nano mechanics and materials (theory, multiscale methods and applications) || lattice mechanics

3

Lattice Mechanics

Crystalline solid states are of particular interest to the nanoscale researcher. Amorphoussystems that display nonrepetitive, irregular patterns of spatial atomic arrangements are inmany respects closer to liquids. The computer modeling and characterization of amorphousmaterials requires the standard particle dynamics formulation of Chapter 2, as in the caseof liquids and gaseous domains. When referring to solid states, one usually implies thosehaving regular crystal structures. Crystalline bodies are usually represented by an individualmonocrystal or by a polycrystal formed by a large number of smaller monocrystals. Crys-talline materials provide further opportunities in terms of computer simulations, analysisand characterization. Another type of spatially regular system is polymer molecules/chains,formed by a large number of repetitive atomic cells. Chemical bonds between atoms ofa given cell as well as cross-cell bonding provide stability for the entire polymer chain.Classical particle mechanics can be made to utilize the intrinsic symmetries and transla-tional invariance of regular lattices , such as crystals and polymer chains. These applicationsof particle mechanics comprise the classical theory of lattice mechanics reviewed in thischapter.

The issues discussed in this chapter relate to the possibility of predicting acoustic anddispersion properties of a regular lattice, finding dynamic and quasi-static solutions forlattices with a vast number of degrees of freedom in closed form, evaluating responsesolutions for a small group of atoms without solving for the entire domain, eliminatingunnecessary degrees of freedom in numerical simulations, and others.

3.1 Elements of Lattice Symmetries

In the general theoretical consideration of regular polymer and crystal lattices, one usuallyimplies an infinite structure in order to eliminate the effects of free surfaces. Mechanicalproperties of a regular lattice are normally associated with two types of lattice symmetries:(1) translational symmetry , which is a spatially periodic pattern of identical groups of atoms,ions or molecules and (2) point symmetry , that is, a geometrical order of the individualperiodic cells.

Nano Mechanics and Materials: Theory, Multiscale Methods and Applications W. K. Liu, E. G. Karpov and H. S. Park 2006 John Wiley & Sons, Ltd. ISBN: 0-470-01851-8

38 LATTICE MECHANICS

a2

a2

a1

a1 a1

a2

a2 a1

Figure 3.1 Ambiguity of the translational vector basis.

The smallest part of the lattice whose spatial repetition forms the entire lattice underanalysis is called the elementary cell . Such a cell may include one or several particles(atoms, ions or molecules). Owing to translational symmetry, each particle from one ele-mentary cell can be associated with an identical particle from another elementary cell. Thespatial locations of such identical particles are determined by lattice vectors, or lattice sitevectors ,

n = ρ +I∑

i=1

niai , ni = 0, ±1, ±2, . . . (3.1)

where I is the dimensionality of the lattice; at I = 1, 2 and 3, the lattice is called one-dimensional (beamlike), two-dimensional (platelike), and three-dimensional, respectively.The triplets of lattice site parameters (n1, n2, n3) uniquely define the elementary cells.The noncoplanar vectors ai are the lattice translation vectors and ρ is the lattice originvector that shows positions of particles in the cell (0, 0, 0) with respect to the origin ofthe Cartesian axes. The components of the vectors ρ and ai are given in groups of threeelements for each particle in the elementary cell; for example, vector ai = (x, y, 0, x, y, 0)

describes a translation of diatomic cells in the x–y Cartesian plane.The entire set of lattice site vectors (3.1) defines the equilibrium positions of all parti-

cles that comprise the given lattice structure. The choice of translation vectors in (3.1) isnot unique; Figure 3.1 illustrates multiple possibilities for ai in a two-dimensional lattice.Typically, the translation vectors are chosen to have the smallest possible absolute values.The lattice translational symmetry manifests itself in the fact the translation vectors do notdepend on the lattice site parameters ni .

3.1.1 Bravais Lattices

Besides translational symmetry, the lattice structure may also feature point symmetries,that is, invariance with respect to reflection in a point (inversion), rotation about an axisat a specific angle, and reflection in a plane. Then, the symmetric structure is said to havea center, axis or plane of symmetry, respectively. The term “point” symmetry is due tothe fact that the relevant symmetry transformation leaves at least one point of a symmetric

LATTICE MECHANICS 39

object unaltered. A complete set of all possible point transformations, including the identitytransformation, forms the symmetry group of a Bravais lattice.

There exist 6 two-dimensional and 14 three-dimensional standard Bravais lattices. How-ever, there is no standard convention regarding one-dimensional lattices, whose pointsymmetry is usually trivial. They may have one transversal and sometimes one longitu-dinal plane or axis of symmetry; see examples depicted in Figure 3.2. In many cases,beamlike lattices have no particular point symmetries, for example, the second example inFigure 3.2.

The standard two-dimensional Bravais lattices are depicted in Figure 3.3. Their pointsymmetry is determined by the symmetry of a parallelogram with the angle α between thesides a and b. Trigonal and orthorhombic lattices are alternatively called the rhombohedral(rhombic) and rectangular lattices ; they have the point symmetry of a rhombus and arectangule, respectively.

