increasing stress carried in the hard grain. The phenomenon is termed load shedding and the investigation provided considerable insight into the experimentally observed effect of cold dwell on fatigue life in titanium alloys. Dunne et al. (in press) carried out rate-dependent, polycrystal plasticity modelling which included length-scale effects through the incorporation of geometrically necessary dislocations into a physically based slip rate equation. A polycrystalline aggregate containing a rogue grain combination—a combination of a hard grain with c-axis near-parallel to the loading direction with adjacent grains having prismatic slip planes oriented at 70° to the normal to the loading direction (where rogue is used to denote potentially damaging rather than a microstructure which is in any way anomalous)—was analysed when subjected to strain-controlled loading both with and without a strain hold (i.e. a cold dwell). Local to the hard grain, very high stresses normal to the basal plane were developed, which decreased with stress relaxation during the dwell, together with high levels of accumulated slip (which increase with dwell). In reversed loading, at the point in the strain cycle of zero macro-level strain, stresses generated normal to the hard grain basal are higher for the cycle with dwell when compared with that without. The results obtained support the assertion that facet fatigue in Ti alloys requires both cold dwell and a particular combination of grain orientations, termed a rogue grain combination. However, it is important to note that it is not the hold in itself which contributes to faceting, but, in fact, the rate dependence of the material which leads, during a hold period, to either load shedding due to stress relaxation under a load hold or plastic strain accumulation under a strain hold. The absence of the material rate dependence would lead to the removal of the dependence on the hold period. Subsequent work suggests, in fact, that a cold dwell comprising a load hold in a cycle is more damaging than a corresponding strain hold (Rugg et al. in press). In the present work, we confine our attention to cycles containing a strain hold.

We show later that it turns out that there is quite a broad range of combinations of orientations, which result in equally damaging basal stress and accumulated slip at the rogue grain combination, to the extent that if this alone were the cause of facet fatigue, it would be observed experimentally far more frequently than is actually the case, and indeed would not be difficult to reproduce in laboratory test samples. In this paper, therefore, we present the results of a systematic investigation using crystal plasticity modelling, of combinations of crystallographic orientation within a rogue-type grain system embedded in an aggregate of grains. In passing, we make quantitative comparisons with the Stroh (1954) model owing to its use, in a qualitative sense, in explanatory models for faceting currently in the literature. We also address the role of the size of the ‘hard’ grain relative to its neighbours in influencing local stress. Importantly, in addition, we also investigate crystallographic and morphological interactions and conclude by postulating an explanation for fatigue facet formation and its volume dependence in Ti alloys.

2. Rate-dependent crystal plasticity

A full presentation of the length-scale-dependent, elastically anisotropic, physically based and rate-dependent hcp crystal plasticity model is given by Dunne et al. (in press), but a summary of the important aspects necessary for the present paper is given here. The alloy considered in the present study is the near-alpha titanium alloy Ti–6Al, which was considered also by Hasija et al. (2003) and has been chosen for a number of reasons: firstly, it is near-alpha and therefore also nearly single (alpha) hcp phase; the beta, bcc, phase can therefore be neglected. Furthermore, it has been observed that it is the orientation of the alpha colony (i.e. that region having uniform alpha crystallographic orientation) within which the facet develops that dominates fatigue life (Bache et al. 1998; Bache 2003). It is comparatively well characterized in terms of its hcp anisotropic elasticity properties (Hasija et al. 2003), and owing to its aluminium content (greater than 5%), does not show significant deformation by twinning (Williams et al. 2002; Xiao 2005). It is reasonable to assume, therefore, that planar glide is the dominant deformation mechanism. This, however, is not the case in a range of hcp metals and particularly in titanium alloys containing smaller amounts of Al. There is some evidence that in these alloys, slip occurs predominantly on the basal (Williams et al. 2002) and prismatic (Suri et al. 1997; Neeraj et al. 2000) planes, since the resolved shear stress necessary to drive pyramidal slip can be approximately three times higher than that for the basal planes, for example. We therefore simplify our model by not allowing slip on pyramidal slip planes.

The plastic velocity gradient is given byEmbedded Image(2.1)which was derived by Dunne et al. (in press) on the basis of planar slip occurring through release and pinning of gliding, statistically stored dislocations with density ρ, with an activation energy ΔF associated with the pinning by sessile, geometrically necessary dislocations with density ρG, over an activation volume b2λp, where Embedded Image. In this equation, b is Burger's vector magnitude; ν is the frequency of dislocation jumps, successful or otherwise; k is the Boltzman constant; T is the temperature; τk is the resolved shear stress on system κ; τc is the critical resolved shear stress; and sκ and nκ are the direction and normal associated with the κth slip system, respectively. Geometrically necessary dislocations on the κth slip system can be represented as those with dislocation line directions parallel to the slip direction, s (screw type), and to the plane normal, n, and m=n×s directions (edge type) such that the density on the κth slip plane isEmbedded Image(2.2)

