## Abstract

In this paper, we examine the problem of deriving local stress measures and associated elastic constants for a region containing a fixed number of atoms that is part of a larger body. By means of an expansion of the local interatomic potential energy, we derive expressions for the local second Piola–Kirchoff stress and the associated elastic constants. We relate these to other previously derived atomic scale stress measures. Numerical calculations are presented for the case of an oversized inclusion embedded in an otherwise perfect face-centred cubic crystal.

## 1. Introduction

A topic that has been of interest for some time now is the calculation of continuum-like quantities, particularly stress and elastic constants, from considerations of interatomic forces. Typically, these calculations are done on a smaller scale than continuum mechanics is able to reach. A stress measure that has been frequently utilized in this context is the so-called atomic stress, apparently first derived in a rather ad hoc fashion by Basinski *et al*. (1971), on the basis of a simple volumetric partition of the bulk or virial stress. Omitting the kinetic contribution, this has the form for a given atom *α*(1.1)where *V* is the interatomic potential. Here, the sum is taken over the *M* atoms that interact with atom *α*, and **R**^{αγ}=**R**^{α}−**R**^{γ} is the relative position vector between atoms *α* and *γ* under reference equilibrium conditions. The quantity *Ω*^{α} is taken to be a representative atomic volume centred about atom *α*. More recently, Cormier *et al*. (2001) have investigated the utilization of a stress measure derived in general form by Hardy (1982), and later in a more precise, albeit somewhat more limited, form by Lutsko (1988). In volume-averaged form, and again with the kinetic contribution omitted, this stress measure is(1.2)where *ω* is the averaging volume and **r**^{αβ} the interatomic relative position vector in the current configuration. The sums are taken over the *N* atoms contained in the averaging volume *ω*, as well as the *M* atoms exterior to *ω* that interact with the interior atoms. Here, is the fraction of the bond length *r*^{αγ} contained within the averaging volume. In its non-volume-averaged form, this stress measure satisfies conservation of linear momentum, a property that the atomic stress lacks.

With regard to elastic constants, there exists a substantial literature on the calculation of bulk constants for crystals, beginning with the pioneering work of Born & Huang (1954). However, because the interatomic forces are sensitive functions of the interatomic spacings, it is reasonable to think that in defect-containing regions where the interatomic spacings differ substantially from the bulk solid, the local elastic constants may also differ from those for the bulk solid. Accordingly, there have been several efforts to calculate local elastic constants in such regions. Kluge *et al*. (1990) carried out such a calculation for a volume containing a grain boundary, based upon previous work by Lutsko (1989) on elastic constants for periodic crystals. Here, the elastic constants were defined as , where is the macroscopic strain. The non-periodic structure of the grain boundary was accommodated in an ad hoc fashion. A later calculation by Alber *et al*. (1992) utilized the work of Martin (1975*a*,*b*) on elastic constants for periodic crystals to calculate local elastic constants, again in the neighbourhood of a grain boundary. The non-periodic nature of the atomic structure under consideration was again taken into account in an ad hoc manner. More recently, Van Vliet *et al*. (2003) have considered both the local stress and the local elastic constants in the highly stressed region beneath a nanoindenter, and have utilized these to predict the initiation of dislocations. Once again, however, an expression for the elastic constants appropriate to a periodic solid was employed.

The purpose of this paper is to establish a consistent set of local stress measures and elastic constants at the atomic scale. In particular, we obtain relations between the previously mentioned atomic stress measures that, in certain circumstances, parallel those existing on the continuum level. We also derive relations for the associated local elastic constants. In contrast to previous work, we will make *no* assumptions with regard to crystal periodicity. Not surprisingly, it will be seen that a lack of periodicity necessitates an approximate treatment for the elastic constants. For simplicity, we restrict ourselves to the low temperature case in which thermal fluctuations may be neglected. Likewise, we will consider only simple pair potentials, although neither of these restrictions is essential.

The initial state of the body is denoted by ( )_{o}. We assume that a self-equilibrating system of surface tractions is applied to the body, so that at locations sufficiently far removed from the surface the body is in a state of uniform macroscopic strain. Within this uniform macroscopic strain region, we consider a subvolume *Ω* that contains *N* atoms. Our approach, which has become fairly standard in the analyses of crystalline solids (e.g. Wallace 1998), seeks an expansion for the *local* interatomic energy density; that is, the interatomic energy density contained within *Ω*, in terms of the components of the macroscopic Lagrange strain tensor E_{ij}. However, our point of departure from the standard approach is that we include what Hardy (1982) terms ‘localization functions’. These are functions that contain the spatial dependence of the energy density within the expansion. The inclusion of these functions adds a previously missing degree of consistency to the results. In any case, by analogy with the continuum mechanics result for the expansion for the strain energy density in the Lagrange strain,(1.3)the local second Piola–Kirchoff (PK2) stress *S*_{ij}, and the local elastic constants *C _{ijkl}* may be identified as the coefficients of the linear and quadratic terms, respectively, in this expansion.