The three-dimensional Bravais lattices are subdivided conventionally into seven sys-tems: cubic, tetragonal, trigonal, hexagonal, orthorhombic, monoclinic and triclinic (seeTable 3.1). The point symmetry of lattices from each of these systems is determined bythe symmetry of a parallelepiped with the edges a, b and c that form the angles α, β

and γ , as depicted in Figure 3.4. Each system includes one or several lattice types: prim-itive (P), base-centered (C ), body-centered (I ) and face-centered (F ). For example, thereexist simple cubic, body-centered cubic (bcc) and face-centered cubic (fcc) lattices. Thecubic lattices have the highest symmetry; they are typical for metallic monocrystals, simplemetallic compounds and frozen monoatomic gases. The simple cubic lattice is observed indiatomic crystals, such as CuZn, CsCl and NaCl. The bcc lattice structure is typical for thealkali elements, such as Li, Na, K and Rb, as well as for Fe, Cr, V and other metals. Thefcc structures occurs in Ca, Ni, Sr, Cu, Pb, the noble metals (Ag, Au, Pd, Ir, Pt), and thefrozen noble gases (Ne, Ar, Kr, Xe, Rn). The complete set of the 14 standard lattices isshown in Figure 3.5.

(a)

(b)

(c)

(d)

n–2 n–1 n+1 n+2n

Figure 3.2 Examples of periodic one-dimensional lattices; index n numbers the elementarycells.


b b

a

a ≠ b, a= p/2

a = b, a= p/2

a ≠ b, a= p/2 a ≠ b, a= p/2 ≠ p/3

a = b, a= p/3 a = b, a≠ p/2 ≠ p /3

aa

b

aab

aa

b

aa

ab

aa

(a) Square

(d) Orthorhombic (e) Centered orthorhombic (f) Monoclinic

(b) Hexagonal (c) Trigonal

Figure 3.3 Two-dimensional Bravais lattices.

Table 3.1 Classification of the three-dimensional Bravais lattices

System Restrictions Lattice types

Cubic a = b = c, α = β = γ = π/2 P, I, F

Tetragonal a = b �= c, α = β = γ = π/2 P, I

Trigonal a = b = c, α = β = γ �= π/2 P

Hexagonal a = b �= c, α = 2π/3, β = γ = π/2 P

Orthorhombic a �= b �= c, α = β = γ = π/2 P, C, I, F

Monoclinic a �= b �= c, α = β = π/2 �= γ P, C

Triclinic a �= b �= c, α �= β �= γ P

3.1.2 Basic Symmetry Principles

A better understanding of the mechanical properties of lattice structures can be obtained bytaking their symmetry into account. The symmetry arguments often provide intuitive testsfor the adequacy of the mathematical description and the accuracy of numerical calculations.The relevant symmetry-based arguments are based upon the following basic principles:

Principle 1. Geometric symmetry of a mechanical system implies the correspondinginvariance of the mathematical models. Mathematical objects, such as functions, matri-ces and operators that are associated with inherent physical properties of the system (thephysical observables of the system) are invariant with respect to the same symmetry


aa

b

c

gb

Figure 3.4 Relative orientation of parallelogram edges.

(a) Cubic P

(d) Tetragonal P

(h) Orthorhombic P

(l) Monoclinic P (m) Monoclinic C (n) Triclinic

(i) Orthorhombic C (j) Orthorhombic I (k) Orthorhombic F

(e) Tetragonal I (f) Trigonal (g) Hexagonal

(b) Cubic I (bcc) (c) Cubic F (fcc)

Figure 3.5 Three-dimensional Bravais lattices.


P P

PP

Figure 3.6 Rectangular lattice under symmetric loading. Empty circles indicate the preloadconfiguration.

transformations that are typical of this system. For example, the potential energy function ofa rotationally symmetric lattice must be invariant with respect to the same set of rotations ofa global coordinate system. Correspondingly, the elastic tensor of an equivalent continuummedia should have the same symmetries as the original lattice structure.

Principle 2. Symmetric causes, when acting on symmetric entities, result in symmetriceffects with the same type of symmetry. One illustration of this principle is depicted inFigure 3.6. Here, a rectangularly symmetric load applied to a rectangular platelike latticeresults in a deformation pattern that has the same symmetry as the original nondeformedBravais lattice, namely, one horizontal and one vertical plane of symmetry, and one centerof symmetry.

3.1.3 Crystallographic Directions and Planes

Translation vectors ai , i = 1, 2, 3, in the equation (3.1) define the major crystallographicdirections/axes of a three-dimensional periodic lattice. In case of a multiatomic unit cell,such a direction is determined with respect to Cartesian axes by three repetitive componentsof ai . The vectors determining the major crystallographic directions provide a basis in athree-dimensional vector space, and the relevant basis vectors are denoted by [100], [010]and [001]. This basis is used to span an arbitrary vector of three integer components [nml]that defines a specific crystallographic axis in the lattice structure.

