## Abstract

In polycrystalline materials, grain boundaries provide an important contribution to the resistance to the propagation of both brittle and ductile cracks. Initially, in this paper, a three-dimensional geometrical model of the brittle fracture of polycrystalline zinc is developed, assuming a single (0001) cleavage plane in each grain. The model predicts that about one-half of the fracture process of the material will be associated with accommodation effects at grain boundaries. In contrast, experimental work over a range of temperatures shows that at low temperatures very little grain boundary failure occurs. There are two reasons for this discrepancy. Firstly, cleavage occurs on (0001) and also on the three variants of the {10-10} planes and secondly, deformation twinning plays a major role in the fracture process. The characteristics of these phenomena have been investigated in detail using focused ion beam microscopy, including subsurface examinations and metallographic techniques. The models were then extended to incorporate these additional mechanisms. Comparisons between the predictions and the experimental observations are discussed and enable information to be deduced about the relative energies of the different fracture mechanisms.

## 1. Introduction

In polycrystalline materials, propagation of a cleavage crack from grain to grain is a complex process (Smith *et al*. 1997, 2002; Argon & Qiao 2002). This is because cleavage occurs on well-defined planes and these planes in neighbouring grains do not usually meet in a line in their common grain boundary. Some additional failure mechanism must therefore be involved. In particular, the two cleavage cracks may be linked by brittle grain boundary fracture, by multiple cleavage in one or both of the grains or by localized ductile failure at the boundary. Traditionally, experimental research has suggested that these accommodation effects play only a minor role in the brittle fracture of polycrystalline materials. However, in recent years, theoretical models and computer simulations have indicated that they may be very significant (Crocker *et al*. 2005). For example, in ferritic steels, the models predict that approximately 30% of brittle failure is by means of these accommodation mechanisms whereas experiments have indicated only a few per cent (Crocker *et al*. 1996). One possible explanation for this discrepancy is that most of the failed grain boundaries will be oriented at a steep angle to the overall fracture surface so that microscopic examination tends to underestimate their areas. Specific experiments involving tilting the fracture surface and examining individual facets have shown that this is an important effect but can only increase the observed accommodation to 20% at the most (Crocker *et al*. 2005). Thus, it appears that alternative accommodation mechanisms make a contribution to the propagation of transgranular cleavage cracks in polycrystalline materials.

The amount of accommodation failure required at each grain boundary is defined by the angle between the traces in the boundary of active cleavage planes in the adjacent grains. This angle depends crucially on the number of cleavage planes available. In ferritic steels, which have the body-centred cubic (bcc) structure, there are three {001} cleavage planes. However, in zinc, which has the hexagonal close packed (hcp) crystal structure with *c*/*a*=1.856, cleavage has been shown to occur, at least predominantly, on the unique basal (0001) plane (Barrett & Massalski 1980). The authors therefore anticipated that zinc would need far more accommodation fracture than is the case for ferritic materials. Hence, in order to explore the amount of accommodation further, it was decided to model brittle fracture in zinc and to carry out detailed experiments to allow a comprehensive description of the fracture to be obtained.

In the past, many theoretical studies of fracture in polycrystalline materials have been based on two-dimensional models, with the grains represented by simple polygons (Crocker *et al*. 1996, 2005). Such models can provide valuable insights into some aspects of fracture but they are incapable of depicting the accommodation fracture required to link together cleavage facets in adjacent grains. For this, three-dimensional models are essential. Such models may consist of a single grain, a neighbouring pair of grains, many grains of restricted shape and orientation, or many randomly shaped and oriented grains. In addition, these grains can have either planar or curved faces and finally, predictions can be obtained using either mathematical analysis or computer simulation.

In the present study, the strategy adopted is to employ several different types of model, each being tailored to investigate a particular aspect of the fracture process. For example, when considering microstructural contributions to fracture within a grain, a model of a single three-dimensional grain is employed. To investigate the propagation of a cleavage crack across a single grain boundary, a model consisting of just two grains is used. To obtain the proportion of different accommodation mechanisms that occur at grain boundaries, a topological model consisting of an array of parallel prisms is used. Such models are employed many times, with grains and interfaces of different orientations relative to the stress axis. The results are then averaged and predictions obtained for the fracture characteristics of polycrystals containing many grains. Thus, by employing simplified targeted models and combining their outcomes with the probability of occurrence of each configuration, it has been possible to obtain quantitative results for the different fracture processes that operate. The approach adopted is consistent with the requirements to describe crack propagation within a large cross-section polycrystal. It has been unnecessary to use the general three-dimensional models that are currently being developed (Smith *et al*. 2004, 2006). Although these models have produced some very encouraging results, they are at present very challenging both to create and to employ.

In this paper, a geometrical model is presented which predicts the amount of accommodation, grain-boundary brittle-failure or an equivalent mechanism that is required to fracture polycrystalline zinc. This model assumes initially that cleavage occurs only on the basal plane. It is more rigorous than previous models used to investigate brittle fracture in ferritic steels (Smith *et al*. 2002; Crocker *et al*. 2005). A summary is then provided of earlier, in some ways inconclusive, experimental work on the fracture of both single crystals and polycrystals of zinc. Specific experiments designed to investigate the fracture of polycrystalline zinc in detail, over the temperature range 77–423 K are then described. For this work, high-resolution focused ion beam (FIB) techniques were used. The results of these experiments are presented. They demonstrate that brittle fracture of polycrystalline zinc is a far more complex process than anticipated. In particular, it is revealed that a significant amount of non-basal cleavage occurs and deformation twins also play a major role. As a consequence, the geometrical models have been extended to incorporate these factors, and the predictions of the resulting more general models are presented. Finally, the theoretical and experimental results are discussed; the implications for brittle fracture in other polycrystalline materials are considered and conclusions are drawn.

