## Abstract

This article presents a basic probabilistic theory for the nucleation of deformation twins in hexagonal close packed (HCP) metals. Twin nucleation is assumed to rely on the dissociation of grain boundary defects (GBDs) under stress into the required number of twinning partials to create a twin nucleus. The number of successful conversion events is considered to follow a stochastic Poisson process where the rate is assumed to increase with local stress. From this concept, the probability distribution for the critical stress to form a twin nucleus is derived wherein the parameters of the distribution are related to properties of the GBDs. The theory is implemented into a multi-scale constitutive model for HCP metals in order to test its predictive capability against measurements made previously on pure zirconium deformed at 76 and 300 K.

## 1. Introduction

### (a) Twinning as a probabilistic event

In hexagonal close packed (HCP) metals, deformation twinning is a highly variable event; the propensity, frequency, density and morphology of twins can vary among grains of the same orientation and under the same loading conditions. These characteristics appear to be highly sensitive to both microstructure (e.g. grain size, orientation, dislocation density, alloy content) and applied conditions (e.g. strain rate, temperature, stress level). While it can be easily appreciated that twinning is statistical in nature, there is little understanding of the physics behind twin nucleation and what governs the variability in twinning.

Most reports from metallographic analyses of twinned microstructures are qualitative. Only a few metallographic analyses of deformation twinning have taken the necessary statistical approach for interpretation. A statistical analysis of twinning was carried out with respect to grain size by Hull (1961) in silicon iron, orientation factor by Reed-Hill (1964) in pure Zr, twin thickness by Priestner & Leslie (1965) in silicon iron, and twin dimensions by Remy (1978) in cobalt–nickel alloys. These analyses, however, only quantified the dispersion in these structural variables and not the relationships between twinning and the local microstructure. Particularly for HCP metals, in which deformation twinning is as important as plastic slip, knowing the role that microstructure plays in twin nucleation is needed for building a physically based predictive constitutive model.

Two recent statistical studies of twinning in pure Zr and Mg (Beyerlein *et al.* 2010; Capolungo *et al.* 2009*c*) correlated deformation twinning with certain microstructural features, thereby revealing the essential factors that a model should capture. The main results can be summarized as follows. The twin variant selected is most probably (but not always) the one with the highest macroscopic Schmid factor and most (but not all) grains oriented for twinning do twin. The number of twins per grain increases with grain size. Twin thickness increased with its Schmid factor but was otherwise independent of grain size. Although these two-dimensional analyses still need to be confirmed with three-dimensional ones, it can be concluded that variability in twinning has largely to do with the processes of twin nucleation and less to do with those determining twin growth.

The observed variation in twinning cannot be predicted unless the material model contains a probabilistic treatment of twin nucleation. This article is devoted to presenting a new probabilistic theory for twin nucleation in HCP polycrystalline metals. As will be explained, the basic concepts are consistent with up-to-date experimental statistical analyses via electron backscattered diffraction (EBSD) (Beyerlein *et al.* 2010; Capolungo *et al.* 2009*c*) and numerical observations from atomistic simulations (Wang *et al.* 2010). In spite of recent progress in experimentation and numerical simulation, a complete understanding of twin nucleation in HCP metals has yet to be achieved. With this in mind, we aim to develop a theoretical framework that is physically sound and sufficiently flexible such that new discoveries regarding twin nucleation can be implemented in a straightforward manner into the formulation.

### (b) Theoretical background

There are seven recognized and observed twin modes in HCP metals, distinguished by twin planes and directions with independent crystallography (Partridge 1967; Yoo & Lee 1991). Owing to the geometry of the HCP crystal, each twin mode has six crystallographically equivalent variants. Twinning is sense-dependent and can only be activated if the resolved shear stress (RSS) on the twin plane is along the twinning direction. The most common HCP twin mode observed experimentally is the mode provided that the loading conditions are favourable. When activated, twinning reorients the *c*-axis of the lattice significantly by approximately 85^{°}, where the exact value depends on the *c*/*a* ratio of the HCP lattice.

Existing meso- and macroscale deformation models that include twinning do not include a physically based mechanism for twin nucleation. Consequently, in constitutive models for HCP metals, much empiricism is invoked to model when a twin forms and in which grain it forms, the twin variant selected, the number of twins per grain, the twin morphology and the relationships between twinning and strain rate, temperature and grain size. Most constitutive models resort to a deterministic critical resolved shear stress (CRSS)-based law for describing nucleation and subsequent propagation, e.g. Tomé *et al.* (2001), Salem *et al.* (2005), Kaschner *et al.* (2006) and Proust *et al.* (2007). Regardless of the length scale to which it is characterized or applied, the critical drawback of the twin CRSS concept that is often overlooked is that it usually amounts to a deterministic treatment of twin nucleation. If a twin CRSS exists, it is doubtful that it is constant. To the authors’ knowledge, no physically based statistical model for a critical twin nucleation stress has been developed.

Twin nucleation is usually conceptualized or numerically modelled as a unit process at the scale of an individual crystal dislocation and below. Apart from a few homogeneous nucleation models (Orowan 1954; Bell & Cahn 1957; Yoo & Lee 1991), most are heterogeneous nucleation models involving dissociations of perfect 〈*a*〉 and 〈*c*+*a*〉 slip dislocations into glissile twinning dislocations (Thompson & Millard 1952; Chyung & Wei 1967; Mendelson 1969, 1970; Yoo & Lee 1991; Capolungo & Beyerlein 2008). Dislocation pile-ups can further drive such dissociations, propagate the twin fault and aid twin formation by generating stress concentrations (SCs) (Mendelson 1969, 1970) and suitably directed stress fields (Capolungo & Beyerlein 2008). Alternative proposals involve the repeated nucleation of twinning dislocations from a grain boundary (Fourie *et al.* 1960; Price 1961) or a free surface (Li & Ma 2009) and in these viewpoints, the role of dislocation slip and/or dissociations of dislocations in these processes is not clear. Common to all the above models is the implicit assumption that twinning can begin with the formation of a single layer twin fault. Recent atomic scale simulations of Mg (Wang *et al.* 2009*a*) contradicted this notion and showed that a monolayer twin is energetically unstable. Wang *et al.* (2009*b*) identified two ways in which a multi-layered zonal nucleus of a twin can develop, both involving *simultaneous* nucleation and glide of partial dislocations. In fact, the more energetically favourable pathway created a twin nucleus composed of 17 crystallographic planes, containing eight partial dislocations whose vector sum equalled the Burgers vector corresponding to a twin. The valuable insight gained from these atomistic studies has yet to be related to heterogeneity in twinning observed at the grain and polycrystal scale.