Any plane orthogonal to the vector [nml] and passing through nonempty lattice sites,is called the crystal plane (nml). Curly brackets around indices, {hkl}, are used in relationto a set of coplanar crystal planes equivalent by translation symmetry. For example, a setof cube faces shown in Figure 3.7 is {100}.

3.2 Equation of Motion of a Regular Lattice

Dynamic solutions for the system of particles that comprise a stable lattice structure can besought according to the Lagrangian formalism presented in the previous chapter. Similarly,


(100) (111) (101)

Figure 3.7 Some important crystal planes in the face-centered cubic lattice.

the lattice atoms, ions or molecules are exerting internal forces, the values of which aredetermined by the instantaneous values of the lattice potential energy. This energy is con-sidered to be a known function of the spatial configuration of lattice particles only. Overthe course of the system’s dynamics, the structure of the interatomic potentials (2.46) isnot perturbed by any possible changes in the internal quantum states of the vibrating lat-tice particles, as well as by any possible electronic excitations. This assumption is knownas the adiabatic approximation , a major approximation of classical lattice dynamics andmechanics.

3.2.1 Unit Cell and the Associate Substructure

In studying the mechanical properties of lattice structures, it is convenient to choose a formalmathematical unit cell in the lattice. The unit cell is an arbitrary part of the lattice, whosespatial translations in arbitrarily fixed directions and distances form the whole lattice underanalysis. The unit cell is a more general concept than the elementary cell (see Section 3.1);it can be larger/smaller than the elementary cell, or identical to it.

In accordance with translational symmetry, each particle from one unit cell can beassociated with an identical particle from another unit cell, just as in the case of elementarycells. The spatial locations of the repetitive unit cells are determined by the lattice sitevectors, n such as (3.1). For a given unit cell, we introduce the time-dependent configurationvector rn(t), which is composed of the radius vectors of all particles in the unit cell thatcorrespond to the lattice site vector, as well as the vector of displacements of particles fromtheir equilibrium positions,

un(t) = rn(t) − n (3.2)

The discussion below concerns the displacement vectors only, and therefore we will referto the subscript lattice site vector simply as a triplet of integer numbers, n ≡ (n1, n2, n3),where ni = 0, 1, . . . , Ni is the lattice site index . In other words, un(t) ≡ un1,n2,n3(t). Insome cases, we will use the following equivalent notations: n ≡ (n, m, l), where n =0, 1, . . . , N , m = 0, 1, . . . , M , l = 0, 1, . . . , L, and un(t) ≡ un,m,l(t). For one- and two-dimensional lattices, we have n ≡ n and n ≡ (n1, n2) ≡ (n, m), respectively. Since thevector n uniquely defines the unit cells, the indices ni can be used for their successivenumbering.

The use of a unit cell, nonidentical with the elementary cell, can be dictated by theprovisos of a given problem. These include the interatomic potential range, domain shape,


(0,1)

(0,1)

(0,0)

(0,0) (1,0)

(1,0)

(1,1)

(0,1) (1,1)

(0,0)

(a) (b) (c)

p/3

(1,0)

a2

a2

a2

a1 a1 a1

(1,1)

Figure 3.8 Three possible choices of the unit cell in a 2D hexagonal lattice. The unit cellsare bounded by the dashed lines.

symmetry of the equations of motion, and convenience in applying boundary and interfacialconditions. For example, consider a hexagonal two-dimensional lattice with a natural choiceof the elementary cell, composed of a single lattice atom, as shown in Figure 3.8(a). Next,assume that some special circumstances require numbering of the repetitive lattice cells tobe applied in two orthogonal directions. One possibility is then to choose a diatomic unitcell, and apply the numbering as depicted in Figure 3.8(b).

According to the unit cell definition, it is not required that any translation of thetype (3.1) should give a physical unit cell. Some of these translations may result in theso-called virtual (empty) lattice sites. It is required, though, that the actual unit cells aregenerated by applying a specific restriction on the values of lattice site indices. For example,for the hexagonal two-dimensional lattice shown in Figure 3.8(c), the actual lattice sitescorrespond to even n1 + n2, and the virtual lattice sites are those with odd n1 + n2.

One current unit cell plus all the neighboring unit cells, whose particles interact directlywith the current cell particles, form the associate substructure, or associate cell . The sizeand shape of the associate cell are determined by the cutoff radius of an interatomic potential(see equation (2.65)). Figure 3.9 shows an example of an associate cell for the hexagonallattice, where the atomic interaction is truncated after the second-nearest neighbor; here,the numbering of unit cells is chosen as in Figure 3.8(c). In the case that the interaction istruncated after the first-nearest neighbor, the associate cell is shaped similar to the hexagondepicted in Figure 3.3(b).