## 2. Geometrical model of brittle fracture in polycrystalline zinc

Initially in the model, it is assumed that zinc cleaves on the unique basal plane of its hcp structure. The lines in which cleavage planes in adjacent grains of the polycrystal may meet their common grain boundary can therefore be at any angle between 0° and 90°, or on average at 45°. The four distinct ways in which this can occur are shown schematically in figure 1. In mechanisms I and II, a cleavage crack in the first grain nucleates a crack in the second grain at the point where it meets the grain boundary. This may be an interior point (I) or an edge point (II) of the common boundary. To separate the two sides of these fracture surfaces, the double-triangle shaded area (I) or the single-triangle area (II) must then fail. In mechanisms III and IV, cleavage cracks are nucleated independently on either side of a grain boundary. The two cracks may then not intersect so that the quadrilateral shaded area (III) has to fail. If they do intersect (IV), the geometry is effectively the same as in I.

It is now possible to estimate the fraction of a grain boundary that, on average, must fail for each of these mechanisms (Smith *et al*. 1997). This average will clearly depend on many factors including the shape of the grain boundary, the location of the point of intersection in I and IV, and the distance of the traces from grain edges in II and III. For example, assuming a circular grain boundary, as in figure 1, and traces 45° apart, mechanisms I and IV give 25% grain boundary failure for a central intersection and 47.5% for an edge intersection, with a weighted average of 32.5%. Similarly, mechanism II gives a minimum of 2.5% and a maximum of 47.5% with an average of 25%. Finally, mechanism III ranges from 9.1 to 100% with a weighted average of approximately 70%.

The next step is to deduce what proportions of mechanisms I–IV actually occur in practice as brittle fracture spreads across the material. On average, when a cleavage crack crosses a grain, it meets six grain-boundaries. Therefore, the propagation of brittle fracture through a polycrystalline material may be represented topologically using a regular array of hexagonal prisms as shown in figure 2. The proportions of the four mechanisms could depend critically on the location of the nucleation site of the first cleavage crack and three special cases will be considered: the centre of a grain, the centre of a grain face and the centre of a grain edge. In figure 3, which gives an axial view of such a model, these are represented by the centre of a hexagon, the centre of an edge of a hexagon and a corner of a hexagon, respectively. In practice, it is expected that the overall average result will be an appropriate mixture of these three cases. A tensile stress is applied parallel to the axes of the hexagonal prisms. It is therefore assumed that in each case when the crack front is projected on to the plane of the diagram, it takes the form of an arc of a circle. As the model is topological, this is considered to be a reasonable assumption.

Figure 3 illustrates in detail the case when the crack nucleates at the centre of a grain, labelled A. As it grows, it meets all six surrounding grains at the midpoints of their vertical faces and therefore propagates by means of mechanism I. Thus, the interfaces between grain A and grains B and C (represented by AB and AC) are of type I. The independent cleavage cracks in B and C then meet boundary BC by means of mechanism III or IV. At this stage, these two mechanisms cannot be separated and so the combination will be represented by VII. The fronts of the cracks in B and C reach the edge between grains B, C and D (BCD) at the same time but at two different points. One of these, selected to be that in B, propagates into D and this results in BD being II and CD VII. The next step is that the cracks moving across B and C reach E and F, respectively, and propagate using mechanism I. As fracture propagates, the following rules define the outcomes of merging and diverging boundaries:The results of several more events are shown in the figure, in which the broken lines represent mirror planes of the scheme that, incidentally, does not have sixfold symmetry.

It is clear from figure 3 that as the fracture process extends, the proportion of mechanism I decreases whereas II and VII increase. This trend has been examined in detail and the results are summarized in table 1. These predictions show that when the crack becomes large, the proportions of these three cases approach zero, one-third and two-thirds, respectively.

The corresponding exercise for the case when the initial cleavage crack is nucleated at the centre of a grain face has also been examined in detail and gives remarkably similar results. After 241 grain boundaries have failed, the proportions are 4, 36 and 60% for mechanisms I, II and VII, respectively. However, a rather different situation arises when the initial crack nucleates at the centre of a grain edge. In this case, no examples of mechanism I arise and the fracture process soon involves equal numbers of II and VII, so the result is 0, 50 and 50%. Therefore, if it is assumed that the average of these three cases represents the general situation, the proportions of mechanisms I, II and VII become 3, 39 and 58%. These figures are of course for large fracture surfaces crossing many grains and do not represent the proportions for cracks crossing only a few grains.