The evolution of geometrically necessary dislocation density is determined geometrically according to Busso's formulation (Busso et al. 2000) fromEmbedded Image(2.3)where the slip rate, Embedded Image, is given in equation (2.1). A non-local finite element approach is necessary in order to determine the spatial gradients of the plastic deformation gradient, and this has been developed within an Abaqus user-defined element as described by Dunne et al. (in press). Eight-noded plane strain elements are used, and we introduce to each such element a further ‘internal’, four-noded linear element for which the nodes are located at the four Gauss points of the eight-noded element. We therefore have full knowledge of the plastic strains at each of the internal element nodes, and use the element shape functions to determine the plastic strain gradients within the element. This enables us to determine the spatial gradients of the plastic deformation gradient in equation (2.3), and in doing so, introduce the length-scale dependence into the finite element model. The plane strain condition does not, in principle, constrain the anisotropic plastic slip which is, in general, fully three dimensional. We allow slip to occur in any plane irrespective of the plane strain condition, and from knowledge of the slip, determine the local, fully three dimensional plastic strain increment, and then impose the plane strain condition by ensuring that the out-of-plane elastic strain increment is equal and opposite to the corresponding plastic strain increment. The plane strain condition only becomes over-constraining for the particular case of a grain having its c-axis in plane and normal to the loading direction, giving a much stiffer response than would be seen in reality. However, in our polycrystal, we avoid such circumstances.

The anisotropic elasticity constants used in our model are those obtained by Hasija et al. (2003) for their Ti–6Al alloy, and the material properties required in equation (2.1) are given by Dunne et al. (in press). There is freedom to choose only the initial densities of statistically stored and geometrically necessary dislocations (1010 and 230×103 m−2, respectively), the Helmholtz free energy and the critical resolved shear stress. These were chosen to ensure that the computed average polycrystal stress–strain behaviour matched that obtained by Hasija et al. (2003) in their experiments (including the strain rate or time dependence). The complete description of the elastically anisotropic, length-scale- and rate-dependent crystal plasticity formulation, together with the material properties, can be found in Dunne et al. (in press). Here, we now introduce the polycrystal analyses and, more particularly, firstly address the role of local crystallographic orientation on grain-level stress.

3. Systematic study of crystallographic orientation, size and morphology in polycrystal deformation

The model polycrystal considered in the present study is shown in figure 1. Twenty-seven grains are considered, initially with random crystallographic orientations. The boundary conditions are shown in figure 1a, and the polycrystal is subjected to strain-controlled loading in the y-direction up to a level of 2% at which a cold dwell (i.e. a strain hold at 20°C) is introduced, as shown in figure 1b. The resulting macro-level stress yy response is shown in figure 1c, including the stress relaxation which occurs during the strain hold. In subsequent sections, we examine results at the unrelaxed and relaxed points in the cycle, as shown in figure 1c. The particular group of grains we examine in detail is shown in figure 1a in which a path is shown through three grains. By analysing larger models with more grains, we have shown that the results obtained in the region of interest (i.e. within the primary and secondary grains) are not influenced in any significant way by the proximity of the boundaries, as discussed by Dunne et al. (in press). The nature of the boundary conditions imposed (whether symmetric or periodic) is not, therefore, of great importance in this study and we adopt the simpler option of symmetric conditions. Typically, at least in this section, the grain indicated by bold boundaries is allocated a crystallographic orientation such that its c-axis is near-parallel to the loading direction. It becomes, therefore, a badly oriented or hard grain, since in the absence of pyramidal slip, very large resolved shear stresses are required to cause basal or prismatic slip compared with a soft grain, whose c-axis is normal to the load, for example. The crystal c-axis is shown in figure 1d, together with potential basal, prismatic and pyramidal slip systems. The two adjacent grains, included within the path in figure 1a, are typically oriented such that their c-axes are normal to the loading direction; therefore, both are comparatively soft grains in which slip on basal and prismatic systems is relatively easy. The combination of the badly oriented or hard gain, together with the two softer grains, is termed a rogue grain crystallographic orientation configuration or simply rogue grain combination for ease. The precise orientation of one of the softer grains, relative to the adjacent hard grain, is the subject of the systematic study considered in §4. Approximately 2500 quadratic plane strain reduced integration elements are used in the finite element model and all elements falling within a given grain are assigned the corresponding crystallographic orientation of that grain. The finite element mesh is shown in figure 1e.

Figure 1

(a) Model polycrystal showing boundary and loading conditions and the path to be examined later, (b) the strain-controlled loading applied, (c) the macro-level stress response showing the stress relaxation resulting from the strain hold (or cold dwell), (d) the crystal c-axis and potential slip systems and (e) the finite element mesh for the polycrystal shown in (a).