The processes of twin nucleus formation in HCP metals proposed by Wang *et al.* (2009*a*,*b*) appear to have both similarities and differences with those in face-centred cubic (FCC) metals. The similarities are that both originate from pre-existing defects and the minimum stable structure is multi-layered. However, the minimum number of layers for an FCC twin is three (Mahajan & Chin 1973; Kibey *et al.* 2007) and for an HCP twin it can be three or greater. The HCP nucleus must be created by the simultaneous glide of multiple partials and a series of shuffles (Wang *et al.* 2009*a*), whereas these complex kinetics have not been posed as necessary for an FCC twin. Unlike the FCC nucleus, the partials included in the multi-layer zonal HCP nucleus are not necessarily equal. These partials can originate from reactions of defects, which may or may not correspond to lattice dislocations of the HCP crystal. In contrast, the formation of an FCC twin starts with a 1/2 〈1 1 0〉 lattice dislocation through various dissociation reactions (Venables 1961; Teutonico 1963; Hirth 1964; Cao *et al.* 2009).

### (c) Twin nucleation from grain boundaries

The complexity of the formation process and the size of a stable twin nucleus in HCP metals strongly support the idea that twins most probably nucleate at grain boundaries because grain boundaries (i) are sources for such large partial dislocations and multiple twinning dislocations, (ii) can support the complex rearrangement and restructuring necessary to form the multi-layered stable twin nucleus, and (iii) favour high SCs that can supply the energy or stress needed to overcome reaction barriers. EBSD evidence shows that twins in Zr and Mg most often start and end at grain boundaries (McCabe *et al.* 2006, 2009; Beyerlein *et al.* 2010; Capolungo *et al.* 2009*c*).

Using atomistic simulations, the grain boundaries in Mg and their role in twin nucleation were studied by Wang *et al.* (2010). As the parameter space of actual grain boundaries is prohibitively large, the authors focused their investigation on symmetrical tilt grain boundaries (STGBs) of varying tilt angles *θ*. A close examination of their atomic structure revealed that for a wide range of *θ*, the STGBs contain arrays of interfacial dislocations, which we will refer to as grain boundary defects (GBDs) (figure 1*a*). Their Burgers vectors and spacing vary with *θ*. Upon introducing a local SC in the form of a pile up (figure 1*a*) to a low *θ* STGB, these GBDs dissociated into a small number of twinning partials (figure 1*b*) and then coalesced into a single twin nucleus (figure 1*c*).

The kinetics in twin nucleus formation are not expected to fundamentally change for actual grain boundaries or HCP metals other than Mg. Actual grain boundaries contain randomly spaced and sized GBDs whose distribution and properties are a consequence of the formation of the grain boundary itself and the various reactions between the boundary and incoming dislocations or other defects. (GBD ‘size’ refers to the grain boundary surface area that is distorted by the GBD.) Under stress, these GBDs can either dissociate or react with other defects to produce one or more twinning partials. A minimum number of such partials must glide simultaneously on adjacent twin planes in order to form a stable twin nucleus. Achieving this critical number within a given portion of the grain boundary depends on the sizes of the GBDs involved in the reactions and their spacing, and the local stress. Thus, there exists a minimum stress that needs to be supplied to form a stable nucleus within a given portion of the grain boundary containing a given configuration of GBDs. The key difference between a grain boundary with a random distribution of GBDs and the STGB shown in figure 1 with a regular array of equi-sized and equi-spaced GBDs would be that the threshold stress required to form a twin nucleus from an arbitrary grain boundary would vary statistically within the grain boundary.

In summary, two random components, one material and the other mechanical, characteristic of actual grain boundaries can be identified as being responsible for the observed variance in the appearance, spacing and morphology of deformation twins in HCP metals. The random material component is the critical stress or energy required to convert GBDs into twin-related partials. The random mechanical component is the distribution of the local stresses in space and magnitude, which is correlated to the random occurrences of local pile-ups, slip bands and other defects terminating along the boundary. Twin nucleation, therefore, relies on the chance that the suitable GBD (or GBDs) encounters a sufficiently high stress. The outcome would be a distribution of twin nuclei in the grain boundary, widely dispersed in space and size, and strongly dependent on local conditions. These statistical concepts will serve as a basis for the twin nucleation model developed here.

### (d) Scope of article

This paper is structured as follows. In §2, the basic formulation of the theory is developed, starting with the underlying physics and assumptions. In §3, ways in which we recommend implementing the theory into large-scale computational polycrystalline models are described. In §4, we implement the theory into a multi-scale constitutive model. In §5, we consider as an application the compression response of clock-rolled pure polycrystalline Zr at 76 K, where twinning is relevant. Predictions of the stress–strain response, texture and relationships between twinning and microstructure are compared against the experimental results reported in Beyerlein & Tomé (2008) and Capolungo *et al.* (2009*c*). We show that capturing local fluctuations is critical for correctly characterizing the macroscopic response of the aggregate, such as the evolution with strain of flow stress, hardening rate, twin volume fraction and texture.

## 2. Probabilistic twin nucleation theory

### (a) Premise and theoretical assumptions

Before describing the model, the assumptions employed in the present work should be listed. (i) Twins are assumed to nucleate from grain boundaries. (ii) Under a local stress state, GBDs in the grain boundaries transform into partials needed for creating a twin nucleus. (iii) The conversion and coalescence process is instantaneous compared with the time scale of the applied deformation. In other words, formation of a stable twin nucleus occurs, without delay, at the instant the appropriate stress is generated in the vicinity of the defect. (iv) The conversion processes taking place within non-overlapping surface regions of the grain boundary are independent of each other.

### (b) Poisson process for transformation of GBDs into twinning partials

As discussed in §1*c*, in actual grain boundaries, GBDs will be distributed in size and space. Under a given stress, only some of these GBDs will transform into twin partials and potentially contribute to the formation of a stable twin nucleus. The number of such transformation events (*N*) will increase with grain boundary surface area (*a*), as well as with the local stress (*τ*). As *N* is subject to the statistical distribution of GBDs in *a*, we can consider *N* to be a random variable following a stochastic counting process {*N*(*a*),*a*≥0}, where *N*(*a*) is the number of transformation events that have occurred in a space of *a*. We can further assume that {*N*(*a*),*a*≥0} is a Poisson process, provided that (i) *N*(0)=0, (ii) it possesses independent increments (the numbers of events in non-overlapping spaces are independent), and (iii) *N*(*a*) follows a Poisson distribution with mean *λa*. Moreover, when the rate of the process *λ*(*τ*) is made proportional to stress *τ* (a scalar quantity), the expected value *E*[*N*(*a*)]=*λ*(*τ*)*a* increases with both *a* and *τ*. It is well known that applying the Poisson approximations to a binomial distribution *B*(*n*,*p*) with *n* trials and probability of success *p*=*λ*/*n* yields a Poisson distribution in the limit of large *n* (and small *p*=*λ*/*n* for fixed *λ*). The latter relationship provides a physical motivation for use of the Poisson process when *n* is identified with all available GBDs in *a*, and *p* is identified with the probability of successfully transforming a GBD into a twinning partial.

