Journal of the Mechanics and Physics of Solids

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1 Journal of the Mechanics and Physics of Solids 89 (26) Contents lists available at ScienceDirect Journal of the Mechanics and Physics of Solids journal homepage:.elsevier.com/locate/jmps Material fields in atomistics as pull-backs of spatial distributions Nikhil Chandra Admal, Ellad B. Tadmor n Department of Aerospace Engineering and Mechanics, The University of Minnesota, Minneapolis, MN 55455, USA article info Article history: Received 6 December 25 Accepted 6 January 26 Available online 26 January 26 Keyords: Microstructures Stress relaxation Stress concentration Atomistics Numerical algorithms abstract The various fields defined in continuum mechanics have both a material and a spatial description that are related through the deformation mapping. In contrast, continuum fields defined for atomistic systems using the Irving Kirkood or Murdoch Hardy procedures correspond to a spatial description. It is uncommon to define atomistic fields in the reference configuration due to the lack of a unique definition for the deformation mapping in atomistic systems. In this paper, e construct referential atomistic distributions as pull-backs of the spatial distributions obtained in the Murdoch Hardy procedure ith respect to a postulated deformation mapping that tracks particles. We then sho that some of these referential distributions are independent of the choice of the deformation mapping and only depend on the reference and current configuration of particles. Therefore, the fields obtained from these distributions can be calculated ithout explicitly constructing a deformation map, and by construction they satisfy the balance equations. In particular, e obtain definitions for the first and second atomistic Piola Kirchhoff stress tensors. We demonstrate the validity of these definitions through a numerical example involving finite deformation of a slab containing a notch under tension. An interesting feature of the atomistic first Piola Kirchhoff stress tensor is the absence of a kinetic part, hich in the atomistic Cauchy stress tensor accounts for thermal fluctuations. We sho that this effect is implicitly included in the atomistic first Piola Kirchhoff stress tensor through the motion of the particles. An open source program to compute the atomistic Cauchy and first Piola Kirchhoff stress fields called MDStressLab is available online at & 26 Elsevier Ltd. All rights reserved.. Introduction Atomistic and continuum models have been extensively used to study material behavior. Atomistic simulations can be used to construct constitutive relations for continuum models, and atomistic and continuum descriptions coexist in multiscale methods that study systems ith to or more length and/or time scales (Tadmor and Miller, 2). Both cases require an exchange of information beteen the atomistic and continuum models, hich requires an understanding of the relationship beteen them. Data in an atomistic model exist in the form of positions and velocities of particles that have to be reinterpreted in the language of continuum mechanics. This is done by developing continuum notions for atomistic systems. Work on this dates back at least to Cauchy in the 82s ith his aim to define stress in static crystalline solids in terms of surface tractions n Corresponding author /& 26 Elsevier Ltd. All rights reserved.

2 6 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) (extended by Tsai (979) to dynamical systems), and then Clausius and Maxell in the 87s ith the definition of the virial stress. Later using the principles of classical nonequilibrium statistical mechanics, Irving and Kirkood (95) derived expressions for continuum fields in atomistic systems that are probabilistic in nature and identically satisfy the continuum balance las. More recently, Hardy (982), Hardy et al. (22) and Murdoch (983) folloed a procedure similar to that of Irving Kirkood but in hich spatial averaging is used instead of statistical mechanics. The relations beteen these various definitions for continuum fields in atomistic systems, and in particular the stress tensor, are discussed in Admal and Tadmor (2), here it is shon that all of the definitions can be obtained ithin a single unified frameork based on the Irving Kirkood procedure ith controlled approximations. The continuum field definitions obtained through the Irving Kirkood and Murdoch Hardy procedures are spatial fields defined in the current configuration. This distinction arises because continuum mechanics is a nonlinear theory here a material can undergo large deformations relative to its reference state. Thus in continuum mechanics a reference configuration is chosen to represent a convenient fixed state of the body to hich the current configuration is compared. Continuum fields can then be expressed materially in the reference configuration, or spatially in the current configuration. A material field can be obtained as a pull-back of the corresponding spatial field ith respect to the deformation map. (See Tadmor et al. (22) for a discussion of material and spatial descriptions in continuum mechanics and the definitions of the terms used in this paper.) Although only the spatial fields have physical meaning (for example the Cauchy stress evaluated in the current configuration is termed the true stress since this is hat the material actually experiences), the material definitions are very useful since through the pull-back operations a continuum boundary-value problem can be recast in the reference configuration here the domain is fixed. This is the so-called Lagrangian formulation. For this reason it is of value to obtain expressions of material continuum fields in atomistic systems. It is the purpose of this paper to obtain rigorous expressions for such fields. To approaches have been applied in the past to extend the Murdoch Hardy procedure (or equivalently the Irving Kirkood procedure) to obtain material fields, both of hich have certain shortcomings: (a) The first approach involves the definition of a pointise deformation gradient using the local environment of particles surrounding the point of interest (see for example Zimmerman et al., 29). The material fields can then be obtained using the deformation gradient, and the continuum pull-back relations beteen spatial and material fields. Hoever, the pointise deformation gradient field is not necessarily a compatible deformation gradient, and therefore the resulting material fields do not generally satisfy the balance las. (b) The second approach involves the application of the Murdoch Hardy procedure in the reference configuration. This as recently done by Zimmerman et al. (2) and revisited by Davydov et al. (23). In this case, although the obtained continuum fields identically satisfy the material forms of the balance las, they do not satisfy the pull-back relations ith the spatial fields obtained separately using the Murdoch Hardy procedure in the current configuration. In this paper e take a different approach. We define a deformation mapping that tracks the particles and then use this mapping to define material fields through an appropriate pull-back operation. We do not attempt to directly pull-back spatial fields since the resulting material fields ould depend on the deformation mapping, hich is not unique. Instead, e first rite the spatial continuum fields obtained in the original Murdoch Hardy procedure as convolutions of the eighting function used for spatial averaging ith respective spatial distributions. This separates the physics from the arbitrary spatial averaging. We then obtain the material distributions as pull-backs of the spatial distributions. Finally, the material fields are obtained as convolutions of the eighting function ith the corresponding material distributions. By construction, the resulting material fields identically satisfy the material balance las. Through this procedure, e obtain material fields for the reference density, momentum density and body force that depend only on the initial and final atomistic configuration of particles, and thus do not explicitly depend on the choice of the deformation map. In other ords, they are uniquely defined. Our procedure also yields an expression for the first Piola Kirchhoff stress, hich in general depends on the deformation mapping, but is shon to converge to a unique definition (proposed by Zimmerman et al., 2) under conditions hen the deformation is sloly varying on the scale of the spatial averaging. An interesting feature of the atomistic first Piola Kirchhoff stress tensor is that it does not include a kinetic part, hich in the atomistic Cauchy stress tensor accounts for thermal fluctuations. We rigorously prove that this effect is implicitly included in the atomistic first Piola Kirchhoff stress tensor through the motion of the particles. Using this result, e address the controversy surrounding the presence of kinetic terms in the atomistic Cauchy stress tensor discussed by Zhou (23), Liu and Qiu (29), and Xu and Liu (29). In particular, e identify that the controversy arises due to the lack of distinction beteen Lagrangian and Eulerian formulations. The paper is organized as follos. In Section 2, e briefly revie the material and spatial descriptions of fields in continuum mechanics. In Section 3, e describe the original Murdoch Hardy procedure to obtain spatial fields, and define the material fields using distribution pull-backs, folloed by a derivation of various expressions for material fields. In addition, e discuss various properties of the resulting material fields, and the controversy regarding the kinetic A deformation gradient field FX ( ) that satisfies curl F = is called a compatible deformation field.