(a) Assessment of rogue grain crystallographic orientation combinations

The rogue grain combination in figure 1a is shown schematically in figure 2 where an indication of the crystallographic orientations is also given. To begin, we examine the distribution of yy stresses and accumulated slip along the path labelled AA in figure 2 such that the hard grain (labelled P for primary) and parts of the two adjacent softer grains (labelled S and T for secondary and tertiary, respectively) are considered. The polygonal grain representations shown in figure 2 need to be differentiated carefully from crystallographic orientation (particularly given the hexagonal crystal structure of Ti–6Al considered here). In figure 2, a line indicating an example set of prismatic slip planes is shown oriented at an angle θ to the x-direction (the normal to the primary grain c-axis). Considering prismatic slip alone, rotational symmetry of the basal plane about the crystal c-axis leads to a period of 60°. This is important in interpreting the subsequent results. In this study, the primary grain is oriented near-parallel to the loading direction, i.e. in particular, its c-axis is in-plane and at 10° to the y-axis, and the secondary grain is rotated, with respect to the reference configuration shown in figure 1d and the coordinate system in figure 2, about the z-axis and fixed. We obtain stress and accumulated slip along AA for a range of angles, θ, made by secondary grain prismatic slip planes with the x-direction shown in figure 2, at both the unrelaxed and relaxed points in the loading cycle. All other grains in the model are assigned crystallographic orientations randomly, but such that they remain soft grains compared with the primary hard grain. The results are shown in figure 3.

Figure 2

Schematic of the three grains of approximate size L, in figure 1, labelled primary (P), secondary (S) and tertiary (T). The primary c-axis is near-parallel to the y-direction (a hard grain); that of the secondary and tertiary grains is parallel to the z-direction (soft grains). A typical set of prismatic slip planes in grain S is shown oriented at an angle θ to the x-direction. The stress, σn, is that normal to the r-direction, where r is measured from the grain boundary as shown.

Figure 3

(a) Unrelaxed and (b) relaxed stress yy, (c) unrelaxed accumulated slip and (d) relaxed slip through the primary, secondary and tertiary grains in figure 2 along the path shown in figure 1 for the orientations of grain S relative to the x-direction shown, and (e) the macro-level stress response for various orientations of grain S (almost indistinguishable), in which the time has been normalized by 0.01 s.

The yy stresses along AA are shown at the unrelaxed and relaxed loading points respectively in figure 3a,b, and similarly for the accumulated slip in figure 3c,d respectively for angles, θ, between a particular prismatic slip plane direction and the x-direction, varying between 30 and 90° as shown. Considering figure 3a, it can be seen that the effect of varying the crystallographic orientation of soft secondary grain S is on the stresses within the grain itself and on those in the primary adjacent grain, but only over a distance a little less than half the grain size. The orientation of secondary grain S seems to have little or no influence on the yy stresses carried by tertiary grain T, as shown in figures 2 and 1a. The yy stress close to the primary/secondary grain boundary varies periodically with orientation of the secondary grain (this is addressed fully in §3b), such that the stresses in the primary grain at the primary/secondary grain boundary vary between approximately 600 and 800 MPa depending on secondary grain orientation. Figure 3a also shows that the highest yy stresses in the primary grain near the grain boundary typically correspond to the lowest stresses in the adjacent secondary grain, but that no effect of orientation is seen in the tertiary grain (whose orientation remains fixed throughout). Figure 3b shows the same spatial distribution of stress but at the relaxed point of the cycle, and it can be seen that accordingly, the local stresses have also relaxed showing reduced magnitudes over the three grains relative to the unrelaxed point shown in figure 3a. The overall pattern and behaviour, however, remain unchanged despite the stress relaxation. Stress decreases of order 100 MPa occur during the cold dwell in the primary grain close to the P–S grain boundary, but this is considerably smaller (approx. 50 MPa) in the bulk of the primary grain. In the secondary grain, however, stress relaxation appears to be more uniform across the grain, resulting from the greater propensity for slip (which accommodates the stress relaxation) in this grain.

The corresponding accumulated slip for the unrelaxed and relaxed points in the cycle is shown in figure 3c,d, respectively. The most striking feature is the comparative absence of accumulated slip in the primary badly oriented grain, and that the stress relaxation is accompanied by an accumulation of plastic slip (typically approximately 10% higher than that from a cycle without a cold dwell). It is also evident that the orientation of the secondary grain does have an effect on the accumulated slip in the tertiary grain, presumably arising from the absence of strain accommodation in the primary grain. The highest levels of accumulated slip in the secondary grain correspond to the orientations leading to the highest stresses in the primary grain. The effect of the secondary grain orientation on the macro-level stress–strain response of the polycrystal is shown in figure 3e; the effect is negligibly small. The same macroscopic behaviour can therefore mask significant local differences at the grain level.