According to the Poisson distribution, the probability that *N*=*m* defects are converted into twinning partials in a given grain boundary of area *a* is
2.1
We identify the Poisson rate *λ* as the expected number per unit grain boundary surface area of GBDs that transform into twin-related partials. Because this process is driven by a local stress *τ*,*λ* is assumed to increase monotonically with *τ* according to the following power law:
2.2
where *α* is a material constant, and *τ*_{0} is a characteristic stress (scalar) that corresponds to a grain boundary surface area *a*_{0}. From the Poisson process model, the expected number of occurrences in a grain boundary of area *a*_{0} is *λ*(*τ*)*a*_{0}, which equals one when *τ*=*τ*_{0}. Accordingly, *τ*_{0} can be interpreted as the stress (or energy) to dissociate on average one GBD in *a*_{0}, and hence *τ*_{0} and *a*_{0} are properties of the GBDs in the grain boundary.

### (c) Critical stress for twin nucleation in characteristic area *a*_{0}

In order to link the discrete probability for the number *N* of conversion events in equation (2.1) (from GBD-to-twin partial) to a continuous probability for twin nucleus formation, how a nucleus would form from *N* such events must be considered. The multi-layered structure predicted by atomistic simulations (Wang *et al.* 2009*a*) suggests that the assembly and coalescence of twinning partials into a stable twin nucleus will be complex (e.g. figure 1*b*,*c*). If so, there exist numerous sequences of activation and coalescence events that can successfully produce a twin nucleus. Accounting for these kinetic processes should lead to a prediction of the size and spatial distribution of twin nuclei produced in a grain boundary surface as a function of local stress state and defect characteristics. The outcome will find that some nuclei may be too small to propagate a twin under the current stress conditions, while the size of others will be more than adequate. For the present work, a relatively simplistic pathway towards twin nucleus formation is employed. It will be implicit that the size of the twin nuclei formed will be sufficient, such that under the appropriate stress conditions the twin can expand and propagate across the grain. The conditions for propagation and the rate at which twins thicken will be treated separately and described later.

Randomly occurring, random strength twin-related GBDs in grain boundary implies randomly varying critical stresses *S* to form twin nuclei. Let *P*(*S*<*τ*) be the probability that the critical stress *S* to form one twin nucleus within a characteristic grain boundary surface area *a*_{c} is less than or equal to *τ*.^{1} Suppose that at least *m** events must occur in *a*_{c} to form a single nucleus. According to equation (2.1), the probability that at least *m** events occur within *a*_{c} is
2.3
All *N*≥*m** defects converted to twin partials within *a*_{c} will coalesce to form one twin nucleus. A conservative approximation for *P*(*S*<*τ*) is obtained by assuming *m**=1. In this case, *P*(*S*<*τ*) is the probability that at least one defect is activated within *a*_{c}, which from equations (2.1) to (2.3) is
2.4
where *τ*_{c}=*τ*_{0}(*a*_{0}/*a*_{c})^{1/α}. We will retain the assumption *m**=1 hereinafter.

Area *a*_{c} represents an important length scale of the model. We have defined it such that all *N*(*a*_{c}) conversion events occurring in *a*_{c} lead to one twin nucleus or alternatively, all twin nuclei formed within *a*_{c} are indistinguishable. Thus, *a*_{c} should be defined large enough such that two adjacent *a*_{c} regions will produce distinct twin nuclei for the length scale of interest. In comparisons with experimental measurement, for instance, it could be identified with the centre-to-centre spacing of newly formed twins.

The mean of the distribution^{2} in equation (2.4) is *τ*_{c}*Γ*(1/*α*+1), where *Γ* is the Gamma function. The variance, which increases as *α* decreases, is . Here, it becomes clear that the exponent *α* in the rate *λ* (equation (2.2)) governs the variation in critical stress *S*.

The total boundary surface area of a spherical grain is *a*_{g}=*πd*^{g2}, where *d*^{g} is the grain diameter. Let *a** be the relevant portion of *a*^{g} from which twins nucleate, i.e. . The number *n** of characteristic areas of size *a*_{c} included in *a** is of course
2.5

As by definition only one distinct twin nucleus can be produced from *a*_{c}, *n** is also the maximum number of nuclei provided from *a**. Suppose for a given *a**, the defect distribution is relatively homogeneous such that these *n** regions are independent and identically distributed (i.i.d.). Associated with the *n** characteristic regions in *a** is a spectrum of random critical stresses, *S*_{i}, *i*=1,…,*n**, required to form a twin nucleus, each following the same distribution in equation (2.4). For simplicity, we consider the minimum and maximum of this set, i.e. and . The random variable is the critical stress to nucleate at least one twin from *a** and the random variable is the critical stress to nucleate all *n** twins from *a**. In other words, is the critical stress associated with the most favourable and the least favourable characteristic region *a*_{c} within *a** for twinning.

The probability distribution for is equal to the probability that at least one of the *n** regions forms a twin nucleus under *τ*. This probability is the complement of the probability (1−*P*(*S*<*τ*))^{n*} that no twin nuclei are formed in any of the *n** characteristic areas:
2.6

A random sampling of *S*_{min} can be obtained from its cumulative distribution function (c.d.f) in equation (2.6) using the standard inverse transformation method. In this method, a uniform *U*[0,1] random variable *Y* is equated to and the corresponding is defined as *F*^{−1}(*Y*) or
2.7
For each *a**, *Y* is randomly assigned a value from a uniform distribution *U*(0,1) in order to generate a random sample of . As a result, the values of produced in this way are distributed according to equation (2.6).

The probability distribution for is equal to the probability that all *n** regions form a twin nucleus under *τ*
2.8
Applying again the inverse transformation method to equation (2.8) generates a random sample of using
2.9
As before, *Y* is randomly picked from a *U*(0,1) distribution to produce a random sampling of for each grain. The values of *Y* for equations (2.7) and (2.9) are individually assigned as the probabilities of and are not the same.

The nucleation model is neither concerned with the processes of twin thickening nor with the possibility that after some amount of strain two neighbouring twin domains may merge into one. The characteristic *τ*_{c} is a nucleation-related stress, unconnected to twin thickening. Likewise, *a*_{c} will affect the number of twins per grain that appear initially but not the thickness of these twins.