In order to divide mechanism VII into its component parts, III and IV, consider figure 4. This is based on two straight lines crossing at an angle *α*, the average angle at which the traces of cleavage planes in adjacent grains intersect their common grain boundary. In the present case, this will be 45° but different values will be needed for materials with a different number of cleavage planes, e.g. 22.5° for ferritic steels with three {001} cleavage planes. The grain boundary is assumed to be a circle of unit radius. If the centre of this circle lies within unit distance of the two lines, both of them will cross the grain boundary. This occurs when the centre of the circle, i.e. the grain boundary, lies within the large rhombus of the diagram. In addition, if the centre of the circle lies within unit distance of the crossing point of the lines, the two traces will intersect within the circle. The corresponding region is the circle shown shaded in the figure. The ratio of the area of the circle to the area of the rhombus is (*π* sin *α*)/4 and this gives the proportion of mechanism IV. When *α*=45° (representing zinc), this is found to be 0.56 leaving 0.44 for mechanism III. These fractions are of course of the 58% of all four mechanisms that have been deduced on average to be of type VII. The final result is therefore approximately 3, 39, 26 and 32% for mechanisms I, II, III and IV, respectively.

It was deduced above that, on average, approximately 32.5, 25, 70 and 32.5% of grain boundary area needs to fail for mechanisms I to IV. These figures can now be combined with the proportions of these mechanisms that arise to give an overall figure of 39% for the average amount of failure. To estimate the proportion of intergranular failure as opposed to cleavage failure, it is now necessary to deduce the relative average area of a grain boundary and of a cleavage plane. Assume that a sphere of unit volume may be used as an approximation to a grain. On average, grains in polycrystals have 14 neighbours and therefore 14 faces. Hence, the average area of a face is found to be 0.35 and the average area of a cleavage plane 0.81, giving a ratio of 0.43. Although, on average, each cleavage plane will meet six grain boundaries, each of these is shared between two adjacent grains so that there are three partially failed grain boundaries for each cleavage plane. Hence, subject to the approximations and constraints of this geometrical model, it is concluded that the percentage of intergranular failure is 0.43×3×39%, i.e. 50%. This is a high value and therefore the assumptions require validation through experiment.

The analysis presented above is entirely geometric. In particular, the results obtained do not depend on the energies of cleavage and grain boundary failures. However, these energies do become crucial in the extended theoretical model of fracture in polycrystalline zinc presented in §6 of this paper.

## 3. Experimental background

Much experimental work on the fracture of single crystal specimens of zinc was carried out during the mid-twentieth century (Deruyttere & Greenough 1953, 1956; Bell & Cahn 1958; Bullen 1963). It was established that cleavage occurs on the (0001) plane unless this is oriented unfavourably with respect to the tensile stress axis. If this is the case, deformation twins form on one or more of the six variants of the {10-12} family of planes (Christian & Mahajan 1995). Within each of these twins, one of the three prismatic {10-10} planes will make an angle of only a few degrees (approx. 4°) with the basal plane of the parent. Therefore, if the basal plane in the parent is well oriented for cleavage, so is a {10-10} plane of the twin (Deruyttere & Greenough 1956). The possibility of cleavage fracture on the three {10-10} prismatic planes was also suggested by the observation of stepped cleavage. This again led to the idea that cleavage in the parent crystal may propagate through twins by means of this alternative mechanism (Bell & Cahn 1958). It should also be noted that the formation of twins prior to fracture of zinc at low temperature is well established, being indicated by load drops on load–displacement curves (Curry *et al*. 1978).

In addition to these observations on single crystals, some work on the fracture of polycrystalline zinc has been carried out (Greenwood & Quarrell 1954; Curry *et al*. 1978). In particular, the effect of stress-state on fracture over the temperature range 77–256 K using uniaxial and triaxial tensile and notch bend specimens has been examined (Curry *et al*. 1978). At low temperatures, the fracture was transgranular cleavage with some evidence of twinning. It was proposed that fracture was primarily (0001) cleavage, but some steps on cleavage planes were also observed suggesting that some {10-10} prismatic cleavage had occurred. In addition, it was proposed that, possibly, cracks had also formed on (0001) planes in the twins. However, no conclusive evidence could be provided for either of these mechanisms. The key points from this work were that the shear stress required to produce cleavage is identical for both uniaxial and triaxial stress states and that the critical value of the shear stress increases with temperature. No detailed fractography was carried out in the experiments, leaving the question of the orientation of cleavage facets unanswered.

## 4. Experimental procedures

In the present experimental work, a 25×25×6 mm coupon of zinc (99.9% pure) was heat treated at 443 K for 2 h and cut into sheets of 0.4 mm thickness. The sheets were deformed and punched to produce 3 mm diameter discs using a Unidisc MKII (Agar Scientific) unit. These deformed discs were then annealed at 523 K for 1 h, producing a grain size of 167 μm (mean linear intercept). Each disc was then precision ground to the final dimension of 0.25 mm using a sequence of 320, 600 and 1200 grit silicon carbide papers. Each specimen was measured to ensure that the finished thickness was known to an accuracy of ±0.005 mm. The miniaturized jig used to test these specimens has been described elsewhere (Vorlicek *et al*. 1995) and was used in conjunction with a universal Hounsfield H10KM machine, equipped with an environmental chamber. These tests were undertaken in the temperature range 77–423 K at a constant cross head speed of 0.003 mm s^{−1}. Temperature was controlled by a chromel/alumel thermocouple positioned in the environmental chamber and a second thermocouple was positioned to measure the temperature of the jig. During testing, load–displacement data were captured at every 4 μm displacement. A correction for the stiffness of the machine/test assembly was applied to each load–displacement curve. In addition, a second 25×25×6 mm thick zinc coupon was sliced into ‘matchstick’ specimens having dimensions of 20×5×5 mm. These were then heat treated for 2 h at 453 K to give a grain size of 600 μm (mean linear intercept). The matchsticks were notched and then fractured at temperatures in the range 77–328 K. The specimens were examined to ensure that a pronounced texture of the type observed by Solas *et al*. (2001), for example, was not present. Examination using a combination of polarized light microscopy, etch pitting and electron backscattered diffraction confirmed little or no texture existed. Moreover, any weak texture would be such that for the present work, the grain orientations could be assumed to be random. In any case, the model can easily be adapted to incorporate effects of crystallographic texture.