The associate cell is the smallest part of the lattice that fully represents its mechanicalproperties. In deriving the lattice governing equations, it is sufficient to consider only oneassociate cell, which is sufficient in deriving the lattice potential energy in a repeatingmanner, as in the equations (3.7) and (3.8). However, the associate cell does not nec-essarily generate the complete lattice by translations. If the associate cell is constructed


(−3,1)

(−3,−1) (−1,−1) (1,−1) (3,−1)

(0,−2)

(−2,0) (0,0) (2,0)

(−1,1) (1,1) (3,1)

Rcutoff

(0,2)

Figure 3.9 Associate cell for the hexagonal lattice. Atomic interaction is truncated afterthe second nearest neighbor.

from the smallest possible unit cells, it usually has the same point symmetry as the corre-sponding Bravais lattice; see the structures depicted in Figure (3.3), and also Figures 3.8(c)and 3.9.

3.2.2 Lattice Lagrangian and Equations of Motion

In order to write the lattice equation of motion, we require the lattice Lagrangian in termsof the particle displacement vectors un. The kinetic energy of lattice particles can be writtenin the matrix form,

T = 1

2

∑n

uTn M un (3.3)

where M is a diagonal matrix of particle masses written for one unit cell. This gives thelattice Lagrangian in the general form

L = 1

2

∑n

uTn M un − U(u) (3.4)

Here, U is the lattice potential energy, and u is a formal notation for all the displacementvectors un in a given lattice.


Substituting (3.4) into the Lagrange equation of motion, written for one unit cell with Sdegrees of freedom,

d

dt

∂L

∂un,s

− ∂L

∂un,s

= f extn,s , s = 1, 2, . . . , S (3.5)

we obtain the lattice equations of motion,

M un + ∂U

∂un= fext

n , n = (n1, n2, n3) (3.6)

where fextn is the vector of external forces exerted on the current unit cell n. In equation (3.6),

un,s and f extn,s are individual components of the vectors un and fext

n , respectively. Utilizingall the lattice site vectors in (3.6) we get a coupled system of generally nonlinear ordinarydifferential equations.

One special form of the lattice equation of motion can be obtained within the harmonicapproximation , which consists of expanding the potential energy in a Taylor series aboutthe equilibrium lattice configuration,

U(u) = U(0) +∑n,s

∂U

∂un,s

∣∣∣∣0un,s + 1

2

∑n,n′,s,s′

∂2U

∂un,s∂un′,s′

∣∣∣∣0

un,s un′,s′ + · · · (3.7)

and truncating the series after the second-order terms. Meanwhile, the first-order termsin (3.7) are trivial, as soon as expanded about the equilibrium configuration, that is, aboutthe global minimum of the lattice potential. The zero-order term can be ignored in (3.7),because a constant shift of the Lagrangian does not alter the equations of motion. Finally,the harmonic approximation gives the lattice Lagrangian,

L = 1

2

∑n

uTn M un + 1

2

∑n,n′

uTn Kn−n′ un′ (3.8)

where the superscript symbol T indicates a transposed vector, and the equation of motionreads as the finite difference matrix form:

M un(t) −∑

n′Kn−n′ un′(t) = fext

n (t) (3.9)

Here, K are the lattice stiffness matrices, or “K -matrices”, which represent linear elas-tic properties of the lattice structure. These matrices are composed of the atomic forceconstants, according to

Kn−n′ = − ∂2U(u)

∂un∂un′|u=0 (3.10)

For a one-, two- or three-dimensional lattice with nearest unit cell interactions, there are upto 3, 9 or 27 nontrivial K -matrices, respectively. Owing to symmetry of the second-orderderivative in (3.10), they have the following general property:

Kn = KT−n (3.11)

One important feature of the equations(3.9) is their invariance with respect to the valueof a current lattice site vector n. This invariance is due to the fact that the K -matrices do not


depend on the individual values of n and n′, but depend only on the difference n − n′, thatis, on the distance between the current unit cell, specified by n, and a neighboring cell n′that interacts with the current cell. Sometimes, a single subscript n is used for brevity, asin equation (3.11). In such a case, it still implies the difference n − n′, that is, n ≡ n − n′.

According to the first symmetry principle discussed in Section 3.1.2, the spatial invari-ance of the K -matrices serves as a consequence of the lattice translational symmetry. In thederivations of K -matrices, it is sufficient to consider the potential energy for one associatecell only.

The lattice equation of motion (3.9) is an ordinary differential equation with a finitedifference with respect to the spatial lattice site indices. This finite difference is given bya discrete convolution sum with a spatially invariant matrix kernel. The effective solutionof this equation involves transform methods that convert the operation of convolution intoordinary matrix multiplication in the transform domain. The relevant solution techniquesare discussed in Sections 3.3–3.5.

3.2.3 Examples

In this section, we construct the harmonic equations of motion for several specific examplesof repetitive lattice structures governed by a pair-wise potential ((2.60) or (2.63)).