3 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) contribution to the atomistic stress tensor. In Section 4, e demonstrate the validity of the obtained first Piola Kirchhoff stress definition through a numerical example involving the finite deformation of a slab containing a notch under tension. We conclude ith a summary in Section Material and spatial description of fields in continuum mechanics A shape or configuration of a body is represented as an open subset of the three-dimensional Euclidean space 3.A reference configuration, denoted by an open set c 3, is chosen to represent a convenient fixed state of the body to hich other configurations are compared. The subscript c in c and in other variables defined for the continuum model is used to differentiate them from their analogs defined for an atomistic model in later sections. The position of an arbitrary material point in the reference configuration is denoted by X. A time-dependent deformation of the body is described by a C continuous deformation mapping φ + t : c c c hich maps the reference configuration c to a current configuration t c at time t. Thus x()= t φ c ( X,, here x( is the position at time t of the material point located at X in the reference configuration. Continuum fields defined in the reference and the current configuration are commonly referred to as the material and spatial fields, respectively (see for example Tadmor et al., 22). The to descriptions are related to each other through a pull-back operation ith respect to the deformation mapping, g ( X, = g ( φ ( X,,, ( 2.) c c c here ğ c denotes the material description of an arbitrary continuum field hose spatial description is denoted by g c. In addition to fields, the balance las of continuum mechanics can also be described in the reference or current configurations. For example, the continuity equation (conservation of mass) in the current configuration is given by ρ c t + div ( ρ v ) =, x c c here ρ c and v c denote the spatial mass density and the velocity fields, respectively. The latter is given by ( ) = ( φ vc x, t vc ( x,,, here v = φ ( X, c c is the material velocity field and φ is the inverse deformation mapping. The c c continuity equation in the reference configuration is given by ρ = J ρ, ( 2.3) c c c here ρ c is the reference mass density (mass per unit reference volume) hich is independent of time, J = detf c c is the Jacobian, and Fc = X φ is the deformation gradient. c The equations of motion in the current configuration are given by ρ σ + = ( v c c div ) x c bc + div x( ρ v v ), c c c t here σ c denotes the Cauchy stress and b c denotes the body force field per unit volume. In the reference configuration the equations of motion are div P + b = ρ a, ( 2.5) X c c c c here P c denotes the first Piola Kirchhoff stress tensor, and a c = φ ( X, denotes the material acceleration field. The relationship beteen the first Piola Kirchhoff stress tensor and the Cauchy stress tensor (called the Piola transformation) is c given by T c c c c P = J σ F. ( 2.6) In the next section, e discuss the material and spatial descriptions of continuum fields for atomistic systems. ( 2.2) ( 2.4) 3. Material and spatial descriptions of continuum fields in atomistics Continuum fields for atomistic systems can be obtained using the Irving Kirkood (Irving and Kirkood, 95) or Murdoch Hardy (Hardy, 982; Hardy et al., 22; Murdoch and Bedeaux, 993) procedures. In this paper, e focus on the Murdoch Hardy frameork, although a similar derivation can be carried out for the Irving Kirkood procedure. An atomistic body is modeled as a collection of N classical interacting point particles. We use the notation m, x and v to denote the mass, position and velocity of particle, respectively. A configuration of the atomistic body is given by the positions of particles in 3. A reference configuration is chosen to represent a fixed state of the atomistic body to hich other configurations can be compared. The position of an arbitrary particle in the reference configuration is denoted by X. A time-dependent deformation of the atomistic body is described by a C continuous deformation mapping φ(,, hich