We next investigate the effect of varying the orientation of the primary (hard) grain while keeping the orientations of the adjacent secondary and tertiary grains constant. Referring to figure 2, the c-axis of the primary grain is initially parallel to the y-axis, and we progressively rotate the c-axis about the x-direction between -10 and 90°. The secondary grain is fixed such that its c-axis is parallel to the z-direction and a prismatic plane makes 70° with the x-direction. The tertiary grain orientation remains fixed as stated previously. We again examine the yy stresses and the accumulated slips through the path shown in figure 1a, labelled AA in figure 2. The yy stresses are shown in figure 4a for a range of primary grain c-axis angles shown. We see again that the peak stresses are developed at or near the grain boundaries and that as expected, the orientation of the primary grain corresponding to the highest stresses in the primary grain is that for which the grain c-axis is at 0° (i.e. parallel to the y-direction). Progressively rotating the primary grain c-axis about the x-direction causes the peak stresses in the primary grain to reduce and the fractional reduction at the P–S grain boundary is approximately 15% in rotating the primary grain c-axis through 90° from it being parallel to the load to normal to it. When rotated such that its c-axis is normal to the loading direction (parallel to the z-direction), shown as the result for 90° in figure 4a, the adjacent secondary grain is in a softer orientation (prismatic plane at 70° to the x-direction) so that even when the c-axes of both grains are parallel, a yy stress difference still exists, though the difference in this configuration is minimum.

Figure 4

(a) Stress yy versus normalized position along path shown in figures 1a and 2 for the case of grain S having c-axis parallel to the z-direction and a prismatic slip plane with angle θ=70° to the x-direction, shown in figure 2, for a range of primary grain c-axis angles generated by rotating the grain about the x-direction. (b) Stress yy at the boundary of the primary and secondary grains in figure 2 for the same range of primary grain c-axis angles.

The sensitivity of the yy stress in the primary grain, local to the P–S grain boundary, to the orientation of the primary c-axis orientation is shown in figure 4b. The result is naturally periodic (with period 180°) and it is noted that while the variation of stress is quite large, it does not vary in any particularly sharp manner with angle. The implication is that many such combinations of crystallographic orientations are almost certain to occur in components in service (or, indeed, test samples in a laboratory) and the absence of observation of cold dwell faceting in laboratory strain-controlled tests cannot be assigned to the smaller volumes of material not generating the worst case combinations of crystallographic orientations. The broad range of angle giving rise to higher grain boundary stresses in figure 4b is certain to arise in practice, even in small volumes of material.

We, therefore, conclude from these analyses that the effect of the cold dwell in strain-controlled loading is to lead to stress relaxation which is pretty uniform in the soft grain and which is highest in the primary grain at the grain boundary, but which decreases with the distance from the grain boundary in the primary grain. Stress relaxation is accompanied by further accumulation of plastic slip in the softer grains. The orientation of the secondary soft grain with respect to the primary hard grain does have an effect on the grain boundary stresses and the accumulated slip in the soft grains. Indeed, a ‘bad’ combination of orientations (e.g. with θ=40°) leads to over a factor of two on the accumulated slip in the soft grain compared to that with a different combination (e.g. θ=90°). Correspondingly, the stresses at the P–S grain boundary increase by approximately 15% in going from the ‘good’ combination to the bad combination. The primary grain having its c-axis near-parallel to the load direction generates the worst stresses, which decrease by approximately 16% as the hard grain c-axis is rotated such that it becomes normal to the loading direction. The dependence of grain-level stress and accumulated slip, particularly at the grain boundary, cannot be considered small, though owing to the various crystal symmetries, nor could the likelihood of such bad combinations existing many times indeed in engineering components. It seems clear that cold dwell facet fatigue cannot be explained in terms of combinations of crystallographic orientations alone, though it may well still be a factor. We next briefly compare qualitatively the results obtained with that predicted by the Stroh (1954) model.

(b) Sensitivity of grain-level stresses to combinations of crystallographic orientation: qualitative comparison with Stroh

In passing, we briefly carry out a qualitative comparison of the results obtained from the anisotropic elasticity, rate-dependent crystal plasticity model and those predicted by Stroh (1954) based on an elastic analysis. We do the comparison owing to the appearance and use of the Stroh model in the recent literature in attempting to explain facet fatigue (Bache 2003; Bache & Evans 2003; Sinha et al. 2006). We again refer to figure 2, but this time, we particularly focus on the primary grain. We examine the stresses developed normal to the x-direction at a normalized distance, r/L, from the P–S grain boundary shown in figure 2, where L is approximately the grain size. Stroh (1954) considered a line of dislocations (a slip line) oriented at an angle θ to the normal to the remote applied stress and, with assumptions of elasticity, determined the stress state at any point (r, θ), where r and θ are given with respect to the line termination (in effect, the grain boundary shown in figure 2), in an infinite medium. The normal stress, σn, was found to be given byEmbedded Image(3.1)