We show that the elastic constants depend upon the details of the atomic displacements resulting from the application of the applied macroscopic strain, which is not the case for the various stress measures that we will discuss. We thus begin with an analysis of atomic displacements under an arbitrary applied macroscopic strain field E_{ij}.

## 2. Atomic displacements

Let us denote by * u*(

*) the continuum displacement field corresponding to a uniform macroscopic strain field characterized by the Lagrange strain tensor E*

**R**_{ij}. If this displacement field is applied to an assemblage of atoms, so that the displacement of atom

*α*is given by

**u**^{α}=

*(*

**u**

**R**^{α}), then the resulting atomic configuration will not generally be an equilibrium configuration, such that the vector sum of the interatomic forces upon each atom will not vanish. In order to ensure equilibrium, an additional atomic displacement, which we will denote by

**δ**^{α}, is required to bring the atoms into an equilibrium configuration. This additional displacement component is sometimes referred to as an atomic relaxation, although we prefer the term ‘non-uniform atomic displacement’. These non-uniform atomic displacements play a prominent role in the determination of the local elastic constants. Therefore, the goal of this section is to determine a linear approximation for the

**δ**^{α}, valid to first order in the strain components E

_{ij}. Unfortunately, this requires a somewhat lengthy, but relatively straightforward, development.

As noted previously, we assume that the interatomic forces may be derived from a pair potential *V*. This is not an essential restriction, and the analysis that we present here may be extended without great difficulty to more complicated multibody potentials. Here, we adopt the point of view of Keating (1966) and Martin (1975*a*), that in order to enforce translational and rotational invariance, the potential energy function *V* must be a function of the scalar products of the relative interatomic spacings, **R**^{αβ}.**R**^{γδ}, where the Greek superscripts denote atomic labels, and **R**^{αβ}=**R**^{α}−**R**^{β} is the interatomic position vector between atoms *α* and *β*, with **R**^{α} the position vector of atom *α* in a Cartesian coordinate system. The scalar products of relative position vectors are invariant under both rigid body rotation and translation. For pair potentials, the foregoing discussion implies that *V* must depend upon the product **R**^{αβ}.**R**^{αβ}=(*R*^{αβ})^{2}. Likewise, one could also consider a dependence upon , and, indeed, we do both. However, the utilization of makes it clear that the arguments are always positive. Thus, in order to make natural contact with the Lagrange strain tensor, which is also invariant under rigid body rotation, we take expansions of the interatomic potential powers of the squares of the relative interatomic spacings. Martin (1975*a*) has pointed out that expansions in the relative interatomic spacings themselves, which lead to the utilization of the infinitesimal strain tensor, will generally agree with the foregoing only in the case of zero stress and strain, and are not otherwise valid.

In the interests of simplicity, we will assume that all atoms are identical. The analysis is easily extendable to the case in which this is not true, and we will give an example of this in a subsequent section. We focus upon a volume *Ω* containing *N* atoms, interior to a much larger body, and assume that there exist a total of *M* atoms exterior to *Ω* that interact with the *N* interior atoms. Under these circumstances, the total energy *Ψ*_{o}, which results from interior and exterior bonds, is(2.1)Here, the sum indicates a sum over all atomic pairs (*α*, *β*) that lie within *Ω*, whereas the sum denotes a sum over pairs (*α*, *γ*) for which *α* is contained within *Ω*, but *γ* is exterior to *Ω*.