4 62 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) maps the position X of an arbitrary particle in the reference configuration to a position x in the current configuration at time t. (The nature of this mapping is the subject of discussion belo.) The velocity of a particle at time t is then given by v()= t φ ( X,. ( 3.) We assume that the particles interact through a potential energy function = ( x,, x ), ( 3.2) N hich is divided into to parts, an internal part int associated ith short-range particle interactions, and an external part ext associated ith long-range interactions. For the remainder of this paper for simplicity, e assume that ext =. The force on particle is given by f = x. ( 3.3 ) Continuum fields can no be obtained using the Murdoch Hardy procedure. 3.. A brief description of the Murdoch Hardy procedure In the Murdoch Hardy procedure, continuum fields are defined as direct spatial averages of microscopic variables. The 3 spatial averaging is performed ith respect to a eighting function : + that has a compact support and satisfies the normalization condition, 3 ( x) dx =. For example, the mass density, momentum density, and the body force fields are defined as ρ ( x, m ( x x ), ( 3.4) p ( x, m v( x x ), b ( x, b ( x x ), ( 3.5) ( 3.6) here x, v and b are functions of time, ranges from to N, and the subscript on the left indicates the role of the eighting function. The body force b in (3.6) is defined as. Since e have assumed =, it follos that b =. The velocity field v is defined as b x ext p ρ ( x,, if ρ ( x,, v( x, otherise. ( 3.7) In order to obtain a definition for the atomistic Cauchy stress tensor, the definitions given in (3.4) and (3.7) are substituted into the equations of motion (see (2.4)) to obtain an expression in divergence form for σ, given by div σ ( x, = f ( x x) div m ( v v ) ( x x ), x rel x rel rel here f is defined in (3.3), and v ( x, v() t v ( x,. From (3.8), it is clear that the atomistic Cauchy stress tensor has to contributions: σ = σ, v + σ,k, ( 3.9) here σ, v and σ,k are commonly referred to as the potential and kinetic parts of the stress tensor, respectively. The kinetic part is given by σ ( x, = m ( v v ) ( x x),,k rel rel ext ( 3.8) ( 3.) and the potential part of the stress tensor satisfies the equation, div σ ( x, = f ( x x ). x v, ( 3.) Clearly, there are many candidates for σ, v that satisfy (3.). The most common is 2 2 The expression for σ v, given in (3.2) is obtained from (3.) in to steps. In the first step, each force f is decomposed into forces f such that (4) is satisfied, to obtain

5 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) σ ( x, = f ( x x ) b( x, x, x ), v,, < here, < N N = = +, the function b is the bond function defined by b( x, x, x ) (( s) x + sx x) ds, and the forces s= f f = f, satisfy the conditions: ( 3.2) ( 3.3) ( 3.4a) f = f. ( 3.4b ) If in addition, the forces f are assumed to be central (i.e. f x x), then the resulting stress tensor is alays symmetric. Although σ, v is commonly referred to as the Hardy stress tensor, e refer to it as the Cauchy Hardy stress tensor to differentiate it from the referential stress tensors defined later. An important feature of the fields ρ, p, and σ, v obtained in the Murdoch Hardy procedure, hich ill be used later in obtaining pull-back quantities, is that they can be ritten as a convolution of the eighting function ith a corresponding distribution. 3 In other ords, 4 ρ ( x, = ρ, p ( x, = p, σ ( x, = σ, ( 3.5) v, v here is the convolution operator, and the distributions ρ and p are given by ρ( x, = m δ( x x ), px (, = m vδ( x x), ( 3.6) ( 3.7) respectively, ith support 5 on the current configuration of the particles { x: =,, N}. Further, σ v in (3.5) is a distribution given by σ ( x, = f ( x x ) δ(( s) x + s x x ) ds, v, s= < ( 3.8) ith support on the current bonds of the system. We define a current bond as a line joining any pair of particles and in the current configuration, such that f. Therefore, the support of σ v is given by { sx + ( s) x : s [, ]}., : f ( 3.9) By riting the spatially averaged fields ρ, p and σ, v in convolution form, e separate out the role of the eighting function, and thereby arrive at the distributions ρ, p and s that capture the physics and by definition exist ithout reference to a eighting function. It is orth noting that the fields v and σ,k cannot be ritten in a convolution form, or in other ords, their corresponding distributions do not exist. (footnote continued) div xσ v, = f ( x x)., < In the second step, the right-hand-side of (3.) is recast as an integral of an anti-symmetric function folloed by an application of a lemma due to Noll (955). See Admal and Tadmor (2) for a derivation of this and various other definitions of σ v,, such as the virial, Tsai or the doubly-averaged stress tensor, and Admal and Tadmor (26) for a discussion on the non-uniqueness of the atomistic Cauchy stress tensor and its interpretation from a continuum mechanics viepoint. 3 A distribution is a linear map from the space of smooth functions ith compact support to the real line. 4 See Hörmander (99) for the formal definition of a convolution of a function ith a distribution. We do not require such a general definition. This is because the distributions ρ, p and σ v, defined in (3.6) (3.8), respectively, are derived from a Dirac delta distribution on hich algebra can be carried out as if it ere a function. Therefore, e use the folloing definition for the convolution of to smooth functions u and v that have a compact support in 3, u v( x) u( x y) v( y) d y, 3 here v is replaced by distributions ρ, p or σ v. 5 Intuitively, the support of a distribution is the closed set on hich the distribution is concentrated. See Hörmander (99) for a rigorous definition.