The normal stress within the primary grain, determined using the current model, is shown in figure 5a against normalized distance, r/L, for a range of angles, θ, made by a single prismatic slip plane in the adjacent secondary grain shown in figure 2, and figure 5b against the angle, θ, for a range of normalized distances, r/L. A quantitative comparison with the Stroh model is not really feasible for a whole range of reasons: Stroh assumes elasticity resulting in a stress singularity at the grain boundary; only a single slip system was considered, whereas the crystal model has the full set of prismatic and basal slip; and the choice of stress with which to normalize is not obvious (here, we take the maximum yy stress at the P–T grain boundary. The crystal model predictions are given in figure 5a,b and the corresponding Stroh results are given in figure 5c,d. Figure 5a shows that for the case of easy slip in the adjacent secondary grain (θ=70 and 80°), the normal stresses in the primary grain reduce with increasing radial position. This is in qualitative agreement with the Stroh results shown in figure 5c. Qualitative agreement is also seen in that the stress dependence on θ is seen to diminish with increasing radial position (i.e. moving further away from the P–S grain boundary into the bulk of the primary grain). However, for the case of a relatively harder adjacent secondary grain (θ=60 and 65°), the crystal results are quite different from those obtained from Stroh. For the case of the dependence of normal stress with angle, θ, shown in figure 5b,d for the crystal and Stroh models, respectively, again qualitative agreement is seen in the dependence of the magnitude of normal stress on normalized position (but this depends on the angle, θ), and the periodic nature of the stress variation with angle. However, the crystal model has rotational symmetry giving a period of 60° compared to that of 180° for the Stroh model, such that figure 5b,d look rather different but in fact do show qualitative agreement. Figure 5a in particular shows that the critical angle leading to the highest primary grain stress very close to the P–S grain boundary is approximately 70°, though it can be seen that this changes with normalized position. However, close to the grain boundary, this is similar to what was predicted by Stroh (1954), at least for locations close to the primary/secondary grain boundary.

Figure 5

Normal stress σn versus (a) normalized distance (r/L) from grain boundary and (b) orientation θ of prismatic slip plane shown in figure 2 obtained from crystal plasticity simulations and (c) and (d) using Stroh's (1954) analysis, respectively.

Therefore, we conclude this section by suggesting that the Stroh model gives a reasonable qualitative description of the role of slip plane orientation and normalized position in determining the stresses in the primary (hard) grain, but that quantitative predictions under conditions of low cycle fatigue (i.e. plastic deformation) are naturally, given the basis of the model, unlikely to be accurate. The same conclusions stand in the context of the application of the Stroh model in explaining facet fatigue. In (c), we address the effect of the size of the primary grain relative to the adjacent grains, and then address the morphological effects in (d).

(c) Effect of primary grain size relative to the adjacent grains

There has been concern, recently, that while optically, a fine, uniform, equiaxed microstructure could be obtained in commercial Ti alloys, examination of crystallography, for example, by EBSD showed that boundaries observed optically did not necessarily always indicate boundaries of crystallographic orientation and from the perspective of crystallography, grains much larger than thought possible from optical microscopy can sometimes occur (Sinha et al. 2006). This is of particular concern in the context of facet fatigue, since if a facet should happen to develop at a particularly large grain (crystallographically speaking; optically, it may not be observable), the propagation time from such a facet is potentially much shorter than that for a small facet. It is therefore of interest to examine the local stresses and the accumulated slip and their dependence on the size of the primary hard grain.

Figure 6 shows a simplified representation of a polycrystal containing nominally 100 square grains, each assigned an independent crystallographic orientation, and each comprising 10×10 plane strain finite elements. The hard grain, which has been allocated an orientation such that its c-axis is parallel to the loading (y-) direction, is shown by bold boundaries in figure 6. Hard grains of different sizes are considered, i.e. between 10 and 50 μm, shown schematically in the figure. All grains other than the hard (rogue) grain are of size 10 μm and are assigned orientations such that their c-axes are parallel to the z-direction and a random rotation, but constrained to be within ±10°, is then applied such that they all remain as soft grains. The boundary conditions are also shown in the figure and, as before, strain-controlled loading is applied to the top boundary as shown up to a strain of 2%. Finally, the broken lines in the figure, crossing the hard–soft grain boundaries for the respective hard grain sizes, are the paths where results are considered in detail.

Figure 6

Schematic of a polycrystal comprising one hard, rogue, grain, of size between 10 and 50 μm, and all remaining grains of size 10×10 μm with random, but soft orientations. Stresses normal to basals are examined along the dashed paths across the grain boundaries shown for square hard grain (shown bold) sizes of 10, 20, 30, 40 and 50 μm, respectively. A displacement in the y-direction of 2 μm is applied at the top boundary.

The macro-level stress–strain response of the polycrystal, as the size of the hard grain increases, is shown in figure 7a. In the bulk elastic regime, increasing the size (and correspondingly, the area fraction) of the hard gain leads to increasing modulus. Once bulk plasticity has set in, little effect of the hard grain size is seen until it starts to become large (40–50 μm) compared to the overall polycrystal. In other words, the area available within the polycrystal for the development of slip diminishes to the extent that the strain gradient effect starts to lead to strain hardening in the plastically deforming regions and hence the macro-level hardening seen in figure 7a. However, such effects are unlikely to be of relevance in facet fatigue. The distributions of accumulated slip and yy stress along the various paths shown in figure 6 for the different hard grain sizes are shown in figure 7b,c, respectively.