## 3. Implementation into simulation

### (a) Local stress measure *τ* and twin variant selection

An important quantity that needs to be discussed is *τ*, the relevant stress measure for twin nucleation. It is commonly assumed that the RSS in the direction of the twinning dislocation is the most important component to activate twinning. As the present nucleation theory considers a scalar form for *τ*, it is reasonable to consider the RSS on the twin plane and in the direction of the twin. As described below, the RSS can be calculated from one of three length scales of stress.

In HCP crystals, there are six variants of the same twin type, *v*=1,…,6. Let *s*^{v} and *n*^{v} be, respectively, the glide direction of the twin lying in the twin plane and the twin plane normal for twin variant *v*. Under a macroscopic applied stress (tensor), the RSS *τ*^{M}(*v*) (scalar) on twin variant *v* is
3.1
where is the Schmid tensor for system *v*, i.e. *P*^{v}=sym(*s*^{v}⊗*n*^{v}). The stress *τ*^{M}(*v*) accounts solely for the effects of crystal orientation. In a polycrystal, however, the average stress in a grain will deviate from , and will differ from grain to grain owing to the constraint of its surroundings and its individual orientation, size, hardening and shape. A more refined estimate for the RSS on twin variant *v* would, therefore, be
3.2
The stress *τ*^{g}(*v*) is the one that can be calculated using standard polycrystal modelling. It is determined by the average properties of the grain and does not account for inhomogeneities in the interior of the grain or at the grain boundaries.

The present nucleation theory assumes that twins nucleate from grain boundaries, and thus the stress *τ* should be related to the stress field in the vicinity of the GBDs. Near the grain boundary, the stress state deviates from by an amount Δ*σ*_{ij} depending on the size and/or proximity of dislocation structures (debris, pile-ups), incompatibility stresses, triple junctions and other defects (voids, microcracks, inclusions). As such, these deviations ±Δ*σ*_{ij}(*x*) fluctuate along the grain boundary and from one grain boundary to another. The relevant scalar stress measure *τ* would best be estimated using . Therefore, for each variant *v*, we can write
3.3
where *τ*(*v*,*x*) is the RSS on the glide plane for variant *v* at point *x* in the boundary. Should non-glide stress components prove to also be critical for activating twinning dislocations (such as the normal or parallel components to the twin plane), their effect can be added to equation (3.3) above, and the definition of *τ*(*v*,*x*) adjusted accordingly.

The most challenging component to characterize in the above formulation is Δ*τ*(*x*) the fluctuation in the RSS. Depending on the source of the heterogeneity, Δ*τ*(*x*) can change with temperature, strain rate, processing history, grain boundary misorientation, deformation mode, alloying content, to name a few. The multitude of these factors and their combinations means that Δ*τ*(*x*) can vary widely in intensity level and spatial distribution.

### (b) Grain boundary surface area *a** from which twins nucleate

The quantity that remains to be assigned to each grain is *a**, the areal fraction of its grain boundary surface from which twins of one variant will nucleate and traverse the grain. Although twin nuclei can appear anywhere in the grain boundary surface, our interest lies in only those nuclei that can potentially traverse the grain should the stress be sufficiently high to allow propagation. Because twinning is directional, a given twin variant can only propagate across the entire grain when it originates from a certain part of the grain boundary surface. Accordingly, for a particular variant, *a** must be less than *a*^{g}, i.e. . Grain boundary areas outside of *a** are regions in which the twinning direction of the variant embodied in the twin nucleus does not allow it to grow into the grain interior.

Every grain is enclosed by a number of grain boundary facets *n*^{f}, where each is shared by one neighbouring grain (figure 2). These facets can vary in size, shape and defect distribution. Each facet has an area *a*^{f(i)}, *i*=1,…,*n*^{f} and thus, *Σa*^{f(i)}=*a*^{g}. Their content of GBD (e.g. atomic structure, misorientation angle), as well as other SCs (e.g. voids, interactions with slip dislocations), can vary statistically. Any differences in defect content will, in principle, affect parameters *τ*_{c} and *α* in the rate *λ*, equation (2.2). In Mg, for instance, the propensity for twinning was found to be higher for boundaries with lower misorientation angles (Beyerlein *et al.* 2010), which may be explained by a dependence of their GBD distribution on grain boundary misorientation (Wang *et al.* 2010). Thus, it is possible that some boundary facets may have certain properties that make them more favourable for twinning than others, in which case it would be reasonable to assign *a** to the total area of such facets and to assign *τ*_{c} and *α* as a property of these facets.

From the above, we can appreciate that appropriate assignment of *a** relies heavily on detailed three-dimensional information of the grain and grain boundary microstructure. One choice for *a**, which is amenable to mean-field polycrystal approaches, is the area of the surface of a spherical cap (*a*_{cap}) defined by the intersection of a cone and the spherical grain (figure 3). This area *a*_{cap} is related to the diameter of the sphere via
3.4
where *θ* is the angle of the cone. Setting *a**=*a*_{cap} would impart a dependence on grain size. Two-dimensional EBSD data (Beyerlein *et al.* 2010; Capolungo *et al.* 2009*c*) reveal that reasonable values of *θ* fall in the range of 30–80^{°}.

### (c) Nucleation criteria

When a grain twins and how many twins it nucleates (*n*_{twins}) will depend on measure of stress (*τ*) and grain boundary surface area (*a**). For simplicity, we arbitrarily choose the following criteria based on and :
3.5
where . According to equation (3.5), the grain forms *n** twins when and if *τ* exceeds some value between and . Otherwise, it remains untwinned.

The critical twinning stresses and in equation (3.5) are related to defect structures within the grain boundary *a** that will probably undergo rearrangement during deformation. Thus, and are subject to change as deformation proceeds. To reflect this change, and could be re-assigned in each strain increment.

The criteria in equation (3.5) introduce a positive size effect of *a** on twin nucleation in two ways. In a direct way, *n*_{twins} increases with *a** as *n** increases with *a**. In a more subtle way, as *n** is proportional to *a**, one can see from equations (2.7) and (2.9) that as *n** (or *a**) increases, the tendency is for to decrease and to increase. As Δ*S* widens with *a**, the likelihood of twinning increases. For demonstration, consider *τ*_{c}=195 MPa and *α*=2. For *n**=5, the average Δ*S* from 10^{4} trials is 209 MPa, whereas for *n**=15, the average Δ*S* increases to 306 MPa.