Monoatomic chain lattice

The monoatomic chain lattice (Figure 3.2 (a)) is the simplest example where all mathemat-ical derivations of the present chapter can be accomplished in closed form.

Assuming a single longitudinal degree of freedom for each atom, the displacementvector becomes a scalar quantity, un = un. Next, we assume that interactions exist onlybetween the nearest lattice atoms; this can be viewed so that the cutoff radius of thepotential function is close to 1.5ρ, where ρ is the equilibrium distance for the potential.The corresponding associate cell is shown in Figure 3.10.

In deriving the K -matrices (3.10), the potential energy of the atomic interactions needsto be written for one associate cell only; this gives

U = V (un − un−1 + ρ) + V (un+1 − un + ρ) (3.12)

where V is a pair-wise potential with the previously defined equilibrium distance ρ. Accord-ing to (3.10) and (3.12), the K -matrices yield

K−1 = −∂2U

∂un∂un+1= k, K0 = −∂2U

∂u2n

= −2k, K1 = −∂2U

∂un∂un−1= k (3.13)

n – 1 n + 1n

Figure 3.10 Associate cell for a monoatomic chain lattice interacting via nearest-neighborinteractions.


n – 1 n + 1n

Figure 3.11 Associate cell for a diatomic polymer chain.

Here, the linear force constant k depends on the parameters of the potential,

k = 36 · 22/3ε/σ 2 (Lennard − Jones), k = 2εβ2 (Morse) (3.14)

Note that all other matrices Kn, |n|>1, are trivial owing to the nearest-neighbor assumption.Substitution of the K -matrices (3.13) into the general equation of motion (3.9) yields

(M is the atomic mass)

M un − k(un−1 − 2un + un+1) = f extn (3.15)

Diatomic polymer molecule

A unit cell for the diatomic polymer molecule (Figure 3.2 (third from top)), is given bya pair of nearby atoms. The lattice associate cell is depicted in Figure 3.11; it consists ofa current unit cell, numbered as n, and two adjacent cells, n − 1 and n + 1, interactingdirectly with the current cell.

As in the previous example, we assume a single longitudinal degree of freedom pereach atom with nearest-neighbor interactions. Then, the displacement vector consists of twocomponents, un = (vn, wn)

T, where v and w are the displacements of the left and rightatoms in the unit cell, respectively. The potential energy of the associate cell is given by

U = V (wn−1 − vn−1 + ρ) + V (vn − wn−1 + ρ) + V (wn − vn + ρ)

+ V (vn+1 − wn + ρ) + V (wn+1 − vn+1 + ρ) (3.16)

where V is a potential of interaction of two adjacent atoms and ρ is the equilibrium distanceof this potential. The corresponding lattice stiffness matrices are found by utilizing (3.10):

K0 =(−2k k

k −2k

), K1 = KT

−1 =(

0 k

0 0

)(3.17)

where k is the force constant (3.14). Substitution of these matrices into

M un − (K1 un−1 + K0 un + K−1 un+1) = fextn (3.18)

gives the equation of motion in matrix form.

Hexagonal lattice

The hexagonal lattice pattern (Figure 3.12) is observed for the (111) crystallographic planeof a face-centered cubic lattice( see Figure 3.7). As a result, this geometry is often used intwo-dimensional modeling of fcc metals.


(n – 1,m – 1) (n + 1,m – 1)

(n + 2,m)(n − 2,m)

(n − 1,m + 1) (n + 1,m + 1)

(n,m)

y

x

Figure 3.12 Associate cell for a two-dimensional hexagonal lattice interacting via nearest-neighbor interactions.

Consider an orthogonal numbering of unit cells in the hexagonal lattice, such as thatdisplayed in Figure 3.8(c). Then the unit cell is represented by a single lattice atom, and thevector un shows in-plane displacements of this atom with the reference to x and y Cartesianaxes: un ≡un,m = (xn,m, yn,m)T. Within the nearest-neighbor assumption, the associate cellis formed by a symmetric group of seven lattice atoms as depicted in Figure (3.12).

Similar to (3.12) and (3.16), the potential energy of the associate cell in the hexagonallattice can be expressed through xn,m, yn,m and ρ. Further, use of the definition (3.10)provides seven nonzero matrices:

K0,0 = −3k

(1 00 1

), K1,1 = K−1,−1 = k

4

(1

√3√

3 3

)

K1,−1 = K−1,1 = k

4

(1 −√

3−√

3 3

), K2,0 = K−2,0 = k

(1 00 0

) (3.19)

Finally, the lattice equation of motion is given by

M un,m −n+2∑

n′=n−2

m+1∑m′=m−1

Kn−n′,m−m′ un′,m′ = fextn,m (3.20)

In equation (3.126) of Section 3.6.3, we also show the lattice stiffness matrices for athree-dimensional fcc crystal.