Figure 7

(a) Macro-level stress for different hard grain sizes versus applied strain, (b) stress yy and (c) accumulated slip along paths across the grain boundaries shown in figure 6 for the different hard grain sizes.

The accumulated slip is plotted against normalized position through a path which is always of length 10 μm, half of which lies in the hard grain and another half in the adjacent soft grain. Figure 7a therefore shows limited, if any, slip in the hard grain and varying slip in the adjacent grain depending on hard grain size. The variation, in fact, depends more on the overall nature of the plastic strain field than directly on the hard grain size. For the case of the hard grain being the same size as all the other grains, at a macro-level, the slip distribution appears very uniform. As the hard grain increases in size relative to the others (and indeed the polycrystal as a whole), the macro-level strain field becomes altered to that expected in a J2 analysis of an elastic particle contained in a continuum plastic matrix. However, the stresses in figure 7c show that as the hard particle size increases, so does the level of strain carried by it (predominantly elastic), so that the yy stresses in the hard grain increase accordingly. The highest stresses carried in the hard particle lead to the lowest stresses in the adjacent soft grain and vice versa. It is perhaps of note that for hard grain sizes remaining significantly smaller than the polycrystal aggregate size, increasing the hard grain size leads to higher yy stresses, but that the increase decreases for fixed size increase. In addition, the stress increase resulting from the hard grain size changing from 10 to 40 μm is over 200 MPa, i.e. an increase of approximately 33%.

(d) Morphology–crystallography interactions

We have so far considered only aspects of crystallography in seeking to understand the conditions likely to cause facet development. We finish by addressing grain boundary morphology and, in particular, we examine the case in which close to the worst combination of crystallography exists a rogue grain combination in which the primary grain has c-axis near-parallel (10° off) to the loading (y-) direction (a hard grain), the adjacent secondary grain has c-axis parallel to the z-direction and a prismatic slip plane oriented at 60° to the x-direction, as shown schematically in figure 8a. Given the particular combination of crystallographic orientations, we now examine the role of the grain boundary direction by considering that part of the bi-crystal shown in the dashed rectangle in figure 8a which is shown enlarged in b. The grain boundary is now assigned an angle α with the x-direction and varied to study its effects on the resulting grain-level behaviour. Angles between 30 and 70° are considered and we include the particular angle of 60° for which the grain boundary orientation coincides with the orientation of the secondary grain prismatic slip plane. The boundary conditions shown in figure 8b are necessarily simple, but trials with others (e.g. free left and right boundaries) do not change substantively the results obtained. We consider the detailed results along the path indicated in figure 8b and also the results for the bi-crystal model as a whole in figure 8b.

Figure 8

Morphology–crystallography interaction with (a) schematic showing a primary grain, with c-axis at 10° to the y-direction, and secondary grain with c-axis parallel to the z-direction and a prismatic plane oriented at 60° to the x-direction. The grain boundary is oriented at an angle α to the x-direction. (b) A simplified finite element bicrystal model representing that in (a) with the same crystallographic orientations and grain boundary angle α.

We first address the distribution of the stresses normal to the basal planes across the grain boundary, shown for a range of grain boundary (morphological rather than crystallographic) angles in figure 9a. The basal stresses (i.e. those normal to the grain basal plane) are determined from σ=(σn).n. The soft (secondary) grain basal stresses are low, and independent of grain boundary angle, reflecting easy accommodation of slip, given the favourable crystallographic orientation. The hard primary grain basal stresses vary with the boundary angle, resulting from the greater proportion of strain being carried by the hard grain with increasing angle (the limit being 90° at which point, for compatibility reasons, the whole of the applied strain is carried by the largely elastic hard grain leading to the highest stresses; the same strain is also carried by the soft grain with much lower stress owing to the deformation being plastic). For each boundary angle, we have extracted the peak, field, basal stress in the hard grain, at the grain boundary. For each grain boundary angle, the peak stress occurs at a slightly different location along the boundary, but it is interesting to examine the sensitivity of the basal stress with the boundary angle. The results are summarized in figure 9b showing a rapid increase in peak basal stress occurring for boundary angles greater than 45°. This largely reflects the greater proportion of strain carried by the hard grain, so is not of particular significance. However, examination of the peak accumulated slip occurring at the grain boundary for different boundary angles is perhaps more significant.

Figure 9

(a) Stress normal to basal plane versus normalized position along the path shown in (b) through the bicrystal for the boundary angle shown, (b) the peak stress normal to the basal plane and (c) the peak accumulated slip, local to the grain boundary versus morphological boundary angle.