## 4. Application to pure zirconium

### (a) Multi-scale constitutive law

Validating the present nucleation theory through comparisons with statistical analyses of EBSD data carried out on deformed polycrystalline HCP metals requires implementing the theory into a polycrystalline plasticity constitutive model. To this end, in this section, we implement the twin nucleation theory into a multi-scale constitutive model that we have been developing for HCP metals. A brief overview is given here and complete descriptions can be found in Beyerlein & Tomé (2008), Capolungo *et al.* (2009*a*,*b*). A schematic of this constitutive model and its components is provided in figure 4. At the largest length scale, the polycrystal scale, the material is represented as an aggregate of plastically anisotropic HCP crystals. A visco-plastic self-consistent (VPSC) approach (Lebensohn & Tomé 1993) is used to relate the aggregate response to the individual crystal response (grain/medium interaction). VPSC predicts texture, stress–strain response and slip/twin activity. The deformation of individual grains takes place by slip and twinning. Reorientation by twinning is modelled through the ‘composite grain’ (CG) model (Tomé & Kaschner 2005; Proust *et al.* 2007), which treats the grain, once it twins, as a composite of alternating twin and matrix lamellae with twin orientation relationship across the interface.^{3} Any number of twin lamella of the same variant can be assigned to a grain. At an even lower length scale is the single crystal hardening model that prescribes characteristic stresses associated with the resistances to slip and twinning on individual systems. For slip, characteristic stresses are based on the storage of dislocations for each slip mode, which evolve according to thermally activated rate laws and hence are explicit functions of temperature and strain rate (Beyerlein & Tomé 2008). The reader is referred to the above cited works to understand how slip, slip–twin interactions and twin reorientation are treated in the present calculations.

For each strain increment, VPSC calculates a homogeneous stress state in each grain that deviates from the macroscopic stress state. The of a grain depends on its individual orientation, shape, hardening history and interaction with the surrounding homogeneous effective medium. In this way, VPSC captures the intergranular stresses (long range) that differ from grain to grain. From , *τ*^{g}(*v*) can be calculated for each variant *v* using equation (3.2)
4.1

### (b) Previous deterministic twin nucleation model

Where the previous efforts and the current work diverge is in the model used for twin nucleation. Previously, the constitutive model described above treated twin nucleation deterministically, wherein the onset of twinning and variant selection are governed by grain orientation, a purely geometric effect. During deformation, the twin shear rate was calculated for each twin variant *v* in each grain using
4.2
where *n*=20, the reference rate is set equal to the effective macroscopic strain rate, *τ*^{g}(*v*) is the RSS on variant *v* from equation (4.1), and *τ*^{twin}_{c} is a CRSS for twinning, which was assigned a constant, rate-independent value. Initially, all grains were untwinned and equal in size. When for the variant *vm* with the highest *τ*^{g}, i.e. , reaches a critical amount of strain *γ*^{crit} in a given untwinned grain, the untwinned grain transforms to a twinned grain. At this moment, it becomes a CG with alternating twin and matrix regions. Rather arbitrarily, four parallel twins corresponding to variant *vm* were assumed to form in each grain. Thereafter, the twin lamellae thicken proportionally with according to equation (4.2). As they thicken, the matrix regions narrow.

Figure 5 shows the stress–strain response predicted by the constitutive model that employs the deterministic twin nucleation scheme just described and compares it with the measured one for pure Zr deformed at 76 and 300 K (Beyerlein & Tomé 2008). For this calculation, the model makes available the four dominant deformation mechanisms in Zr reported in McCabe *et al.* (2006): prismatic 〈*a*〉 slip, first-order pyramidal 〈*c*+*a*〉 slip, twinning (tensile in Zr), and twinning (compressive in Zr). The clock-rolled Zr sheet was compressed in-plane at 10^{−3} s^{−1}, a direction with respect to the material texture that promoted twinning (figure 6). As shown in figure 5, at room temperature, the model performs well. The calculated twin fractions are low (<5%) and deformation is primarily accommodated by prismatic slip. At liquid nitrogen temperature (76 K), however, in which twin activity is more pronounced,^{4} the model underestimates the flow stress and hardening rate. Twinning evolves too rapidly in the calculation and is exhausted in the first couple of per cent strain. Revisiting this previous calculation in figure 5 demonstrates that while the hardening laws for activating individual slip systems are adequate, the deterministic criterion for activating the twin systems is not.

In the sections that follow, the probabilistic twin nucleation theory will be implemented into this multi-scale constitutive framework (replacing the deterministic criterion for nucleation) and the same simulations will be carried out. Unlike in the above, a statistical twin microstructure will be predicted with dispersions in the effects of grain size and orientation that can be directly compared with statistical data analyses.

### (c) Implementation of probabilistic twin nucleation

In implementing the new theory, the following simplifying assumptions are made. (i) The stress *τ* is uniform over the area *a**. (ii) All grain boundaries are treated equally, regardless of misorientation angle and the relative slip and twin activities between the two neighbouring grains. (iii) Only one twin variant is allowed to nucleate per grain. (iv) Twin thickening and secondary twinning are not treated as probabilistic events. Twin thickening proceeds and secondary twins are activated as in the previous model according to equation (4.2).

The initial microstructure is set as follows. The polycrystal model imports an initial material texture measured by EBSD, which is represented by 8095 orientations, each with a distinct area fraction *A*^{g}/*A*^{tot}, where *A*^{tot} is the total area of all measured orientations and *A*^{g} is the measured area of the grain. All grains are initially untwinned. As shown in figure 6, the starting material is strongly textured, where the basal poles tend to be aligned with the normal to the plate. The distribution of *A*^{g} associated with these 8095 orientations is shown in figure 7*a*. The initial grain shape is spherical with diameter *d*^{g}, which is approximated from *A*^{g}, i.e. . The resulting distribution for *d*^{g} is shown in figure 7*b*. For each grain, the area *a** is assigned the partial area on a sphere of diameter *d*^{g} intersected by a cone with *θ*=60^{°}, which according to equation (3.4) is *πd*^{g2}/4. The characteristic area is taken to be *a*_{c}∼0.5*d*^{g} (μm^{2}) to place the length scale of the twins in the nucleation model on the same length scale of both the VPSC/CG model and the EBSD analysis, with which we will compare our results. As a result, *n**=*a**/*a*_{c}=*πd*^{g}/2.