3.3 Transforms

The concept of a functional transform can be explained in analogy with the definition ofa function. While the function f assigns to every element x from set x a unique ele-ment y = f (x) from set y , where x and y are numbers, the functional transform A is asimilar operation, but with x and y being functions themselves, y = A{x}. Thus, the func-tional transform assigns a new (transform) function X for each original function x suitable


Original problem

Problem intransform space

Difficult solution

Functionaltransform

Easy solution

Inversetransform

Solution of theoriginal problem

Solution intransform space

Figure 3.13 General idea of the transform solution methods, Arfken and Weber (2001).

for the given transformation. Examples include X(t) = ddt

x(t), X(β) = ∫ b

ax(t)Q(β, t) dt ,

and so on. The second of these examples is known as the integral transform with the kernelfunction Q; a suitable candidate x for the integral transform has to be a function integrableon the interval [a,b].

Integral transforms take on physical significance in the time-frequency relations estab-lished by the Fourier transform, as well as the Laplace transform, which are reviewed in thissection. Another type of functional transform used in the following sections is the discreteFourier transform (DFT) accomplished over discontinuous sequences, such as the displace-ment vectors un and the matrices Kn; these sequences are viewed as discrete functions ofthe lattice site indices ni . The discrete Fourier transform establishes relations between thequantities in the real-space quantities and the space of wave numbers. The common ideabehind the transform solution methods is explained in Figure 3.13. The transform methodsare effective in applications to ordinary differential equations with a finite difference of thetype (3.9).

The Laplace, Fourier and discrete Fourier transforms outlined in this section are linear ,that is, they satisfy the following general property,

A{C1x + C2y} = C1A{x} + C2A{y} = C1X + C2Y (3.21)

where x and y are functions, and C1, C2 are constant numbers.The inverse transform, A−1, is defined as the following:

A−1{A{x}} = A−1{X} = x (3.22)

The inverse of a linear transform is also a linear transform,

A−1{C1X + C2Y } = C1x + C2y (3.23)

3.3.1 Fourier Transform

The Fourier transform is used to transform a time-dependent function into the frequencyspace,

X(ω) = F t→ω {x(t)} =∫ ∞

−∞x(t) e−iωt dt (3.24)


Fourier transforms obey the time-derivative theorem,

Ft→ω

{(∂

∂t

)n

f (t)

}= (iω)nF (ω) (3.25)

The inverse Fourier transform is defined as

x(t) = F−1ω→t {X(ω)} = 1

2π

∫ ∞

−∞X(ω) eiωtdω (3.26)

3.3.2 Laplace Transform

The Laplace transform of a continuous function in time gives

X(s) = Lt→s{x(t)} =∫ ∞

0e−st x(t) dt (3.27)

where s is known as the complex frequency. The Laplace transform has the followingvaluable properties: the convolution theorem,

Lt→s

{∫ t

0x(t − τ )y(τ ) dτ

}= X(s)Y (s) (3.28)

and the time-derivative rules,

Lt→s{x(t)} = sX(s) − x(0−) (3.29)

Lt→s{x(t)} = s2X(s) − sx(0−) − x(0−) (3.30)

Here, the notation “0−” stands for a value of the argument t “right before” the zero point.Mathematically, x(0−) and x(0−) are limits of the functions x(t) and x(t), as t approacheszero from negative values.

The convolution theorem implies that the transform of a convolution integral of twofunctions is equal to the product of the transforms of the individual functions. The time-derivative rules imply that the transform of a time derivative is equivalent to multiplicationof a function by s. This property allows reducing differential equations to simple algebraicequations in the transform domain, and therefore it is useful in finding analytical solutionsto the classical equations of motion; see the example below in this section.

The inverse Laplace transform is defined as

x(t) = L−1s→t {X(s)} = lim

b→∞1

2πi

∫ a+ib

a−ib

estX(s) ds, a, b ∈ R (3.31)

However, this general mathematical definition is rarely used in practical applications. Insome cases, the inverse Laplace transform for X(s), that is, the function x(t), can be foundin the standard tables, such as Oberhettinger and Badii (1973). In Table 3.2, we provide ashort listing of some standard Laplace transforms. This table includes power, exponentialand trigonometric functions, as well as two special functions: the Dirac delta,

δ(t) ={ ∞, t = 0,

0, t �= 0;∫ ∞

−∞δ(t) dt = 1,

∫ ∞

−∞δ(t − t0) x(t) dt = x(t0) (3.32)


Table 3.2 Standard Laplace transforms

x(t) X(s)

1. x(at) 1aX

(sa

)2. e−at x(t) X(s + a)

3. tnx(t) (n = 0, 1, . . .) (−1)nX(n)(s)

4.∫ t

0x(τ) dτ

1

sX(s)

5. x(t − t0) (t > t0) e−st0X(s)

6. δ(t) 1

7. δ(t − t0) e−st0

8. tn (n = 0, 1, . . .)n!

sn+1

9.√

t1

2s

√π

s

10.1√t

√π

s

11. e−at1

s + a

12. ln t −γ + ln s

s(γ = 0.57722 . . .)