Detailed fractography was carried out using an FEI Strata FIB201 FIB workstation. The accelerating voltage used in all cases was 30 kV, the current of the gallium ion beam being varied according to the imaging requirements: low beam currents (less than 70 pA) were used for high-resolution imaging and lower specimen damage. Details of the milling and imaging capabilities of this technique when applied to fractographic examination of bcc and hcp polycrystalline specimens have been described elsewhere (Hughes *et al*. 2005). Additional information was obtained by etch pitting the fracture facets using a solution of CuCl_{2}.2H_{2}O : HCl : H_{2}O in the ratio 7 : 3 : 90 (Barrett & Massalski 1980). This etches {10-1*l*} planes, where *l* is a small positive or negative integer. It therefore produces irregular hexagonal-shaped etch pits on (0001) surfaces.

## 5. Experimental results

### (a) Small punch-test results

Figure 5*a*,*b* are load–displacement curves obtained from the small punch specimens at 77 and 423 K. These temperatures correspond to the brittle region and the upper transition, or even fully ductile, region of the ductile to brittle transition curve (Agnor & Shank 1950). At the lower temperature, there is limited displacement arising from ductility prior to fast brittle fracture. At the higher temperature, the curve has the characteristic shape described previously for ductile materials (Baik *et al*. 1986; Mao *et al*. 1987; Campitelli *et al*. 2004). For the lower temperature, after the elastic deformation, there is evidence for load instabilities consistent with the occurrence of twinning events (Christian & Mahajan 1995). This indicates that twins form prior to the onset of brittle fracture.

### (b) Temperature dependence of the fracture mechanism

‘Matchstick’ specimens of zinc were fractured at 77, 293 and 328 K covering the brittle to the upper transition region. Fractographic examination was carried out using FIB-induced electron images and four distinct fracture mechanisms were recognized: transgranular, grain boundary, ductile and twin boundary. The relative percentage areas of these mechanisms at each temperature were estimated from the projected images, and the results are given in table 2. These were obtained using a grid consisting of 10×10 points placed over the image and the dominant fracture mechanism at each point of the grid was noted. Ion-induced secondary electron images for the three temperatures are shown in figure 6.

At 77 K, the dominant failure mechanism is transgranular cleavage (82%) together with a significant proportion of twin boundary fracture (14%) and a small amount of intergranular fracture (4%). Figure 6*a* shows that, at this temperature, the principal cleavage plane has three sets of straight steps at approximately 60° to each other. This suggests that multiple cleavage has occurred on the expected (0001) basal plane and also on the three variants of one of the prismatic planes, either {10-10} or {11-20}. In §5*c*, this observation is examined in detail. Clearly, the stepped transgranular cleavage has the effect of limiting the amount of grain boundary fracture that is required as cracks propagate from one grain to the next. It is noteworthy that fracture surfaces from grain to grain are typical of those associated with randomly oriented grains.

As the temperature is raised, the amount of twinning decreases as slip increases. Associated with this is the removal of twin boundary fracture. In addition, cleavage cracks in either basal or prismatic planes tend to propagate through individual grains without deviation, resulting in large individual facets. Furthermore, there is an increase to 28 and 29% of brittle intergranular fracture at 293 and 328 K, to accommodate cleavage crack propagation across grain boundaries. Moreover, as the temperature is increased from 77 to 328 K, there is a progressive increase in the intervention of ductile fracture, i.e. the transition from brittle to ductile fracture is encountered.

### (c) Detailed study of prismatic cleavage and twin boundary failure

Direct observation of the crystallographic orientations of features on fracture surfaces using standard diffraction pattern techniques is not easy because the surface facets are tilted. Fortunately, however, it is possible to establish conclusively the active multiple cleavage planes in zinc specimens fractured at 77 K, by making a detailed study of specific areas of the fracture surfaces. This study has used a combination of trace analyses of specific features, FIB cross-sectioning at identified locations, and etch pitting of the fracture surfaces. For example, figure 7*a* shows the ion-induced secondary electron image of one region of a fracture surface, which has been deliberately tilted by 20°, about a horizontal axis, from being parallel to the plane of the image. This was done in order to reveal the lightly coloured horizontal cleavage plane near the top of the figure. The fracture surface is also shown schematically in figure 7*b*, where the main features are labelled. In addition, figure 7*c* gives a standard (0001) hcp stereographic projection for zinc (*c*/*a*=1.856) with the major poles of planes under consideration indicated.