Under the action of the uniform macroscopic strain field, the relative position vector for each atomic pair is altered from **R**^{αβ} to **r**^{αβ}. We expand the potential energy given by equation (2.1) in a Taylor's series about the initial equilibrium state. We obtain(2.2)

By a simple chain rule relation, we obtain(2.3)Substitution into equation (2.2) gives(2.4)Thus,(2.5)and(2.6)

We must be specific about the nature of the atomic displacement field. Let **U**^{α} be the displacement of atom *α*, so that **r**^{α}=**R**^{α}+**U**^{α}. In general, **U**^{α}=**u**^{α}+**δ**^{α}, where **u**^{α} is the part of the displacement arising from the uniform macroscopic deformation field * u*(

*), and*

**R**

**δ**^{α}is an additional non-uniform contribution that will be required to bring the atom into an equilibrium position. In the Lagrangian approach, all particle coordinates are referred to the reference configuration. Hence,

*(*

**u**

**R**^{α})=

**u**^{α}, and(2.7)

By Taylor expansion(2.8)

As is customary, lower case Latin subscripts indicate vector components, which range in value from 1 to 3. Here, we adopt the usual summation convention for repeated Latin subscripts. The displacement gradient in equation (2.8) derived from the uniform part of the displacement field is assumed to be constant, or essentially so, over the length-scale of interest. Hence, subsequent terms in the expansion vanish, and we write(2.9)Then(2.10)where E_{ij} is the continuum Lagrange strain tensor (Fung 1965) associated with the macroscopic deformation field. Even though the Lagrange strain tensor is valid for arbitrary values of strain, we assume that . Under these circumstances, the Lagrange strain is essentially identical to the usual infinitesimal strain tensor.

As noted, the deformed configuration must be an equilibrium configuration, and the quantities **δ**^{α} are determined by simply enforcing this condition. This leads, to first order, to a set of linear equations for the **δ**^{α} having the form **A*** δ*=

*. We refer the reader to appendix A for a detailed analysis, in which expressions for the coefficient matrix*

**c***and the right-hand side vector*

**A***are derived. The right-hand side vector, in particular, depends both upon the macroscopic strain field E*

**c**_{ij}and the non-uniform displacements

**δ**^{γ}for the

*M*exterior atoms. In order to obtain a computable result, we now consider the nature of these latter quantities. If we were dealing with a periodic crystalline arrangement of atoms, then these would be determined exactly by considerations of periodicity (e.g. Wallace 1998). However, for a non-periodic solid, an exact knowledge of these displacements would require precise knowledge of the atomic displacements in an annulus surrounding the

*M*exterior atoms, which, in turn, would require a knowledge of the atomic displacements in a still more distant annulus, and so on. Evidently, an exact solution would require the solution of the atomic equilibrium problem for the entire body, subject to a specified set of self-equilibrating surface forces. This is usually not possible, and therefore some approximate description for

**δ**^{γ}is required. There are several possibilities here, from which we choose the simplest, assuming that the atomic displacements of the

*M*exterior atoms are just those resulting from the macroscopic strain field. Thus,(2.11)and hence(2.12)

We discuss the limits of applicability of this assumption further below. With this approximation, the solution has the general form (see appendix A)(2.13)where(2.14)Then,(2.15)

We note that , and hence and , is symmetrical in the indices . Likewise,(2.16)

We now substitute equations (2.15) and (2.16) into equation (A 1). With equation (2.12), this yields(2.17)

Equation (2.17) gives, to quadratic order in the E_{ij}, the total interatomic potential energy within the volume *Ω*, as well as contributions from those exterior atoms that interact with atoms inside *Ω*.

## 3. Local stresses and elastic constants

We now focus upon the local energy density *ψ* for the volume *Ω*, seeking to expand this quantity in a fashion similar to the expansion of the total energy *Ψ* carried out in §2. In order to do this, we assume that *ψ* has the form(3.1)where * r* is the position vector. The function

*Δ*(

*,*

**r**

**R**^{α}) contains the spatial variation of the energy density, and is termed the ‘localization function’ (Hardy 1982). It has local support about

**R**^{α}, implying that it is non-zero only within a limited distance of

**R**^{α}. It is also subject to the normalization requirement that(3.2)where the integral is taken over a volume sufficiently large so as to enclose the region over which

*Δ*is non-zero. Evidently,

*Δ*has units of vol.

^{−1}. The localization function is often taken to be the Dirac delta function (e.g. Irving & Kirkwood 1950; Schofield 1966; Zubarev 1974). However, as Hardy (1982) pointed out, this choice is not unique, and other forms of the localization function are possible. For example, Zimmerman

*et al*. (2004) recently utilized the radial step function and a cubic polynomial in this context.

We begin by expanding *ψ*_{o} (* r*) in powers of as in §2. Noting that the localization function does not involve the relative interatomic spacings, we have(3.3)

We now substitute equations (2.10) and (2.15) into equation (3.3), and collect terms of up to quadratic order in E_{ij}. The result has the general form(3.4)The steps outlined in §2 leading up to equation (2.17) now yield(3.5)and(3.6)

Equation (3.5) gives a local stress measure *S*_{ij} that, by analogy to continuum mechanics usage, we will term the local second Piola--Kirchoff (PK2) stress. Equation (3.6) yields the local elastic constants .