13. sin ata

s2 + a2

14. cos ats

s2 + a2

15. sinh ata

s2 − a2

16. cosh ats

s2 − a2

17. Jn(at) (n = 0, 1, . . .)(√

s2 + a2 − s)n

an√

s2 + a2

18.1

tJn(at) (n = 1, 2, . . .)

(√

s2 + a2 − s)n

nan

and the Bessel function,

Jn(t) = 1

π

∫ π

0cos(t sin φ − nφ) dφ (3.33)

For a more comprehensive list of Laplace transforms, we refer the reader to mathematicsliterature.

More generally, the inverse Laplace transform can be computed numerically. One effec-tive numerical algorithm applicable to a wide range of functions was proposed by Weeks


(1966). This algorithm can be summarized as follows:

x(t) = ec t− t2T

A∑γ=0

bγ Lγ (t/T )

bγ = 2 − δγ,0

2T (A + 1)

A∑α=0

(ReX(sα) − cot

zα

2ImX(sα)

)cos(γ zα)

zα = π(2α + 1)

2(A + 1), sα = c + i

2Tcot

zα

2,

T = tmax

A, c = 1

tmax

(3.34)

Here, Lγ are Laguerre polynomials, X is the Laplace transform of x, tmax is the maximumrequired value of the argument for the function x, and δ is the Kronecker delta,

δi,j ={

1, i = j

0, i �= j(3.35)

The value A controls the accuracy of the result, and its choice depends on tmax and thebehavior of f (t) at t ∈ [0, tmax]. For the kernel functions considered in this section and inthe subsequent sections of this book, a sufficient value for A is of the order 100. Weeks’smethod has been found by Davies and Martin (1979) to give excellent accuracy for theinversion of a wide range of functions. Other numerical inversion algorithms are discussedin Davies and Martin (1979) and Duffy (1993).

Example: harmonic oscillator

Consider the solution to the equation of motion of a harmonic oscillator subject to theinitial conditions:

y(t) + ω2y(t) = 0, y(0) = B, y(0) = 0 (3.36)

Applying the Laplace transform and taking into account (3.21) and (3.30), one obtains asimple algebraic equation for the transform solution Y(s),

s2Y(s) + ω2Y(s) = sB (3.37)

This gives

Y(s) = Bs

s2 + ω2(3.38)

The original for this function can be found in Table 3.2; application of the inverse Laplacetransform to (3.38) yields

y(t) = B cos ωt (3.39)

3.3.3 Discrete Fourier Transform

In lattice mechanics, the DFT is used to transform functional sequences, such as the dis-placement vector un or matrix Kn from real space to wave number space.


A key issue in solving the lattice equation of motion (3.9) is the deconvolution of theinternal force term. This is accomplished naturally with the discrete Fourier transform,

x(p) = Fn→p {xn} =∑

n

xn e−ipn, p ∈ [−π, π ] (3.40)

xn = F−1p→n {x(p)} = 1

2π

∫ π

−π

x(p)eipndp (3.41)

which obeys the convolution theorem,

Fn→p

{∑n′

xn−n′yn′

}= x(p) y(p) (3.42)

Here, the calligraphic symbol F denotes the Fourier transform operator, and the hattednotation is used for the Fourier domain function.

The inverse transform can be computed numerically by applying a trapezoidal or mid-point scheme to the integral (3.41). The trapezoidal scheme yields

xn = F−1p→n {x(p)} = 1

N

(N−1)/2∑p=(1−N)/2−1

x(2πp/N)ei2πpn/N (3.43)

Here, N is the total number of integration steps, which gives an effective size of the real-space domain, and the value p is an integer wave number, varying from −N/2 to N/2 − 1.The midpoint scheme gives an alternative symmetric form,

xn = F−1p→n {x(p)} = 1

N

(N−1)/2∑p=(1−N)/2

x(2πp/N)ei2πpn/N (3.44)

where the half-integer wave number p varies from (1 − N)/2 to (N − 1)/2 with a unitstep.

Provided the function x is simple enough, the integral (3.44) can be used for analyticalinversion of the DFT. Meanwhile, solutions involving a standard summation (3.43) or (3.44)are called semianalytical.

A final valuable property of the DFT is the shift theorem,

F {xn+h} = x(p)eihp, h ∈ Z (3.45)

3.4 Standing Waves in Lattices

In this section, we consider the solution of the equation of motion of a free lattice,

M un(t) −∑

n′Kn−n′ un′(t) = 0 (3.46)

that is, with all trivial external loadings, fextn = 0, as compared to (3.9). A general solution

to this equation can be found as a superposition of standing waves in the polymer or crystallattice. Each of these waves represents one normal mode of the free lattice vibration, forexample, thermal vibrations. For a given normal mode, the lattice atoms oscillate with aunique frequency around their equilibrium positions. We demonstrate below the structureof such a solution by utilizing Fourier transform methods.