As shown in figure 7*b*, two lenticular {10-12} twins, T_{1} and T_{2}, intersect the main cleavage plane and the traces make an angle of approximately 60° with each other. To satisfy this condition, the cleavage plane must be either (0001) or one of three planes with very large indices, the latter seeming most unlikely. In addition, FIB sections indicate that the twins intersect the cleavage plane at an angle of approximately 47°, the angle at which {10-12} planes meet (0001). It is concluded therefore that the predominant cleavage plane is the (0001) basal plane and that the traces lie along 〈1-210〉 directions. This is of course the anticipated result. The minor cleavage planes in figure 7*b* also make traces in the main cleavage plane at angles of approximately 60° with each other, as indicated for example by _{1} in figure 7*b*. In addition, the traces of C_{1} and C_{2} are parallel to the traces of T_{1} and T_{2} and, therefore, also lie along 〈1-210〉. Finally, plane C_{1} makes an angle of approximately 90° with the main cleavage plane _{2} in figure 7*b*. Only {10-10} planes satisfy these conditions and this result has again been confirmed by FIB milled cross-sections. The secondary cleavage planes must therefore be of {10-10} type. In particular, the alternative {11-20} prismatic planes, the poles of which are labelled P_{1}, P_{2} and P_{3} in figure 7*c*, are not consistent with the observed cleavage planes because their traces make angles of 30° with the twin traces.

The fact that the main cleavage plane is (0001) has also been confirmed by an analysis of pits produced by chemical etching these facets. The etchant noted in §4 was used and generated hexagonal pits as shown in figure 8. When the pits are isolated, they have a fairly regular hexagonal shape. However, when they interact more complex shapes can arise but still the basic hexagonal form is clear. The three-dimensional form of the pits can also be complex as the etchant selects several different {10-1*l*} planes, where *l* is a small integer. However, examination of figure 8 suggests that most of the sidewalls are of {10-11} type.

Although alternative electron backscattered diffraction techniques offer the potential to orient fracture facets, the geometrical constraints of the systems combined with the unknown inclination of each facet to the incident electron beam means that this would not be a trivial exercise. Indeed, it can only be achieved by the use of detailed two surface analysis techniques if the solutions are to be unambiguous (Slavik *et al*. 1993).

An example of the accommodation that occurs when a cleavage crack meets a grain boundary is illustrated in figure 9. The crack in grain A is essentially planar, although near the grain boundary, it cuts through some twins and is deviated by approximately 5°. Then, as it crosses the boundary into grain B, it initially follows an intergranular path, but soon a complex system of finely stepped cleavage forms. This acts to minimize the amount of grain boundary fracture needed to accommodate the mismatch between the adjacent grains. As the cleavage propagates further into grain B, the steps tend to merge and produce much larger cleavage facets on basal and prismatic planes.

### (d) Interaction between cleavage cracks and twins

The contrast effects of FIB microscopy allow the identification of changes in crystallography at the surfaces of cleavage facets. This has been used to study the possible interactions between cleavage cracks and twins. In figure 10*a*, twins intersect the major (0001) cleavage facet. In most cases, the cleavage crack has propagated across the twins without significant deviation. However, at X, the crack meets the twin and propagates down the twin boundary. A similar event has occurred in the adjacent grain at Y.

Figure 10*b* shows an image of a FIB cross-section of a heavily twinned region. The surface of the facet shows a deal of complexity, with cleavage cutting across twins. In addition, a complex stepped region has formed at the surface, with further twinned regions at the surface towards the right of the large central twin. Looking at the subsurface information from the face of the cross-section, a complex array of twins has formed below the surface within the twinned region. The array of twins makes a contribution to the stepping observed on the surface. A similar effect is seen in figure 10*c* where a relationship can be seen between subsurface twinning and surface steps.

## 6. Extended theoretical model of brittle fracture in polycrystalline zinc

### (a) Introduction

As a result of the above experimental observations of basal plus prismatic cleavage and the formation of twins, it is now necessary to extend the simple model for describing brittle fracture of polycrystalline zinc presented in §2. The first stage is to introduce the alternative potential cleavage planes into the model. Cleavage will then be allowed to occur on a single cleavage plane, either basal or prismatic, in each grain. The second stage is to allow cleavage to occur on two or more planes in some or all of the grains. Finally, the possible interactions of a cleavage crack with a twin boundary are considered.

### (b) Additional cleavage planes

It has been shown that brittle fracture in polycrystalline zinc occurs on the three variants of the {10-10} family of prismatic planes in addition to the unique (0001) basal plane. Therefore, on average, the smallest angle between the traces of cleavage planes in adjacent grains in their common grain boundary will be 11.25° rather than 45°. However, one of these planes may not be oriented favourably relative to the stress axis so that the second best angle of 33.75° will be selected. If it is assumed that the occurrence of these two cases is inversely proportional to these angles, i.e. 3 : 1 (Smith *et al*. 1997), the average angle between the traces of the two cracks will be approximately 17°. This angle can now be used, as in §2, to estimate the percentage of grain boundary that, on average, must fail for mechanisms I–IV of figure 1. The results are approximately 12.2, 9.4, 70 and 12.2%, respectively.

These figures now have to be combined with the proportions of mechanisms I–IV that occur in practice. For I and II, these are approximately 3 and 39%, as in §2. However, the remaining 58% for mechanism VII divides differently into its component parts, III and IV, because the angle *α* between the two traces in figure 4 is now approximately 17° and not 45°. Hence, one obtains 45% for III and 13% for IV. Combining the percentages of areas and the percentages of mechanisms now gives an overall average figure of 47%. This is the average percentage area of those grain boundaries that fail partially, if cleavage cracks are to be linked together, when no other accommodation mechanism is available. As in §2, the proportion of intergranular fracture to cleavage fracture is now given by 0.43×3×47%, i.e. approximately 61% rather than 50%, when only basal cleavage occurs. This result may seem surprising as one might expect less grain boundary failure when more cleavage planes are available. It arises owing to the higher proportion of mechanism III (45% rather than 32%) and this has a much larger amount of grain boundary failure (70%) than mechanism IV (12.2%).