At this point, we want to contrast the local PK2 stress measure given by equation (3.5) with a local stress measure derived by Hardy (1982) and in a more restricted formulation by Lutsko (1988). Up to the inclusion of arbitrary solenoidal terms, this later stress measure satisfies conservation of linear momentum and thus might be termed (again, with reference to continuum mechanics usage) the local Cauchy stress. If *g*(* r*−

**r**^{α}) is the localization function for the momentum density

*, i.e. , then Hardy (1982) showed that this is related to the localization function*

**p***f*(

*,*

**r**

**r**^{α},

**r**^{αβ}) for the local Cauchy stress (Hardy utilizes the term ‘bond function’) by(3.7)

If one substitutes *g*(* r*−

**r**^{α})=

*δ*(

*−*

**r**

**r**^{α}) and evaluates the integral, the result is (Cormier

*et al*. 2001)(3.8)where

*Θ*(

*x*) is the Heaviside step function, and

**p**_{1}and

**p**_{2}are orthogonal unit vectors that are mutually orthogonal to

**r**^{βα}. The resulting local Cauchy stress, in the current notation, is(3.9)where(3.10)

It is worthwhile to point out here that the signs of the stresses differ from those in some previous work (e.g. Lutsko 1988; Cormier *et al*. 2001). The sign is dependent upon whether or not one defines the divergence of the stress or its negative as the linear momentum flux. Here, we prefer the continuum mechanics convention that takes the positive sign.

## 4. Volume-averaged local quantities

Generally, in order to obtain computable results, and certainly to make contact with continuum level results, a volume averaging procedure over *Ω* is required. At this point, in the interests of comprehensiveness, we make a specific choice for the localization function; namely, the Dirac delta function, *Δ*(* r*,

**R**^{α})=

*δ*(

*−*

**r**

**R**^{α}). This choice yields(4.1)

Likewise,(4.2)All of the terms in equation (4.2) evidently have the index symmetries . Moreover, all the terms, except(4.3)are also symmetrical under interchange of the index pairs *ij* and . A detailed analysis of this term (appendix B) shows that it is symmetrical in the index pairs *ij* and , so that .

For , the volume-averaged local PK2 stress and the local elastic constants are related by the equation (Wallace 1998)(4.4)where denotes the change in stress resulting from the imposition of a strain and(4.5)The stresses on the right-hand side of this equation are those existing prior to the imposition of the strain (i.e. the stresses in the initial state). It is noteworthy that . This relation is, of course, only approximate if the approximate elastic constants given by equation (4.2) are utilized.

The volume-averaged local Cauchy stress given by equation (3.9) may likewise be expressed in volume-averaged form (Cormier *et al*. 2001),(4.6)Here, *ω* is the volume in the current configuration and , , is the fraction of the bond length *r*^{αγ} contained within *ω*. Structurally, the volume-averaged local PK2 and Cauchy stresses are similar. Apart from differences in reference configuration, the principal difference is that the Cauchy stress includes the fractional bond length between interior and exterior atoms contained within *Ω*, whereas the PK2 stress does not.

We now want to establish a relationship between the volume-averaged local Cauchy and PK2 stresses similar to that existing between the corresponding continuum mechanical quantities. To do this, we consider large averaging volumes so that we approach the continuum length-scale. Now, as the region *ω* grows large, the number of exterior atoms *M* interacting with the *N* interior atoms, and contained in an annulus about *ω*, becomes small compared with *N*. Hence, as *ω* becomes large, the second sum in equation (4.6) becomes negligible compared with the first, and(4.7)with a similar result for the volume-averaged local PK2 stress given by equation (4.1). Now,(4.8)and from equation (2.10)(4.9)where *R*^{αβ}=*R*^{αβ}*n*^{αβ} and the *O*(E_{ij}) terms arise from the non-uniform atomic displacements through equation (2.15). Hence, from equations (4.8) and (4.9),(4.10)

The continuum-deformation gradient corresponding to the uniform part of the displacement field is given by *F*_{ij}=∂*r*_{i}/∂*R*_{j}, so that(4.11)where, again, the *O*(E_{ij}) terms arise from the non-uniform part of the displacement field. Hence, from equations (4.7) to (4.11),(4.12)where *J*=*ω*/*Ω*. Hence,(4.13)The first term in equation (4.13) will be recognized as the standard continuum mechanics transformation between Cauchy and PK2 stress (Fung 1965).