6 64 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) The distributions defined in (3.5) play an important role in obtaining the material definitions of continuum fields for atomistics. In the next section, e sho ho the spatial fields obtained using the Murdoch Hardy procedure can be pulledback to obtain referential forms. We focus on the reference mass density ρ and the first Piola Kirchhoff stress tensor P appearing in (2.5) A notion of a material point in atomistics In continuum mechanics, material fields are obtained as a pull-back of the spatial fields ith respect to the deformation mapping. In order to pull-back the spatial fields obtained in Section 3., e need a continuum deformation mapping that maps an open subset representing the reference configuration of the atomistic body to an open subset representing the current configuration. In other ords, e need a definition for the shape of the atomistic body. Murdoch (26) proposed a definition for the shape of the body at time t, given by t 3 { x : ρ ( x, }. ( 3.2) Clearly, the set given in (3.2) is an open set that includes the positions of all the particles of the system. In addition, Murdoch proposed a deformation mapping based on his interpretation of a material point for atomistic systems. (We describe this approach here, but adopt a different line of reasoning in our on ork as explained belo.) According to Murdoch, an arbitrary point X defines the position of a material point in the continuum associated ith the reference configuration of the atomistic body. As the atomistic body evolves from its reference configuration, the corresponding t continuum body evolves to at time t. An evolution for a material point X is given by a mapping φ^ (, : 3 that satisfies the ordinary differential equation φ^ ( X, = v (^ φ ( X,,, ( 3.2a) φ^ ( X, ) = X, ( 3.2b) here v, defined in (3.7), depends on the evolution of the atomistic body. A solution to the evolution equation given in (3.2b) has the folloing interpretation: φ^ ( X, gives the position of a material point that started at X at time t¼. The velocity of the point positioned at φ^ ( X, at time t is equal to a local spatial average of velocities of the particles in the neighborhood of φ^ ( X, given by v ( φ^ ( X,,. It is important to note that does not track the motion of particles, i.e. e φ^ do not necessarily have the equality φ^ ( X, = x. If (3.2a) has a solution, then e can obtain a material description for the continuum fields defined in Section 2 by pulling-back the spatial fields obtained in Section 3. ith respect to the deformation mapping φ^. Wedo not follo this approach for the folloing reasons. First, it is not a priori clear under hat conditions the solution maps the set φ^ to t. Second, the solution to (3.2) depends not only on the initial and final configuration of the atomistic body, but also on the path traversed. For example, consider a non-trivial deformation indexed by time t [, ], such that x( ) = x( ) = X ( =,, N). In other ords, the system is deformed and brought back to its original configuration. Using the field generated by this deformation in solving (3.2) results in a deformation that does not necessarily satisfy the condition φ^ φ^ ( X, ) = φ^ ( X,) because φ^ does not track the motion of particles. Finally, from a practical point of vie, solving (3.2) can be computationally expensive. In the next section, e obtain referential distributions as a pull-back of the spatial distributions ith respect to a deformation mapping that tracks particles henever such a deformation mapping exists in principle. We see that some of the resulting material distributions are independent of the choice of the deformation map, and depend only on the reference and current atomistic configuration of particles Derivation of the reference fields In this section e derive referential atomistic distributions, such as the reference mass density ρ, atomistic Piola Kirchhoff stress P, reference linear momentum density p, reference force f, and the reference body force b Pull-back of atomistic spatial fields To proceed e introduce a C deformation mapping φ(, : 3, that tracks the motion of the particles 6 : φ( X, = x (. ( 3.22) This implies that φ ( X, = v(, and φ ( X, = a(. We note that constructing one such mapping can be highly non-trivial hen the atoms are disordered in an atomistic deformation. This is due to lack of simple interpolation theorems for multivariate functions compared to functions of a single variable (de Boor and Ron, 99). Although the condition given in 6 Recall that (3.22) is not satisfied by the deformation mapping φ^ defined in Section 3.2.

7 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) (3.22) does not result in a unique φ, e sho belo that certain distributions are independent of the choice of φ. Given the deformation mapping, e define the pull-backs of the spatial mass density, linear momentum, force, body force, and the potential part of the Cauchy stress distributions: mδ( φ( X, x) if X, ρ ( X, otherise. ( 3.23) px (, m vδ( φ( X, x ) if X B, otherise. ( 3.24) maδ( φ( X, x) if X B, f ( X, t ) otherise. ( 3.25) mbδ( φ( X, x) if X B, bx (, t ) otherise. ( 3.26) σ ( φ( X X σ ( X, = v,, if, v otherise. ( 3.27) We no define the atomistic referential distributions using definitions (3.23) (3.27), and the folloing relations from continuum mechanics: ρ J ( X, if X ρ, otherise, ( 3.28) ( ) JpX, t if X, p otherise, ( 3.29) JbX, t if X, b otherise, ( 3.3) Jf X, t if X f, otherise, ( 3.3) σ T J vf if X, P otherise, ( 3.32) here J( X, det Xφ( X, is the Jacobian. Note that in continuum mechanics the relations (3.29) and (3.3) appear in the folloing alternate forms: ρ v = J ρ v, ( 3.33) c c c c c ρ a = J ρ a, ( 3.34) c c c c c respectively. Therefore, contrary to the analogy beteen (2.3) and (3.28), the analogy beteen (3.33) and (3.29), and (3.34) and (3.3) is not perfect since e do not have a distributional definition for v and ă. Note that the distributions in (3.23) (3.32) are defined using a deformation map. In the next theorem, e sho that some of these distributions (ρ, p, f and b) are independent of the choice of φ. Theorem. Let φ : + 3 be a mapping that satisfies (3.22). The distributions defined in (3.28) (3.3) satisfy the relations ρ ( X) = m δ( X X ), ( 3.35)