The above model is however based entirely on geometry and does not depend in any way on the energies, *E*_{b} and *E*_{p}, of basal and prismatic cleavage, respectively. In practice, it is clear that prismatic cleavage is more difficult than basal cleavage and, therefore, although a {10-10} plane may be oriented favourably for fracture propagation, owing to its higher energy, a less favourable (0001) plane may be selected. Indeed, if the ratio *R*=*E*_{p}/*E*_{b} of these energies, is large, very little prismatic cleavage will occur and the prediction will be much closer to 50% than to 61%. This ratio is not known and therefore, initially, the cases of *R*=2, 4 and 8 will be considered here. Assume that the applied stress ** σ** is uniaxial and makes an angle

*θ*with the normal to the (0001) basal plane within a zinc grain. The normal stress on this plane is, therefore,

*σ*cos

^{2}

*θ*. As the prismatic planes are perpendicular to the basal plane, the maximum possible normal stress on these planes is

*σ*sin

^{2}

*θ*, which occurs when

**lies in one of the {11-20} planes. The equation**

*σ**Rσ*cos

^{2}

*θ*=

*σ*

^{2}sin

*θ*or simply tan

^{2}

*θ*=

*R*then gives the boundary in the {11-20} plane between basal and prismatic cleavage being favoured. Similarly, the minimum possible normal stress on the prismatic planes occurs when

**lies in one of the {10-10} planes and**

*σ**σ*sin

^{2}

*θ*cos(

*π*/6) resulting in tan

^{2}

*θ*=2

*R*/√3. These two limiting values of

*θ*enable the proportions of the different cleavage planes that are favoured to be deduced, and the results are presented in table 3.

The ratio of *E*_{p}/*E*_{b}=2 is close to that suggested by the present experimental observations. In addition, Miller *et al*. (1969) have calculated that the free energy of the {10-10} planes is 1.87 times greater than that of the basal (0001) plane. It seems reasonable therefore to consider the case of *R*=2 in greater detail. In particular, it is important to estimate the average fraction of grain boundary accommodation failure for this energy ratio. Using 50% for *E*_{p}/*E*_{b}=∞ and 61% for *E*_{p}/*E*_{b}=1, the percentages in table 3 suggest that the appropriate figure is approximately 59%. In addition, for *R*=2, the probability of basal cleavage being followed by basal cleavage is then 0.58^{2}=0.3243 and, similarly, prismatic by prismatic is 0.1853, basal by prismatic is 0.2452 and prismatic by basal is 0.2452.

### (c) Multiple cleavage

The analysis in §6*b* is based entirely on a single cleavage plane, either basal or prismatic, operating in each grain. In practice, it is clear from the experimental work presented in §5 that a large number of cleavage facets are stepped indicating that multiple cleavage occurs in many grains. If three independent cleavage planes are available, an appropriate combination of them can generate an average fracture surface of any crystallographic orientation. It is now known that this is the situation in zinc. The three prismatic planes all contain the unique [0001] axis and therefore provide two independent systems and basal cleavage provides the third. In principle, multiple cleavage could occur in zinc on any plane and, in particular, on the plane in each grain that is perpendicular to the stress axis. In this case, no grain boundary failure would be required. However, in general, these planes would not be atomically smooth and have much higher energies than the low index individual cleavage planes. In practice, a compromise has to be reached between these conflicting considerations.

The details of this compromise will of course depend on the relative orientations of the two grains and their common grain boundary. However, figure 9 provides an excellent example where single cleavage fracture has occurred on the basal plane in the grain at the right and meets the boundary with the grain at the left that is badly oriented for single cleavage. Therefore, localized fine-scale multiple cleavage occurs in this second grain to minimize the proportion of grain boundary failure. However, as the crack propagates, the higher energy of the resulting rough fracture surface is soon reduced by facets merging to generate much larger steps. In addition, near the grain boundary, there are three facet orientations but one of these disappears as fracture progresses. In addition, one particular facet orientation soon becomes predominant. The projected angles of approximately 120°, produced on these major facets by the intersections of pairs of minor ones, indicate that the failure is dominated by basal cleavage.

In order to make theoretical predictions about the rate at which multiple cleavage degenerates to double cleavage, many assumptions about the experimental conditions, the crystallographic relationship of the two grains and the energies of the different interfaces would have to be made. Unfortunately, at present, available information of this kind is far from complete and therefore no attempt has been made to make any such predictions. However, in principle, a detailed study of specimens such as that shown in figure 9 could provide valuable information on the relative energies of the different fracture mechanisms involved.

### (d) Interactions of cleavage cracks with twin boundaries

To investigate the way in which basal and prismatic cleavage can propagate either through a twin or cause failure along a twin boundary, a model consisting of a spherical grain was adopted. The crystallographic orientation of this grain was selected randomly and, as shown in figure 11, the stress axis was fixed in the vertical direction. A single twin boundary was introduced passing through the centre of the grain to give two subgrains S1 and S2. Subgrain S1 retained the original orientation of the grain and S2 was the mirror image of this in the twin boundary. This boundary is along one of the six variants of the {10-12} family of planes, which in zinc make an angle of 46.98° with the basal plane.