For each untwinned grain, the scalar stress measure *τ* is calculated in each strain increment and used to determine whether the grain twins (equation 3.5), which variant it activates (equation 4.3) and the number of twins it nucleates (equation 3.5). While determination of *τ*^{g}(*v*) is straightforward using equation (4.1), direct calculation of the local fluctuations in the RSS Δ*τ* at the grain boundary is not. For the present application, a sample of Δ*τ*(*v*) for a given variant *v* is randomly picked from a uniform *U*(−Δ*T*,+Δ*T*) distribution, where ±Δ*T* are prescribed upper (+) and lower (−) bounds on Δ*τ*. A different Δ*τ*(*v*) is selected for each variant in each untwinned grain in each strain increment. Changing Δ*τ*(*v*) with each deformation step reflects a dynamically changing localized defect structure. For simplicity, any spatial variation of Δ*τ*(*v*) within *a** is neglected. The variable Δ*τ*(*v*), therefore, results from the difference between the uniform local stress prevailing over grain boundary surface area *a** and the uniform stress in the interior of the grain.

The variant nucleated is the one with the highest *τ*^{g}(*v*)±Δ*τ*(*v*) at the time of nucleation. Therefore, the relevant stress measure *τ* is
4.3
and the variant *v*_{max} is the corresponding variant. Accordingly, the deviation of Δ*τ*(*v*) from *τ*^{g}(*v*) governs the selection of twin variants across the microstructure. The larger Δ*τ*(*v*) is relative to *τ*^{g}(*v*), the less probable the twin variant selected corresponds to the one with the highest *τ*^{g}(*v*). In this respect, a useful quantity is the SC at the time of nucleation, given by
4.4

### (d) Twin propagation

As in our earlier works, the twin nucleus will grow provided that the RSS on the variant nucleated is positive and is close to a characteristic propagation stress *τ*_{prop}. Once nucleated, the *n*_{twins} twins thicken proportionally with the shear rate , given by the same power law in equation (4.2)
4.5
where *τ*_{prop} is a characteristic propagation stress. The important distinction made here is that unlike twin nucleation, twin growth is driven by *τ*^{g}(*v*) based on the homogeneous grain stress (equation (4.1)) and not by *τ*, which includes the short-range local fluctuations Δ*τ*(*v*) (equation (4.3)), presumably present only locally at the grain boundary.

### (e) Twin nucleation algorithm and material parameters

Table 1 summarizes the twinning model parameters used in the present application to Zr. For twin nucleation, the parameters are *α*, *τ*_{c}, *a*_{c} and Δ*T*. The only parameter for twin propagation is *τ*_{prop}. The other hardening parameters are related to slip and slip–twin interactions and are associated with the dislocation density model described in Beyerlein & Tomé (2008). The parameters specific for Zr can be found in Beyerlein & Tomé (2008) and Capolungo *et al.* (2009*b*).

Each simulation is carried out using the following generic algorithm. At the beginning of each simulation, information regarding the material, temperature, strain rate, loading direction and initial microstructure is provided. For every grain representing the texture, three Euler angles and the area *A*^{g} from EBSD data are read and *d*^{g} and *a** are calculated (equation (3.4)). The initial ellipsoidal shape of each grain is assigned. All grains are initially untwinned. During simulation, in each strain increment, the following procedure is executed for every *untwinned*grain. A random number between 0 and 1 is generated and assigned to *Y* in order to obtain a sample of from equation (2.7). A second random number is generated to obtain a sample of using equation (2.9). Using the VPSC-calculated grain stress *σ*^{g} calculated for the present increment, *τ*^{g}(*v*) for each variant is determined (equation (4.1)). Each variant is also assigned a fluctuating stress Δ*τ*(*v*) randomly picked within the range [−Δ*T*,+Δ*T*]. The stress *τ* is set to the maximum of *τ*(*v*)=Δ*τ*(*v*)+*τ*^{g}(*v*) over all six variants (equation (4.3)). If the nucleation criterion is satisfied (i.e. in the present application, equation (3.5)), the twin nucleation algorithm tags the grain as twinned. The newly twinned grain becomes a CG containing *n*_{twins}=*n** equally spaced twins. The rate at which these twins thicken is determined by equation (4.5). If, on the other hand, the twin nucleation criterion is not satisfied (i.e. ), the grain remains untwinned and for this grain, this twin nucleation algorithm is repeated in the next strain increment with a new sampling of , and Δ*τ*(*v*), *v*=1,…,6.

## 5. Comparisons with measurement

Deformation simulation results obtained with the constitutive model presented above are compared with two kinds of accessible experimental information. One type is information associated with the aggregate as a whole, such as stress–strain response, texture evolution and twin fraction evolution. The other kind includes recent results from statistical analyses of EBSD data for Zr (Capolungo *et al.* 2009*c*).

### (a) Polycrystal scale properties

Figure 8 compares the stress–strain curves at 76 K. Incorporating the new twin nucleation model leads to predictions of flow stress and hardening rate in good agreement with the measurement, a significant improvement from previous ones derived from the deterministic twin model (figure 5). Figure 8 also reports a hypothetical prediction with twinning suppressed for the 76 K case. A comparison with the prediction with twinning demonstrates that twinning does not influence the macroscopic stress–strain behaviour before 5 per cent strain. A similar simulation was carried out for the same material and loading conditions, but at a deformation temperature of 300 K. As shown in figure 8, agreement with the measurement has also improved. In this case, the model predicts that the contribution of twinning is not as relevant as for 76 K.^{5} As will be demonstrated in the following, improvement in agreement at both temperatures can be attributed to the fact that the present model accounts for the statistical nature of twin nucleation.

The first important result revealed here is that twin nucleation occurs before one observes substantial impact on the stress–strain response and texture evolution. Figure 9 compares the texture predictions with EBSD measurements from McCabe *et al.* (2009) at 4, 9, 14 and 19 per cent strain in the form of basal pole figures. Both sets consistently show that reorientation of the basal poles along the compression axis as a result of twinning is not apparent at 4 per cent, starts appearing at 9 per cent and becomes significant at 14 per cent strain and higher. Artificially ‘delaying’ twin activation in the model until about 5–10% strain in order to reproduce the bulk texture measurements would lead to inaccurate characterization of model parameters.

Specifically, the model indicates that twins have nucleated and grown some amount before deformation reaches 5 per cent. The evolution of twin fraction predicted from the model is shown in figure 10*a*. The fraction of material twinned over the entire polycrystal grows from 0.03 at 5 per cent strain to 0.16 at 10 per cent strain, an evolution that is consistent with area fractions (symbols) taken from EBSD data (Capolungo *et al.* 2009*c*; McCabe *et al.* 2009). Beyond the available data, the model predicts that the growth rate decreases after 30 per cent. As a related statistic, figure 10*b* shows the evolution with strain of the fraction of twinned grains, that is, grains containing at least one twin. The prediction in figure 10*b* not only compares well with the experiment, but also shows that twinning increases rapidly in the first 5 per cent strain (nearly 60% of the grains have nucleated twins), although the usual signatures of twinning on bulk properties have not yet become apparent.