8 66 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) px (, = m vδ( X X), ( 3.36) f ( X, t ) = m δ ( )= δ ( ) a X X f X X, ( 3.37 ) bx (, t ) = bδ ( ) X X. ( 3.38 ) Therefore, the above distributions are independent of the choice of φ. Proof. Let D denote the distribution given on the right-hand-side of (3.35). Eq. (3.35) is satisfied if and only if D = ρ, ( 3.39) for all smooth functions ith a compact support. We then have 7 D( X, = m ( X Y)( δ φ( Y, x )( J Y) dy = m( X φ ( y, ) δ( y x ) dy φ( ) = m( X X ) = ρ ( X), here e have performed a change of variable from Y to y ( = φ( Y, ) in the second equality, folloed by the substitution φ ( x, = X in the third equality. Eqs. (3.36) (3.38) follo from a similar argument. The spatial averages associated ith the distribution pull-backs defined in (3.23) (3.26) are ρ = ρ, p = p, f = f, b = b. ( 3.4) We make the folloing observations regarding these measures:. Although ρ, p, f, and b are obtained by convolving ith their distribution pull-backs, Theorem does not imply that ρ = ρ ( φ( X, ), p = p ( φ( X, ), f φ = f ( ( X, t )), or b = b ( φ ( X, t )). Thus ρ, p, f and b are not pull-backs. 2. The relation given in Theorem does not generally imply that Jρ = ρ. This relation only holds if φ is a uniform deformation mapping. This follos from (3.35), and the equality J( ρ ) = ( J ρ ), here J is no a constant. Similarly, the equalities Jp = p, Jf = f, and Jb = b are satisfied only for a uniform deformation mapping. Theorem 2. Let φ : + 3 be a mapping that satisfies (3.22) and the condition that all current bonds are in the image of φ at time t, i.e. { sx ( + ( s) x ( : s [, ]} φ(,., : f ( 3.4) The distribution P, defined in (3.32), satisfies the relation PX (, = f δ(( φ ϒ )( s, X)( φ ϒ ) ( s, ds,, s= < ( 3.42) here ϒ ( s, = ( s) x + sx represents a point indexed by s on the line joining particles and at time t, the relation s, t s, t, t ( φ ϒ )( ) = φ ( Υ ( ) ) is a function composition, and the prime denotes differentiation ith respect to s. Proof. From the definition of σ v in (3.27), e have 7 Note that the integration bounds for the convolution have been changed from 3 to the body in the reference configuration, since D is zero for X.

9 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) σ ( X, = f ( x x ) δ(( s) x + sx φ( X, ) ds v, s= < = f δ( ϒ ( s, φ( X, ) ϒ ( s, ds,, s= < ( 3.43) here e have used ϒ ( s, = x x. Using (3.4), it can be easily shon that the Dirac-delta distribution in the integrand of (3.43) satisfies the relation Jδ( ϒ( s, φ( X, ) = δ( φ ϒ( s, X). ( 3.44) From (3.43), and using (3.44), e obtain, s= < Jσ ( X, = f δ( φ ϒ ( s, X) ϒ ( s, ds. v ( 3.45) Post-multiplying both sides of (3.45) ith T F T, and using, s= < Jσ ( X, F = f δ( φ ϒ ( s, X)( φ ϒ ) ( s, ds v F ϒ ( s, = ( φ ϒ ) ( s,, e obtain the desired result: = PX (,. ( 3.46) By construction, the distributions f, b and P, defined in (3.28) (3.32), satisfy the balance equations in (2.5). Zimmerman et al. (2) have taken an alternate approach to obtaining referential atomistic fields. In Zimmerman et al. (2), the authors carry out the Murdoch Hardy procedure on the reference configuration. Similar to the original Murdoch Hardy procedure, they begin by proposing definitions for ρ, p, f and b for the reference configuration, and obtain a definition for the first Piola Kirchhoff stress tensor that can be expressed in terms of the folloing distribution: uni P = f ( X X ) δ(( s) X + s X X ) ds., s= < ( 3.47) The superscript uni indicates that this definition is correct in the limit of uniform deformation as shon next. Corollary. For a uniform deformation mapping φ, e have uni P = P. ( 3.48) Proof. Under uniform deformation, e observe that φ Υ = ( s) X + sx, i.e. the inverse image of the line joining x and x is equal to the line joining X and X. Therefore, (3.42) simplifies to (3.47). We list a fe properties of the definitions given in (3.42) (3.47).. The support of the distribution P is the inverse image of the set of current bonds defined in (3.9) under the mapping φ. On the other hand, P uni is supported on the reference bonds of the system. We define a reference bond as a line joining any pair of particles and in the reference configuration, such that f. Since f is calculated using the current configuration of particles, the reference bonds change ith time. Therefore, the support of P uni is given by { sx + ( s) X : s [, ]}., : f For non-uniform deformation mappings, the supports of P and P uni differ. Moreover, P and P uni are non-symmetric. uni 2. The spatially averaged fields P P and P P uni are given by P ( X, = f b ( X, X, X ), uni, < P ( X, = f ( X X ) b( X, X, X ),, < here the scalar function b is defined in (3.3) and the vector function b is ( 3.49) ( 3.5) ( 3.5)