Fracture was initiated at point I, in subgrain S1, on the equator of the spherical grain (figure 11). As in §6*a*, the fracture energies *E*_{b} and *E*_{p} of the basal and prismatic cleavage planes were put equal to 1 and 2, respectively, and the experimental results suggested that it would be appropriate to use 4 for the twin boundary energy, *E*_{t}. In a particular simulation, the cleavage plane with the most favoured combination of orientation and energy was selected. This may be the basal plane (B) or one of the three prismatic planes (P1, P2, P3). The crack on the selected plane will intersect either the grain boundary or the twin boundary. In the latter case, the best option for propagation was selected from the four cleavage planes in S2, and the twin boundary, again using their orientations and fracture energies as criteria. Thus, if one of the four possible cracks in S1 intersects the twin boundary, there are five possible propagation mechanisms and therefore a total of 20 combinations. Four of these involve twin boundary fracture and the remaining 16 cleavage in S2. The traces of the cleavage planes in the twin boundary are either 〈11-20〉 or 〈22-43〉. The angle between the pair of traces is 0° in six cases and therefore no accommodation at the twin boundary is required. These are the four pairs of crystallographically equivalent cleavage planes linked by reflection across the boundary, and the basal and prismatic planes that meet the boundary from either side in the same close-packed 〈11-20〉 direction. Using the above abbreviations, these are the B–B, P1–P1, P2–P2, P3–P3, B–P*n* and P*n*–B combinations, where *n*=1, 2 or 3. For the remaining 10 cases, the possible acute angles between the traces are 68.50° for B–P*n* and 43.00° for P*n*–P*m*, where *m* and *n*=1, 2 or 3 and *m*≠*n*. These are the angles between 〈11-20〉 and 〈22-43〉 and between two variants of 〈22-43〉, respectively.

Simulations were carried out for 50 different random orientations of subgrain S1 relative to the stress axis. In addition, for each of these orientations, all six variants of the {10-12} twin boundary were considered. The results are summarized in table 4.

Cleavage in S1 occurred on the basal plane in 80% of these 300 simulations and on one of the three prismatic planes in the remaining 20%. In 6% of the cases, the cleavage plane met the grain boundary and not the twin boundary and these cases were all of basal type. In a further 6% of cases, fracture did not propagate into subgrain S2 but continued along the twin boundary. Again, these were all cases in which the cleavage in subgrain 1 was basal, but in principle, it could have been prismatic. In the remaining 88% of cases, the cleavage fracture propagated across the twin boundary and this divided into 72% prismatic and 16% basal. It can also be divided into 68% direct propagation without accommodating fracture of the twin boundary and 20% with accommodation.

In practice, of course deformation twins have two approximately parallel interfaces and, in the model, the cleavage cracks that penetrate the twin will reach the second boundary. They will then have nine choices, the same as in table 4 except that B–G and P–G are no longer available. This means that approximately 94% of them will propagate by cleavage of which 75% will be basal and 19% prismatic. The remaining 6% will fail along the second twin boundary. The total amount of twin boundary failure is therefore predicted to be approximately 12%, close to the experimental value of 14% reported in table 2 for experiments carried out at 77 K. This suggests that at that temperature the value of 4 adopted for the ratio of the energies of twin boundary and basal failure is reasonable. At higher temperatures, as indicated in table 2, no twin boundary fracture was observed.

## 7. Discussion

The present paper provides an account of an integrated theoretical and experimental study of the propagation of brittle cracks in polycrystalline zinc. It was prompted by the fact that in the past, reports on the brittle fracture of hcp materials, such as zinc, have been dominated by cleavage on its unique basal plane, in contrast to the three cube planes adopted by ferritic materials. In many grains of polycrystalline zinc, the single cleavage plane will then be badly oriented relative to the stress axis. In addition, when an active cleavage plane meets a grain boundary, it will often make a large angle with the trace of the single cleavage plane in the adjacent grain. Hence, propagation may only proceed if a large proportion of the grain boundary fails or if some other equivalent accommodation mechanism occurs. As a consequence, it would be anticipated that the propagation of brittle cracks across grain boundaries in polycrystalline zinc would be much more difficult than in ferritic materials, such as steels. Indeed, under these conditions, the boundaries in zinc would provide significant resistance to cleavage crack propagation.

To theoretically investigate the propagation of cracks across grain boundaries, it is essential to use three-dimensional models. In practice, most theoretical research on the fracture of materials has been two-dimensional and only in recent years have successful three-dimensional models been developed (Smith *et al*. 2006). However, these are demanding to use and in the present project, as far as possible, results have been obtained either by using a topologically equivalent, pseudo-three-dimensional model, consisting of a regular array of hexagonal prisms, or by considering the interactions of cleavage cracks with the boundaries of a single grain. In both cases, the grains have a random crystallographic orientation, although a preferred orientation could be imposed to accommodate any contribution from texture. The present results were obtained for polycrystalline zinc with no evidence of pronounced texture. Hence, appropriate averages are then taken over a large number of separate predictions. This has proved to be a convenient and very effective method. In particular, the model developed here is more rigorous than those adopted in earlier work on steels and capable of extension to many other applications. The results indicated that for brittle fracture of polycrystalline zinc, if only one cleavage plane was active in each grain and no other accommodation mechanism operative, about one-half of the failure would be intergranular.