Before moving forward with the statistical details, it is important to examine the relative role of twinning with respect to that of the other deformation mechanisms underlying the stress–strain response in this case. From the model, the relative contribution of each mechanism to accommodating the applied deformation can be determined at every strain increment. Figure 11*a*,*b* plots the strain evolution of deformation activity for each mode within (figure 11*a*) the parent matrix and (figure 11*b*) twinned regions. With respect to the sense and direction of macroscopic loading, most of the grains are oriented well for both prismatic slip and tensile twinning and hence both mechanisms are active in most grains. Within the matrix domains, as prismatic slip hardens and becomes more difficult, its activity decreases, while, in contrast, twin activity increases (figure 11*a*). Within the reoriented twin domains, the basal poles are placed in compression, which promotes secondary compressive twinning. Activity profiles for the twin domains in figure 11*b* show that activity of secondary compressive twins increases concomitantly with primary twin volume fraction. The prediction in figure 11*b* is supported by EBSD evidence of secondary twinned domains after 17 per cent strain (McCabe *et al.* 2009).

### (b) Microstructural properties

In this section, we focus on the microscale twin characteristics and the dependence of twin nucleation on grain crystallographic orientation and size. In order to compare model predictions of orientation effects directly with EBSD data, we will use the macroscopic twin Schmid factor of a given variant *v*, defined by
The twin Schmid factor distribution predicted by the model for the twin population nucleated by 5 and 10 per cent strain is shown in figure 12*a*,*b*. Every twin, including multiple twins belonging to the same variant in a given grain, is accounted for when calculating this distribution. In agreement with the distribution obtained by EBSD at 10 per cent strain (figure 12*c*), the predicted frequency of twins with a given Schmid factor *m* increases with *m*. As in the measurement, the calculated *m*-distribution is broad, even containing twins with *m*<0.3.

The variance in *m* can partly be attributed to the dispersion in the twin variants selected. As listed in table 2, the model finds that the twin variant with the highest macroscopic Schmid factor is found in only 60–63% of the twins and with the second highest in 21–23% of the twins. These fractions are in good agreement with those found via EBSD analysis, which are also provided in table 2. Variant selection is determined largely by the stress fluctuations Δ*τ* assumed to prevail at the grain boundary, which for the present calculations were randomly picked from a *U*(−40 MPa, 40 MPa) distribution (table 1). Figure 13 shows the SCs (equation (4.4)) at the time of twin nucleation for twins that have nucleated by 10 per cent strain. The SCs are heavily weighted between 1 and 2. Apparently, the fluctuations Δ*τ* do not have to deviate too much from *τ*^{g}(*v*) to explain the observed variant selection and lower the preference to nucleate the variant with the highest macroscopic Schmid factor *m*_{(1)}. If in the model these fluctuations were set to zero, i.e. Δ*T*=0 and , then 82 per cent of the twins would correspond to the one with the highest and 8 per cent to the second highest.

The model predicts that twinning occurs most often in grains preferentially oriented for twinning, albeit with a non-negligible dispersion. Consider the highest macroscopic twin Schmid factor *m*_{(1)} in the grain among the six possible variants as a measure of a grain’s suitability for twinning. As shown in figure 14, the fraction of grains containing at least one twin (a ‘twinned’ grain) increases with *m*_{(1)}. While this result alone may not be surprising, the prediction that the dispersion is high, particularly for the lower strain of 5 per cent, is. At 5 per cent strain, not all grains most favourably oriented for twinning (*m*_{(1)}>0.4) nucleated twins. At 10 per cent, however, all ‘suitable’ (i.e. *m*_{(1)}>0.4) grains have twinned. Direct comparison with the result from the EBSD analysis (figure 14*c*) at 10 per cent strain finds that even more variance is observed in the measurement. For instance, in figure 14*c*, only approximately 90 per cent of the suitable grains with *m*_{(1)}>0.4 have twinned. The relatively higher dispersion measured by EBSD may result because the two-dimensional section scanned by EBSD happened not to intersect the twins formed in some grains. On the other hand, the relatively lower variability predicted by the model may arise because the model did not include additional microstructural dependencies, such as the small preference of twinning found in Zr for low angle grain boundaries (Capolungo *et al.* 2009*c*).

An important microstructural feature observed in many experimental studies to affect twinning is grain size (Armstrong *et al.* 1962; Chun *et al.* 1969; Okazaki & Conrad 1973; Ecob & Ralph 1983; Song & Gray 1995; Meyers *et al.* 2001; Agnew *et al.* 2003; Barnett *et al.* 2004), where it is generally believed that twins form ‘more easily’ in large grains. As shown in figure 15, both the experiments and model find that whether or not a twin forms in a grain is independent of its size, meaning that small grains are just as likely to form *at least one twin* as large grains. A similar result has been reported for twins in pure Mg (Capolungo *et al.* 2009*c*; Beyerlein *et al.* 2010). In both studies, size effects were studied among grains belonging to an inhomogeneous microstructure, and not to two distinct samples of the same material with different average grain sizes.

The most important grain size effect predicted and also reported by EBSD (Capolungo *et al.* 2009*c*; Beyerlein *et al.* 2010) is that the number of twins nucleated per grain (*n*_{twins}) scales with grain size. As shown in figure 16, the model predicts, in agreement with the data, that the rate of increase in *n*_{twins} with *A*^{g} is high, rising from five per grain for *A*^{g}=100 μm^{2} to approximately 17 per grain for *A*^{g}=500 μm^{2}.

Overall, the good agreement in figures 15 and 16 indicates that the manner in which the probabilistic twinning model accounts for grain size effects is physically sound. It would be of interest to determine if the same statistical correlations with grain size are observed for other twin modes, in other low-symmetry metals possessing different crystal structures, and for the same polycrystalline metal with vastly different *average* grain sizes.

## 6. Discussion

Grain boundaries play a key role in twin nucleation. Much of the variability in twinning has to do with variations in the local conditions (defects and stress states) at the grain boundaries that control twin nucleation. Grain boundaries contain both the large defects that can act as sources for twinning partials and the large stress states needed to produce the critical number of twinning partials required for a stable twin nucleus from these defects. As both GBDs and local stress fields are statistical quantities, twin nucleation is a random event that depends on the chance that the appropriate stress state will encounter the appropriate GBD or configuration of GBDs. In order to further advance understanding of twin nucleation, studies of the stresses and defects at grain boundaries are needed.