The results are shown in figure 9c. Again, as the boundary angle changes, the grain boundary location of the peak accumulated slip changes slightly. However, we see that a large peak in slip occurs at a grain boundary angle of 45° which later drops off. A secondary peak in accumulated slip is seen to occur with increasing boundary angle with the maximum value occurring at a boundary angle of 60°. We recall that this coincides with the orientation of the secondary grain prismatic slip plane. Therefore, it appears that there exists a crystallographic–morphological interaction when the slip plane orientation and grain boundary orientation coincide. Furthermore, we note that the interaction occurs over a comparatively small range of boundary angle of approximately ±5°. A potentially more important aspect of the interaction becomes clear only on examination of the field variations of accumulated slip. These are shown for three particular grain boundary angles (30, 55 and 60°) in figure 10ac. For the case of 30° (and indeed all angles up to approximately 55°), the hard grain largely deforms elastically and a generally uniform slip distribution exists in the soft grain. At 55°, however, shown in figure 10b, falling within the secondary peak region in figure 9c, the slip distribution becomes much more localized and, in particular, a small zone of localized slip penetrates into the harder grain. The magnitude and localization of the slip peak at precisely the angle at which the orientations of the grain boundary and prismatic slip plane coincide, as shown in figure 10c. In addition, the range of grain boundary angle over which the slip localization and penetration into the hard grain occur is rather narrow, namely within 10°. Outside this range, no such effects or interaction are observed.

Figure 10

Field plots of accumulated plastic slip for the bicrystal with crystallographic orientations shown in figure 8 for a grain boundary angle of (a) 30°, (b) 55° and (c) 60°.

4. Hypothesis for facet formation and volume dependence

The results obtained enable us to hypothesize on the mechanism of facet formation and the crystallographic and morphological configurations required for it. We have systematically studied the combinations of crystallographic orientations leading to the worst case localized stresses and plastic slip. We concluded that the worst cases of stress and slip existed over quite broad ranges of angles of orientation and that crystallography alone cannot explain the volume effect observed experimentally in facet fatigue. We find that the effects of morphology can also be important, but that their effects alone are not sufficient to explain the formation of facets nor the volume effect. However, we see that a crystallographic–morphological interaction occurs over a small range of grain boundary angle and we therefore hypothesize that facet formation requires:

  1. the existence of a rogue grain combination, i.e. a primary hard grain having c-axis near-parallel to the loading direction, with adjacent secondary grains having c-axes near normal to the loading direction with a prismatic plane oriented at approximately 70° to the normal to the loading,

  2. the boundary between the primary grain and a secondary grain having a morphological orientation coincident with a crystallographically active slip plane in a secondary grain, and

  3. a loading cycle which leads to load shedding (under load control) or stress relaxation leading to increased accumulated slip (under strain control), though the former has been shown to be worse (Rugg et al. in press), resulting from the rate dependence of the material. A cold dwell (strain or stress controlled) is likely to enable this.

We also argue that it is the need for the morphological–crystallographic interaction in cold dwell which also provides an explanation for the volume effect. In other words, because the crystallographic combination is required together with a very specific morphology, which exists over a rather narrow range in order to ensure the interaction, it starts to become reasonable to argue that such combinations occur sufficiently rarely that they become difficult to observe in laboratory samples and manifest themselves more frequently in large structural components. To conclude, we hypothesize on the mechanism of facet formation on the basis of our computational studies, reinforced by independent experimental data (Bieler et al. 2005a,b; Sinha et al. 2006).

We have shown that particular crystallographic combinations lead to stresses normal to the primary grain basal plane significantly in excess of those in the absence of that particular crystallography, that under strain-controlled loading, cold dwell can lead to significantly enhanced levels of accumulated slip at the primary–secondary grain boundary, and that finally, the morphology–crystallography interaction can lead to a very particular localization of the accumulated slip in the secondary grain combined with a very significant penetration of plastic slip into the adjacent primary hard grain. We therefore hypothesize that it is the localized penetration of slip into the hard grain which, under conditions of cyclic loading, results in the generation of a persistent slip band and sub-grain-level microcrack generation. Under the action of the high basal stresses, normal to the grain c-axis, which exist in the circumstances outlined, the nucleated defect propagates through the grain, parallel to the basal plane, until it hits the boundaries of the adjacent softer grains. Here, the crack blunts owing to the considerable slip and its ease in the adjacent grains and does not easily propagate further under conditions of low cycle fatigue, owing to the vastly lower normal stresses which exist in the softer grains. However, the size of the hard grain is important because, in effect, it determines the size of the initial facet. More conventional crack propagation analysis then enables the determination of subsequent lifetime.

5. Experimental verification of model prediction

We note that in the experimental hot deformation studies carried out by Bieler et al. (2005a,b) on Ti–6Al–4V, a clear correlation was found to exist between the locations of cavity nucleation and growth, and those where the rogue grain combinations of crystallographic orientations occurred. Bieler et al. also found that cavitation occurred at locations where adjacent colonies had c-axes perpendicular to one another, irrespective of whether one of the colonies had its c-axis parallel to the loading direction. The latter study is not the one we have carried out in this investigation, but the former, while for hot deformation rather than fatigue, provides considerable experimental support for the conclusions drawn from our calculations.