A more general choice of the localization function (e.g. one that was non-zero over a finite radius about **R**^{α}) would lead to results different from those given by equations (4.1) and (4.2) only in contributions from localization functions that are non-zero over regions lying partially within and partially without the averaging volume. Such contributions might also lead to difficulties in satisfying the expected index symmetries in the elastic constants. However, an argument similar to that given above indicates that these contributions become negligible as the averaging volume becomes large.

## 5. Numerical example

We first examine the effect of averaging volume upon the elastic constants for a perfect face-centred cubic (FCC) crystal, using an array of 2048 atoms and assuming periodic boundary conditions, with the interatomic potential described by a slightly modified Lennard–Jones (LJ) potential (Broughton & Gilmer 1983). Here, because of centrosymmetry, the non-uniform displacements vanish, and the elastic constants are simply given by the first term in each summation in equation (4.2). Figure 1 shows the computed bulk modulus and shear modulus *C*_{1212} for a perfect crystal for different averaging volumes, where the averaging volume is taken to be a sphere about the central atom in the array. Units are LJ units. By way of reference, the lattice parameter for the crystal, again in LJ units, is approximately 1.55. Figure 1 indicates that for averaging radii smaller than a lattice parameter, the computed values show substantial oscillations. These oscillations are probably owing to the discrete nature of the body. On the other hand, averaging volumes whose radii are greater than lattice parameters, yield values for the bulk and shear moduli that are in good agreement with the values for an infinite bulk crystal. As expected, the agreement improves as the radius of the averaging volume increases, and the computed values are essentially identical with the bulk values for averaging radii in excess of two lattice parameters.

Next, we consider the local elastic constants given by equation (4.2) in the region surrounding an oversized inclusion in an otherwise perfect FCC lattice. Numerical results for the volume-averaged local Cauchy stress for a similar situation have been given previously (see Cormier *et al*. 2001). Here, we consider again an array of 2048 atoms subjected to quasi-periodic boundary conditions. The inclusion was placed at the centre of the array, and was simulated in simple fashion by doubling the LJ parameter *ϵ* for this atom. The equilibrium atomic positions were then determined by an energy minimization procedure, holding the position of the inclusion fixed.

Figure 2 shows the same quantities computed from equation (4.2) with the spherical averaging volume centred at the inclusion. In order to make the * A* matrix non-singular, the rows and columns of this matrix corresponding to the inclusion were deleted. For a fixed averaging volume, the effect of the inclusion is generally to raise both the bulk and shear moduli. This increase is especially notable at the smaller averaging volumes. This result is expected, because smaller volumes include a proportionally larger number of atoms in close proximity to the inclusion. The oscillations in the computed values are somewhat more pronounced than for the undefected solid.

## 6. Discussion

Utilizing the simplifying assumption of pair potentials, we have derived here a set of local, atomic scale-stress measures and approximate elastic constants for a non-periodic assemblage of atoms based upon a particular choice of localization function for the interatomic potential energy and for the momentum density, the commonly utilized Dirac delta function. This choice leads directly to particular forms for what we have termed the local PK2 and Cauchy stresses, along with the (approximate) associated local elastic constants. The inclusion of localization functions (i.e. functions that contain the spatial dependence of the quantities in question) is not common in the literature, but as we have shown, it is important for obtaining a clear picture of the relationship between the various atomic-scale stress measures and their relation to the elastic constants. In particular, the standard approach taken in the analysis of periodic crystals, of obtaining the energy density by simply dividing the total energy by the averaging volume (the volume of the periodic unit cell), fails in the non-periodic case because it provides no clear answer to the question of how the atomic bonds that are intersected by the averaging volume are to be treated.

The local Cauchy stress corresponding to the utilization of the Dirac delta function as a localization function in the conservation of the linear momentum equation has been identified as a stress measure considered earlier by Lutsko (1988) and Cormier *et al*. (2001). Volume-averaged versions suitable for computational purposes are obtained by averaging over the volume of interest *Ω*. On a suitably large length-scale, we have shown that, up to *O*(E_{ij}), the volume-averaged local PK2 and Cauchy stresses are related by an equation similar to that existing between the PK2 and Cauchy stresses at the continuum level. In common with its continuum counterpart, the local PK2 stress does not satisfy conservation of linear momentum. However the local Cauchy stress does, in fact, have this property (Hardy 1982; Lutsko 1988).