The theory developed here treats nucleation in a probabilistic manner and growth in a deterministic one. Nucleation and growth are assumed to be controlled by different stress states—localized fluctuating stress states at the grain boundaries for nucleation and long-range intergranular stress states for growth. Success in predicting the evolution of twin volume fractions with these considerations suggests the following regarding twin growth. (i) The twin growth process appears to be less variable than twin nucleation. (ii) Twin growth by propagation across the grain and thickening normal to the twin plane is dictated by the long-range stress state generated in the grain. (iii) A threshold CRSS on the twin plane and direction must be met in order to keep twinning dislocations in motion (once nucleated), to expand the twin fault loop in the twin plane and to propagate it across to the other side of grain. The primary microstructural feature governing twin growth is crystallographic orientation. Therefore, while inappropriate for twinning nucleation, a Schmid law that introduces a CRSS for twin propagation seems to be appropriate for twin growth. The last point is supported by recent model interpretations of twinning in single crystal Mg (Capolungo *et al.* 2009*a*), wherein the authors compared two models for twin thickening—one based on a constant CRSS and the other on dissociations of lattice dislocations at twin boundaries—and found the former to be more consistent with experimental observation. Last, we mention the two relevant statistical aspects of twin growth that are not included in our model: merging of two parallel and adjacent twins and percolation of twins across grain boundaries (Beyerlein *et al.* 2010).

For pure Zr, we find that there is an incubation period in the early stages of straining in which twins nucleate and grow, but the impact on the polycrystal stress–strain response and texture is small and undetectable by bulk measurement techniques. As a consequence, there is an apparent ‘delay’ in twinning activity from a macroscale viewpoint. This delay is a well-known characteristic of Zr and has often been attributed to the need for plastic slip prior to twinning to either supply defects or high stresses or both. From the present study, we find that there are two factors that contribute to this delay, both of which are related to the postponement of twin growth after nucleation: (i) the twin variant nucleated under SC is not always the one with the highest RSS and (ii) stress fluctuations (or concentrations) aid twin nucleation but not propagation. The twin variant selected in nucleation is not necessary the most favourably oriented for growth.

The stress–strain response of HCP metals that deform by both slip and twinning is known to contain multiple transitions in hardening rate. A physically based, multi-scale constitutive law has the potential to predict the various stages of hardening and the mechanisms that govern them. The present constitutive model predicts that the response of pure Zr at 76 K up to 30 per cent is governed by a combination of prismatic slip, twin nucleation, twin growth, slip–twin interactions and secondary twinning within the primary twins, each taking part within a distinct strain regime:

— The initial portion of the stress–strain response up to approximately 5 per cent is where prismatic slip dominates and most twin nucleation occurs. Although many grains have nucleated at least one twin, they are thin and because of this, they do not significantly impede plastic slip nor affect bulk texture evolution and stress–strain response.

— The subsequent stage from 5 to 10 per cent involves both twin nucleation and growth. As the twins grow, matrix regions shrink and thus the mean free path of dislocation slip in the matrix decreases. In this stage, a change in hardening rate from that expected by slip alone becomes apparent.

— The response after 10 per cent is characterized by a rapid hardening rate, a common macroscopic signature of the presence of twinning. However, it would be inaccurate to attribute the onset of this stage to the onset of twinning. This stage in fact involves twin growth and little nucleation, because at this point, a majority of the grains are twinned (approx. 80% at 0.1 strain). The increasing rate of hardening beyond 10 per cent strain is dictated by (i) the continuous reduction of the mean free path of prismatic slip within the matrix regions as the twins grow and matrix regions shrink and (ii) the rise in secondary twin activity within the twins.

It is important to emphasize that the quantitative characteristics discussed above apply to pure Zr at 76 K and its particular starting microstructure. Clearly, these details should not be generalized to other HCP metals or even to Zr at other temperatures or strain rates. Only a well-tested, physically based multi-scale constitutive model has the potential for broad application to a wide range of materials and loading conditions and to predict the same level of detail as in the above example.

## 7. Conclusions

This article formulates a new probabilistic theory of twin nucleation in HCP polycrystalline metals. Conceptually, it is more consistent with the observed statistical nature of twinning than previous deterministic treatments of deformation twinning. The theory is implemented into a multi-scale constitutive model for HCP metals in order to test its predictive capability against measurements made previously on pure Zr deformed at 76 and 300 K. Unlike a deterministic criterion for twinning, the present nucleation theory adequately predicts the dispersion in the onset of twinning and variant selection with respect to crystallographic orientation. The theory also explains the proportional scaling in the number of twins per grain with grain size. It is clearly demonstrated that capturing the substantial variability in the incidence of twinning is critical for correctly predicting the strain evolution of flow stress, hardening rate, twin volume fraction and texture.

Our understanding of the mechanisms behind twinning in HCP metals is not yet complete. The present formulation is an analytical, nano- to micron-scale description intended to provide a bridge between atomic-scale information and polycrystalline-scale models. The theory can profit greatly from three-dimensional microstructural characterization and atomic-scale studies of grain boundaries and the kinetics of twin nucleus formation. The latter simulations, for instance, would provide a way of characterizing nucleation parameters (*a*_{0}, *τ*_{0} and *α*) and permit development of formulas for *a*_{c} and *τ*_{c}, both as a function of grain boundary structure. In this way, the theory can address important issues such as the type of defects in grain boundaries that lead to twinning, their spacing and critical stresses (or energy barriers) required for nucleus formation and the link between the propensity for twin nucleation and grain boundary properties, such as grain boundary misorientation angle.

This work contributes to the greater goal of understanding and predicting the constitutive response and microstructural evolution of metals with an HCP crystal structure, such as Zr, Mg, Ti, Be, Hf, Cd and Zn. The nucleation theory has broad applicability to many HCP twin types and metals, and can easily be implemented into other continuum level numerical schemes for polycrystalline deformation.

## Acknowledgements

The present work was performed with support from Office of Basic Energy Sciences, Project FWP 06SCPE401, under U.S. DOE Contract no. W-7405-ENG-36.

## Footnotes

↵1 Implicit in this definition is the assumption that a uniform

*τ*acts over area*a*_{c}.↵2 The distribution in equation (2.4) is a Weibull distribution with scale parameter

*τ*_{c}and shape parameter*α*.↵3 Owing to subsequent slip in the matrix and/or twin lattice, the initial twin orientation relationship at nucleation can change with further deformation in the model.

↵4 At 76 K, twinning was profuse, most of the twins were , and twins within the same grain were of the same twin variant (Beyerlein & Tomé 2008; Capolungo

*et al.*2009*c*).↵5 We do not have EBSD statistical data for the 300 K case.

- Received December 20, 2009.
- Accepted February 15, 2010.

- © 2010 The Royal Society