10 68 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) b( X, X, X) ( X φ ϒ)( φ ϒ) ds. s= ( 3.52) The definitions in (3.5) and (3.5) satisfy the equations of motion given in (2.5). This can be easily shon using our earlier results (see Admal and Tadmor, 2) on the atomistic stress tensor obtained for curved paths of interaction). 3. Comparing our approach to that of Zimmerman et al. (2), e note that unlike in Zimmerman et al. (2), e do not prescribe definitions for ρ, p, f, and p. Instead, e obtain them as pull-back of the spatial distributions obtained in the Murdoch Hardy procedure. 4. Comparing Eqs. (2.6) and (3.32) e note that the right-hand side of (3.32) includes only the potential part of the atomistic Cauchy stress, hereas the right-hand side of (2.6) includes the entire continuum Cauchy stress. This discrepancy can be addressed by noting that in continuum mechanics, for a body in equilibrium, the deformation gradient F c is independent of time. In contrast, F in (3.32), oscillates about a mean value due to thermal vibrations of the particles. It is precisely these oscillations that lead to a thermal contribution to P. This observation also addresses the controversy surrounding the presence of kinetic terms in the atomistic Cauchy stress tensor. For example, see Zhou (23), Liu and Qiu (29), and Xu and Liu (29) for various discussions on this topic. Based on the results of this ork, e can conclude that much of the confusion arises due to the lack of distinction beteen the Lagrangian and Eulerian formulation. For instance, Zhou (23) argues that the atomistic stress should not include the kinetic contribution, and Liu and Qiu (29) and Xu and Liu (29) argue that the kinetic contribution to the atomistic stress tensor is non-objective. 8 There have been many orks since the publication of Zhou (23) that highlight the role and importance of the kinetic contributions to the atomistic Cauchy stress tensor (see for example Admal and Tadmor, 2; Subramaniyan and Sun, 28). The controversy regarding the presence of the kinetic terms in the atomistic stress tensor can be laid to rest ith a simple experiment that as performed by Zimmerman et al. (2) in hich a face-centered-crystalline copper block is modeled using an EAM potential at a positive temperature. The system is heated to K at constant volume. After equilibrium is attained, the system is under a uniform hydrostatic compressive stress. Since the deformation gradient is identity, e expect the Cauchy and the first Piola Kirchhoff stresses to coincide. Zimmerman et al. (2) have shon that although the kinetic contribution to the atomistic Cauchy stress is quite significant, the total atomistic Cauchy stress and the atomistic first Piola Kirchhoff stress indeed coincide. In other ords, the experiment described above clearly shos that the atomistic first Piola Kirchhoff stress includes the kinetic part through fluctuations in forces. In Appendix A, e sho that the numerical results obtained by Zimmerman et al. (2) can be proved mathematically for a simple case of an ideal gas enclosed in a container. In particular, e sho that P uni and σ exactly agree in a calculation of the Tsai traction for T > and F = I. Note that both P uni = P uni and P = P qualify as definitions for the atomistic first Piola Kirchhoff stress because they satisfy the equations of motion given in (2.5). Although P uni does not in general satisfy the pull-back relation in (3.32), it has an advantage over P in that it does not depend on the choice of φ hose construction can be highly non-trivial. The use of P uni can be justified under conditions here the deformation is close to uniform (i.e. constant deformation gradient F) on the scale of the eighting function. In this case, P uni approximately satisfies (3.32) up to some order of the gradient of F. It is easy to see that the definition of the first Piola Kirchhoff stress tensor also results in the folloing definition for the second Piola Kirchhoff stress tensor S,, s= < SX (, = f ( X X) δ(( s) X + sx X) ds, here f ( X, F f is the pull-back of the force (3.53) satisfies the condition S = FP uni. Since f depends on the mapping φ, so does S. f ( 3.53) under a mapping φ that follos particles. The definition given in 4. Numerical example In this section, e compare the atomistic first Piola Kirchhoff stress tensor P uni given in (3.5), to the continuum first Piola Kirchhoff stress P c using a numerical example. In order to capture the asymmetry of the first Piola Kirchhoff stress e 8 The authors in Liu and Qiu (29) and Xu and Liu (29) claim that the kinetic part of the atomistic stress expressed using average velocities of particles is non-objective by noting that it depends on the choice of the eighting function. They observe in their simulation that for small averaging domain sizes, the kinetic part is quite large and it seems to decrease and converge as the averaging domain size is increased. Based on this observation, the authors conclude that the atomistic stress depends considerably on the choice of the averaging domain, and therefore it is non-objective. We believe this is not a conclusive evidence because the authors have not explored the effect of the averaging domain on the potential part of the stress tensor, and the spatial variation of the potential and kinetic components of the total stress field. In our earlier ork (Admal and Tadmor, 2), e studied the delicate balance beteen the kinetic and potential parts of the atomistic stress tensor. We demonstrate (see Fig. 9 in Admal and Tadmor, 2) ho the kinetic part of the Tsai stress tensor oscillates as e move across the lattice planes. At the same time, the potential part also oscillates in a ay so that the total stress remains nearly constant.

11 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) Fig.. A schematic of the geometry of the studied boundary-value problem, ith displacement boundary conditions enforced on parts of the boundary shon in red. (For interpretation of the references to color in this figure caption, the reader is referred to the eb version of this paper.) choose an example ith large geometric nonlinearity. We study a system that exhibits this property a to-dimensional plane strain slab containing a rounded notch under tension. The slab is a 2 A 2 A square ith a notch of size 4 A as illustrated in Fig.. It consists of single crystal Al in the face-centered cubic (fcc) structure ith the x, y, z axes oriented along the [ ], [ ], [ 2] crystallographic directions (see Fig. ). The notch is stretched in tension by displacement boundary conditions applied at the top and bottom left corners of the model as shon in red in Fig.. Rigid-body motion is prevented by constraining a portion of the right edge of the slab. Periodic boundary conditions are applied in the out-of-plane (z) direction. The system is studied at zero temperature ( T = K). The above problem is simulated in to different ays: () an atomistic molecular statics simulation is performed from hich P uni is computed; (2) a continuum finite element simulation using a constitutive relation consistent ith the atomistic model in () is performed from hich P c is computed. 4.. Atomistic simulation of a notched slab In the atomistic simulation, the notched slab system is modeled using a collection of Al atoms. 9 An embedded-atom method (EAM) interatomic potential due to Ercolessi and Adams (994) archived in OpenKIM (Elliott, 24a,b; Tadmor et al., 2) is used to model the interaction beteen the Al atoms. The equilibrium fcc lattice parameter at K calculated for this potential is a = A. The reference (unloaded) configuration of the notched slab (shon in Fig. ) is obtained by cutting out a 2 A 2 A square from a perfect fcc single crystal ith the orientation described above and removing all atoms that fall inside the notch. The thickness in the z direction is taken equal to the minimum crystallographic repeat distance in that direction hich is a 3/2 = A. The resulting system consists of,56 atoms. The boundary conditions shon in Fig. are applied by identifying the atoms lying on the part of the boundary on hich the displacements are applied. The maximum magnitude of the displacements ( A) on the upper-left and the loer-left corners of the system is reached in 5 steps of.2 A each. At each time step, the displacement boundary conditions are updated, and the system is evolved by minimizing its potential energy ith respect to the positions of the atoms using a conjugate gradient algorithm (see for example Tadmor and Miller, 2). This corresponds to a molecular statics simulation. The atomistic first Piola Kirchhoff stress field is obtained using the expression given in (3.5). The forces f in (3.5) are given by f = r EAM x x, r here EAM denotes the EAM potential energy hich is a function of the distances beteen atoms r = x x, and x and x are the positions of atoms and in the current configuration obtained at the end of the atomistic simulation. We compute to stress fields, P uni and P uni, corresponding to to different uniform eighting functions, ( X ) = ^ ( X,2.5A ) and ( X ) = ^ ( X,5A, ) here cr if X < R ϵ, ^ R X ( X, R ) = cr cos ϵ π if R ϵ< X < R, 2 otherise. ( 4.2) The support of the eighting function is a sphere of radius R, and c R in (4.2) is chosen such that ^ is a normalized function. ( 4.) 9 Given the physical nature of the system, e sitch from the more generic term particles used so far to atoms.