As we noted at the outset, the so-called atomic stress given by equation (1.1) has seen extensive utilization as a stress measure at the atomic scale. A comparison of equations (1.1) and (4.1) shows that the atomic stress is just the volume-averaged local PK2 stress with the averaging volume *Ω* centred about a single atom *α*, and enclosing only this atom. If this volume is chosen so that it is space filling when periodically repeated (e.g. as a Voronoi polyhedron; Alber *et al*. 1992), then the utilization of this volume will give precise results both for the stresses and the elastic constants for a uniformly strained crystal composed of periodically repeating unit cells containing just a single atom each. However, this property is lost for a non-periodic arrangement of atoms; the circumstances in which the atomic stress is usually applied. As has been shown both here and by Cormier *et al*. (2001), both the calculated values of stress and elastic constants are sensitive to the averaging volume chosen, and exhibit large oscillations with averaging volume for smaller averaging volumes. For this reason (among others), averaging volumes considerably larger than the atomic volume seem to be required to obtain results comparable to those obtained by continuum analyses. In fact, in the limit, as the averaging volume becomes large, one may utilize the same arguments presented earlier to demonstrate that the volume-averaged local PK2 stress approaches the virial, or bulk, stress. In any case, despite its frequent utilization in the literature, in most circumstances, there appears to be no convincing reason to prefer the atomic volume to any other choice of averaging volume.

Previous work on the calculation of local elastic constants has relied on the ad hoc utilization of results based upon the assumption of crystalline periodicity. Here, we have presented expressions that avoid this assumption. Although this is certainly an improvement, an approximate treatment is still required. This is because in the absence of any assumption regarding periodicity, an exact computation requires a knowledge of not only the displacements of the *N* atoms interior to *Ω*, but also of the motion of the *M* exterior atoms that interact with the interior atoms. As we have pointed out, the displacements of these exterior atoms can only be obtained by solving the equilibrium problem for the entire assemblage under the action of the given surface tractions. In most cases, this is numerically infeasible. Here, in order to obtain computable results, we have assumed that the displacements of the *M* external atoms external to *Ω* are *known*. In particular, we have assumed that the motion of these atoms is simply due to the uniform strain field, and that the non-uniform atomic displacements for the exterior atoms vanish. One might improve upon this assumption for a non-centrosymmetrical crystal by including, for example, the non-uniform displacement due to the uniform strain field in a periodic crystal (e.g. Keating 1966; Martin 1975*b*; Lutsko 1989). Evidently, the degree to which this approximation is satisfied is dependent upon the circumstances. For example, the non-uniform atomic motions might be expected to be reasonably large with respect to the uniform motions in the immediate neighbourhood of a defect. For this reason, the results for smaller volumes might be suspect, whereas the values for larger volumes would probably be more accurate. In any case, the results are dependent upon the volume chosen by virtue of the discrete atomic nature of the body; a fact that is clear from a consideration of figures 1 and 2. This discreteness also introduces a considerable amount of granularity into the results, which is also apparent from figures 1 and 2. Moreover, it is clear that the local elastic constants have an inherent degree of non-locality, in the sense that they depend not only on quantities inside the volume under consideration, but also upon the atomic displacements within an exterior annulus whose size is approximately equal to the interaction distance. These, in turn, depend upon the motion of still more distant atoms, and so on.

Evidently, there exists some degree of arbitrariness in the definition of local stresses and elastic constants. This is largely, although not entirely, due to the fact that there exists a great degree of latitude in the selection of localization functions (Hardy 1982). With only some weak mathematical restrictions, these are constrained mostly by considerations of physical reasonableness. We have utilized the previously employed Dirac delta function in this context; however, other choices are possible (e.g. Hardy 1982; Zimmerman *et al*. 2004). This arbitrariness may be a reflection of the fact that stress and elastic constants are really continuum mechanical concepts, and translate down to the atomic scale only with some difficulty. However, as we have noted for sufficiently large averaging volumes (the continuum limit), the volume-averaged stresses and elastic constants become insensitive to the choice of localization function because of the local support and normalization properties of these functions. Hence, it is only on the local, atomic scale that this choice is significant. On this scale, physical reasonableness seems to be the only available guide.

## Acknowledgments

The author is grateful to J. M. Rickman and R. LeSar for helpful discussions.

## Footnotes

- Received April 20, 2004.
- Accepted October 4, 2004.

- © 2005 The Royal Society