More particularly, Sinha et al. (2006) investigated experimentally fatigue facet formation in a near-alpha titanium alloy undergoing cyclic loading with a cold dwell under load control. They determined the locations of facet formation and the orientations of the grain in which the facet formed and the surrounding grains, and also investigated the propensity for slip along basal and prismatic systems in the grains. By means of serial sectioning, orientations and propensity for slip were determined in three dimensions. A number of notable experimental observations were made.

  1. The inclusion of a dwell at room temperature reduced the fatigue cycles to failure by a factor of approximately 54. Therefore, there is no doubt experimentally that cold dwell is significant.

  2. The facet formed in a grain with basal plane near-normal to the loading direction and adjacent grains had c-axes about normal to the faceted grain. This is precisely the worst combination of orientations predicted by our model.

  3. Prismatic slip was found to be feasible in the adjacent grains but not in the faceted grain. This is also just as predicted in our model in order to generate the ‘rogue’ grain orientation combination.

  4. The facet was found to have near-basal orientation. Our crystal plasticity predictions tell us that this is the direction normal to the highest direct stresses and is expected from our model.

  5. The qualitative statement that the misorientation between the basal planes in adjacent grains (one of which is the faceted grain) of approximately 85° leads to stress build-up at the boundary which can assist crack initiation is made. This is in agreement with the experimental observations of Bieler et al. (2005a,b).

The three-dimensional data presented by Sinha et al. (2006) enable us to determine, in addition to crystallographic orientation data and in a rather approximate way, the morphological grain boundary angle between the grain that facets and the adjacent soft grain. We therefore investigate whether it is close, or otherwise, to the experimentally determined orientation of the active prismatic slip system in the adjacent soft grain to look for the possibility of a crystallographic–morphological interaction. We examine grain boundary angles in planes parallel to the loading direction, as in the crystal plasticity modelling. We have traced the grain boundary location using the ‘slices’ provided by Sinha et al. (2006) and have calculated a grain boundary angle with knowledge of its in- and out-of-plane location. We find that over a length which is a significant fraction of the faceted grain size, the grain boundary angle is approximately 23° to the basal facet, whereas the crystallographic orientation of the adjacent prismatic slip plane makes an angle of 30° to the basal facet. This is, therefore, within 8° of the critical interaction angle discussed in §3d, though there are insufficient data to draw any significant conclusions.

Sinha et al. (2006) identified a further potential rogue grain combination. Its c-axis was found to be near-parallel to the loading direction, but the adjacent grains had c-axes oriented typically at approximately 60° to this. We have shown previously that this particular configuration leads to considerably lower loading direction stresses at the boundary between the hard and soft grains (as shown in figure 3a), and we note that in the experiments, faceting was not seen at this location.

6. Conclusions

Polycrystal plasticity calculations have been carried out to study, in a systematic manner, how the combinations of crystallographic orientations in neighbouring grains influence local stress and accumulated slip distributions. In particular, we have shown that the ‘worst’ combination is that in which a hard grain with c-axis near-parallel to the loading direction is adjacent to a softer grain with c-axis near-normal to the load and a prismatic slip plane at approximately 70° to the normal to the load. The configuration is worst in the sense that the stresses normal to the grain basal, and the accumulated slip in the adjacent grain, are highest. We term such a combination a rogue grain combination. In passing, we show that the Stroh (1954) model provides qualitative descriptions of some of the behaviour observed, but that it is not sufficient to explain facet formation.

We have shown that a morphological–crystallographic interaction occurs under rather extreme circumstances. A rogue grain combination is required which, in addition, needs to have a grain boundary between hard and soft grains with (morphological) orientation which coincides (within ±5°) with that of an active slip system in the soft grain. The interaction leads to very localized accumulated slip at the grain boundary within the soft grain, which peaks when the orientations coincide exactly, and a penetration of slip into the hard grain, which does not exist outside the ±5° range given previously.

The experimental observations of Sinha et al. (2006) of fatigue facet formation in a near-alpha titanium alloy support the significant conclusions of the crystal plasticity investigations. In particular, faceting was found to occur under just those ‘rogue grain’ combinations predicted to lead to the worst case stresses. There is, unfortunately, very little experimental information as yet to verify the requirement for a crystallographic–morphological interaction in facet formation. Nor, however, is there evidence to contradict the requirement.

Acknowledgments

F.P.E.D. would like to express gratitude to the Royal Society for funding, and Steve Williams, Rolls-Royce plc, for his support.

Footnotes

  • Currently on leave at Rolls-Royce plc, PO Box 31, Derby DE24 8BJ, UK.

    • Received August 18, 2006.
    • Accepted February 15, 2007.

References

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