12 7 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) We use to eighting functions ith different supports ( R = 2.5 A and R = 5 A, ith ϵ =.2 R) to explore the effect of eighting function on the atomistic stress at the surfaces Continuum simulation of a notched slab In order to obtain the continuum first Piola Kirchhoff field P c, a continuum simulation must be performed. The comparison to the atomistic results is only meaningful if the continuum constitutive relation is consistent ith the atomistic simulation. To this end, e apply the local Quasicontinuum (QC) method of Tadmor et al. (996). Local QC corresponds to a finite deformation (nonlinear) finite element simulation in hich the constitutive relation is obtained from an atomistic model using the Cauchy Born rule (see for example Tadmor and Miller, 2). Briefly, the constitutive response at an integration point is obtained by applying the deformation gradient at that point to a periodic unit cell of the underlying crystal structure and computing the energy from the resulting current configuration of atoms using an interatomic potential energy function. The strain energy density follos as: Wc( Fc) = Eunit( Fc ), Ω ( 4.3) here Ω is the unit cell volume in the reference configuration, and E unit is the energy per unit cell subjected to a uniform (affine) deformation F c. In our case, E unit is computed using the same EAM potential used in the atomistic simulation. The Fig. 2. A plot of the xx, yy and zz components of the continuum and atomistic first Piola Kirchhoff stress tensor fields in units of ev/a 3. The atomistic stress is given for to eighting functions, and.

13 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) first Piola Kirchhoff at the integration point is obtained by differentiation, = W c Pc. F ( 4.4) c See Tadmor et al. (996) or Tadmor and Miller (2) for more details and specific local QC expressions for W c and P c for the EAM potential. For the notched slab boundary-value problem, a to-dimensional local QC simulation is performed. As in the atomistic simulation, periodic boundary conditions are applied in the out-of-plane direction. The finite element mesh consists of linear triangular elements. The mesh in the reference configuration is uniform and symmetric about the y¼ plane. The nodes of the mesh coincide ith the positions of the atoms. In QC terminology this is referred to as a fully-refined mesh. The local QC simulation is performed in a similar manner to the atomistic molecular statics simulation. The boundary conditions are incremented in steps of.2 A and the potential energy of the system is minimized using the previously converged result as an initial guess. At the end of the simulation, the first Piola Kirchhoff field is computed at the integration points (element centroids) using (4.4) Comparison of the continuum and atomistic first Piola Kirchhoff stress The stress profiles obtained from the continuum solution are compared to those obtained from the atomistic solution in Figs Based on these results e make the folloing observations: Fig. 3. A plot of the xy, yz and xz components of the continuum and atomistic first Piola Kirchhoff stress tensor fields in units of ev/a 3. The atomistic stress is given for to eighting functions, and.

14 72 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) Fig. 4. A plot shoing the asymmetry of the continuum and atomistic first Piola Kirchhoff stress tensor fields in units of ev/a 3. The atomistic stress is given for to eighting functions, and.. Figs. 2 4 compare the atomistic first Piola Kirchhoff stress tensor corresponding to the eighting functions and ith the continuum first Piola Kirchhoff stress. Fig. 5 shos an error plot of the atomistic first Piola Kirchhoff stress relative to the continuum first Piola Kirchhoff stress in the bulk region of the system (i.e. aay from surfaces). From Figs. 2 5 e see that the atomistic first Piola Kirchhoff stress tensors are in good agreement ith the continuum first Piola Kirchhoff stress in bulk regions of the system. Moreover, since the support of is larger than the support of, the noise in P uni is loer than the noise in P uni. 2. The effect of surfaces on the atomistic first Piola Kirchhoff stress are clear in the xx, yy and zz plots shon in Fig. 2. In particular, e see that the presence of surfaces results in a localized surface stress ith a magnitude that is comparable to the bulk stress. We also see from Figs. 2 4 that the localization decreases as the support of the eighting function increases. 3. The surface effect on the xx component is negligible on planes perpendicular to the x-axis, such as the region at the rounded surface of the notch and the outer boundary surfaces at x =± A. Similarly, the effect of surfaces on the yy component is negligible on the y =± 2 A surfaces of the notch. From the above observations, e conclude that the definition of the atomistic first Piola Kirchhoff stress P uni given in (3.5) is in good agreement ith continuum first Piola Kirchhoff stress. This definition can be used to obtain the first Piola Kirchhoff stress tensor in atomistic simulations ithout having to define a deformation mapping as long as the deformation is close to uniform on the scale of the eighting function. Here, e are referring to noise as the magnitude of high frequency components of the atomistic stress that do not exist in its continuum version.