This prediction was found to be different from the result obtained from the present experiments. In particular, at low temperatures, very little of the failure was found to be intergranular. There were two apparent reasons for this. Firstly, the cleavage cracks were stepped in an intricate way indicating that several other cleavage planes were operating in addition to the basal plane. Secondly, many deformation twins were formed before fracture providing cleavage planes with different orientations to the parent material and also brittle fracture paths along the twin/matrix boundaries. Detailed investigations of these features were therefore carried out using FIB microscopy, including subsurface examinations, together with chemical etch pitting. It has been demonstrated that a substantial amount of cleavage occurs in polycrystalline zinc on the {10-10} prismatic planes. Several variants of this mode can be active in addition to basal cleavage, which means that the overall resultant fracture plane can approximate to the plane perpendicular to the stress axis. The steps tend to be small close to the grain boundaries and this tends to minimize the amount of grain boundary failure that is needed. However, as one moves away from the boundaries, the steps tend to merge and therefore become larger. These phenomena have been noted previously (Curry *et al*. 1978; Argon & Qiao 2002; Hughes *et al*. 2006), but no detailed crystallographic analysis was carried out for hcp materials.

The role of twinning in the brittle fracture process can, potentially, be dramatic. There are six variants of the twinning mode and each twin may cleave on a basal plane and on three prismatic planes, giving a total of 24 possible cleavage possibilities per grain. In practice, of course not all of these will be active but clearly the presence of twins has to be taken into account. Twins have been shown to form prior to the initiation of a cleavage crack, or in the process zone ahead of a growing crack tip. The experimental work has demonstrated conclusively that cracks may follow twin boundaries. At 77 K, where twinning occurs readily, 14% of the failure was found to be of this type. As the temperature increases from 77 K, there is as expected less twinning and twin boundary failure. There is also less cleavage but of a more planar character and an increasing amount of ductile failure, and more intergranular fracture, reaching 29% at 328 K. Of crucial importance in all the experimental work is the ability to investigate specimens in three dimensions using FIB microscopy coupled with local milling.

As a result of these unexpected experimental results, it was necessary to extend the theoretical models. In particular, the fact that cleavage occurs on the prismatic planes as well as the basal planes of grains in polycrystalline zinc had to be introduced. This had the advantage that, in principle, the modelling procedures could also be readily applied to other materials with different crystal structures and hence different numbers of possible cleavage planes. However, it introduced the problem of not knowing the relative energies of the basal and prismatic cleavage mechanisms. Different possible ratios were therefore used and the predictions compared with the corresponding experimental results. This suggested that prismatic cleavage is approximately twice as difficult as basal, consistent with the energies derived by Miller *et al*. (1969). An attempt has also been made to consider multiple cleavage and the size of the resulting cleavage facets on different fracture surfaces, but more detailed experimental work is required before an adequate understanding can be reached.

The theoretical models have also been developed to accommodate the influence of twin boundaries on the propagation of cracks. Eleven distinct possibilities arise. These link basal or prismatic cleavage in the parent grain to grain-boundary or twin-boundary failure, or to basal or prismatic cleavage in the twin, with or without accommodation. The case in which the twin boundary failure energy is twice the prismatic cleavage energy and four times the basal cleavage energy was investigated in detail. A striking feature of the results is that although roughly 80% of the cleavage is basal in the parent grain, owing to geometrical constraints, most of it propagates as prismatic cleavage in the twin. In addition, approximately 6% of the cracks are diverted along the first twin interface and a further 6% along the second. The total being near the present experimental observations on zinc fractured at 77 K.

Overall, the present approach, which incorporates several three-dimensional models of aspects of the brittle fracture of polycrystalline zinc provides the understanding and description of the crack propagation process. Indeed, it addresses the resistance to the propagation of both the grain boundaries and the twin boundaries. As a consequence, these details can be input into a large three-dimensional model similar to that described by Crocker *et al*. (2005). Such a model then allows the overall propagation and resulting interactions to be taken into account, together with probabilistic considerations of fracture in the brittle fracture regime for polycrystalline zinc.

## 8. Conclusions

The availability of three-dimensional models, albeit sometimes in a simplified form, greatly enhances our ability to simulate fracture of polycrystalline materials.

The most effective use of such geometric models is when they are developed and employed in conjunction with related experimental work.

Experimental techniques that are now available enable much more detailed crystallographic information to be obtained than when earlier work on zinc was carried out in the mid-twentieth century. In particular, FIB microscopy enables a wealth of new observations to be made.

The experimental results on polycrystalline zinc presented here demonstrate that several phenomena, which have only been suggested tentatively in earlier work, are important mechanisms. These include prismatic cleavage, intricately stepped cleavage planes, accommodation of fracture at grain boundaries and the role of twin boundaries in the propagation of cracks.

The new experimental results on randomly oriented polycrystalline zinc made it necessary to extend the models to incorporate prismatic cleavage, multiple cleavage and the role of twin boundaries, and this has provided additional insights into the fracture processes.

The research presented is considered to provide a quantitative basis for the description of brittle crack propagation relevant to polycrystalline materials.

## Acknowledgments

The authors are indebted to the EPSRC for funding this conjoint work. P.E.J.F would like to thank British Nuclear Group for allowing time to work at the University of Bristol. Finally, we would like to thank Prof. J. F. Knott FRS FREng for his invaluable comments on this paper.

## Footnotes

- Received November 28, 2006.
- Accepted May 11, 2007.

- © 2007 The Royal Society