15 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) Fig. 5. An error plot of the xx, yy, zz, xy, yz and xz components of the atomistic first Piola Kirchhoff stress tensor fields relative to the continuum stress tensor fields in units of ev/a 3. In order to remove the effect of surfaces, the plot only shos errors in the bulk. 5. Conclusion In this paper, e obtained material fields for atomistic systems by defining referential distributions as pull-backs of the spatial distributions obtained in the Murdoch Hardy procedure ith respect to a deformation mapping that tracks particles. The fields resulting from this procedure are the reference mass density, reference momentum density, reference body field,

16 74 N. Chandra Admal, E.B. Tadmor / J. Mech. Phys. Solids 89 (26) and the first Piola Kirchhoff stress. These fields identically satisfy the material form of the balance las and the pull-back relations ith their respective spatial fields. One of the key features of this procedure is that the reference mass density, reference momentum density and the reference body force fields are independent of the choice of the deformation map, and only depends on the reference and final configuration of the atomistic system. In addition, e sho that although the first Piola Kirchhoff stress distribution obtained in this procedure depends on the choice of the deformation map, under small variations in the deformation gradient, it is close to an alternate definition proposed by Zimmerman et al. (2) that only depends on the initial and final configuration of particles. In other ords, e have obtained referential fields appearing in the balance las ithout explicitly constructing a deformation map. An interesting observation from our derivation is that the atomistic first Piola Kirchhoff stress tensor does not have a kinetic contribution, hich in the atomistic Cauchy stress tensor accounts for thermal fluctuations. In Appendix A, e sho through an example of an ideal gas that this effect is included in the atomistic first Piola Kirchhoff stress tensor through the motion of particles. In addition, since the bond function present in the atomistic first Piola Kirchhoff stress is defined using the reference configuration, it can be evaluated once and can be reused as the system evolves. This provides a significant computational advantage relative to the Cauchy stress hich requires tracking of changes in the forces, bond function, and the velocities of the particles. Instead evaluating the first Piola Kirchhoff stress tensor only requires calculation of the average forces. If the Cauchy stress is needed, it can be computed from (3.32) using a deformation gradient suggested by Zimmerman et al. (29). The validity of our proposed definition for the first Piola Kirchhoff stress is demonstrated in Section 4 ith a numerical example involving the finite deformation of a notched Al slab under tension. The first Piola Kirchhoff stress in (3.5) computed from a molecular statics simulation is found to be in good agreement ith the results of a nonlinear finite element simulation using a Cauchy Born constitutive relation (i.e. a local QC simulation) for the same geometry and loading. A possible application of the atomistic first Piola Kirchhoff is its use in specifying traction boundary conditions for large deformation problems. For instance, consider an atomistic system deformed using traction boundary conditions defined ith the Cauchy stress. Since the Cauchy stress describes traction per unit area in the current configuration, the forces imposed on the boundary atoms have to be continuously altered as the area of the boundary changes. This requires the definition of the area of an atomistic boundary, hich results in an extremely non-trivial control of boundary forces. In contrast, tractions specified using the atomistic first Piola Kirchhoff stress are easier to impose since they are measured ith respect to a fixed reference configuration. An open source Fortran 9 program called MDStressLab for computing the first Piola Kirchhoff stress tensor (and other useful stress measures) is available at (along ith documentation and examples). Given the positions and velocities of the atoms and a suitable KIM-compliant interatomic model (see MDStressLab calculates fields of the Cauchy and referential (Piola Kirchhoff) versions of the virial, Tsai and the Hardy stress tensors. In addition, MDStressLab computes a decomposition of the atomistic stress tensor into an irrotational part and a solenoidal part. Acknoledgments This ork as partly supported by the National Science Foundation under Aards no. PHY and DMR-482. Dr. Admal ould like to gratefully acknoledge the support provided by the Institute for Pure and Applied Mathematics at the University of California Los Angeles here a part of this ork as carried out. Appendix A. Thermal contribution to the atomistic first Piola Kirchhoff stress tensor in an ideal gas Recall that the atomistic Cauchy stress σ defined in (3.9) has a kinetic contribution hereas the definition of the first Piola Kirchhoff P uni in (3.5) does not. This leads to an apparent contradiction hen considering the relation beteen these to measures. In particular, hen a system is not deformed, e expect σ = P uni. To investigate this issue, e consider the example of an ideal gas enclosed in a container at non-zero temperature. Since an ideal gas is modeled as a collection of non-interacting particles, the potential part of the Cauchy stress tensor aay from the container boundaries is identically zero. This enables us to focus on the kinetic contribution alone. We sho belo that although P uni does not have an explicit kinetic-like term, the time-average of P uni recovers the kinetic contribution. Consider an ideal gas confined ithin a cubic container. The ideal gas particles do not interact ith each other, but do interact ith the particles of the container through a strong short-range potential hich keeps the gas particles strictly inside the container ithout diffusing into the container alls. The particles are placed in contact ith a thermal reservoir at the set temperature (for example using a numerical thermosta and are evolved until equilibrium is established. Suppose e are no interested in calculating the stress at the center of the cube. In order to do this, e choose a eighting function hose support is centered at the center of the cube. Assume that the size of the support is small, so that it is aay from See Admal and Tadmor (26) for the significance of this decomposition.