3.4.1 Normal Modes and Dispersion Branches

In deriving the free lattice solution, we first utilize the DFT in space over both sidesof equation (3.46) for all components of the vector n. Further use of the convolutiontheorem (3.42) gives an equation in the space of wave numbers,

M ˆu(t, p) − K(p) u(t, p) = 0 (3.47)

Here, each wavevector p gives a triplet of wave numbers, p = (p1, p2, p3).We next apply the continuous Fourier transform in time (3.24), and employ the time-

derivative theorem (3.25). This gives the equation of motion in the frequency/wave numberFourier domain, (

ω2M + K(p))

U(p, ω) = 0 (3.48)

As compared to the original time/space equation (3.46), this equation is simply an algebraicequation with small parametric matrices M and K(p) of the size S × S, where S is thenumber of degrees of freedom per unit cell. The solution of this equation is straightforward.

At first, note that equation (3.48) has nontrivial solutions only if

det(ω2M + K(p)

) = 0 (3.49)

This characteristic equation imposes a functional dependence of the normal frequencyupon the wave vector, ω(p), known as the dispersion law . For an S -component vector U,there exist S linearly independent functions ωs(p), s = 1, 2, . . . , S, called the dispersionbranches .

The dispersion branches are divided into two classes: acoustic branches with ω(0) = 0and optical branches with ω(0) �= 0. They refer respectively to a particular value of thefunction ωs(p) as an acoustic or optical frequency. Normally, the total number of acousticbranches is equal to I , the dimensionality of the lattice; then, the total number of opticalbranches is S − I . Dispersion properties of periodic lattices are discussed in greater detailin solid-state physics literature, for example, Kittel (1976). Some simple examples are alsoconsidered in Section 3.4.2.

For a finite lattice, there exists a discrete set of characteristic normal frequencies ωp,s

corresponding to each of the dispersion branches. Thus, a solution of equation (3.48) canbe nontrivial only for ω = ±ωp,s :

U(p, ω) =∑

s

ap,s dp,s

(δ(ω − ωp,s) + δ(ω + ωp,s )

)(3.50)

where ap,s is a participation coefficient, δ is the Dirac’s delta function (3.32), and dp,s is acomplex parametric eigenvector that satisfies

−M−1K(p) dp,s = ω2p,s dp,s (3.51)

We next apply the inverse Fourier transform (3.26) for ω, and the inverse DFT (3.43)for the components of p, onto the equation (3.50) to get

un(t) = 1

2πN

∑p,s

ap,s cos(ωp,s t

)dp,s e

i p·n (3.52)


where p = 2π(p1/N1, p2/N2, p3/N3) and N = N1N2N3. Obviously, the temporal parts ofthe normal modes in (3.52) may also contain some arbitrary initial phases φ. Furthermore,the constant (2πN)−1 can be viewed as the scaling factor for the participation coefficients.Thus, we rewrite the solution (3.52) in the form

un(t) =∑p,s

ap,s cos(ωp,s t + φp,s

)dp,s e

i p·n (3.53)

Here, each term of the summation represents a normal mode of the lattice. For each valueof the wave vector p, there exist s linearly independent normal modes. The oscillatoryquantities ap,s cos(ωp,s t + φp,s) are normal coordinates and the values ap,s and ωp,s arenormal amplitudes and normal frequencies of the lattice vibration. The structure of solu-tion (3.53) implies that the vector dp,s determines the direction of polarization of a normalmode vibration. Also, the form of this solution indicates that the normal modes representstanding waves in the lattice.

Note that because of (3.11) the Fourier transform matrix K(p) is Hermitian, that is, ithas the property

K∗(p) = KT(p) (3.54)

where the asterisk stands for complex conjugation, and also

K∗(p) = K(−p) (3.55)

Indeed,

K∗(p) =∑

n

Kn ei p·n = K(−p) =∑

n

K−n e−i p·n =∑

n

KTn e−i p·n = KT(p) (3.56)

As a result of (3.55), the eigenvectors in (3.51) obey

d−p,s = d∗p,s (3.57)

Taking into account this symmetry, we can construct two real-valued solutions to the latticeequation of motion (3.46):

u+n (t) =

∑p,s

a+p,s cos

(ωp,s t + φ+

p,s

) (cos (p · n)Re dp,s − sin (p · n) Im dp,s

)(3.58)

and

u−n (t) =

∑p,s

a−p,s cos

(ωp,s t + φ−

p,s

) (sin (p · n)Re dp,s + cos(p · n) Im dp,s

)(3.58′)

Here, the operators Re and Im give the real and imaginary parts of the polarization vectors.The normal amplitudes and phases can be different in (3.58) and (3.58′) for the same pair pand s; however, they should satisfy

a+−p,s = a+

p,s , a−−p,s = −a−

p,s , φ±−p,s = φ±

p,s (3.59)

Owing to the symmetries (3.57) and (3.59), any pair of terms in (3.58) correspondingto −p and p are identical. Thus, the solutions (3.58) are in total composed of SN linearly


independent real-valued normal modes of the lattice structure that correspond to a completeset of vibrational and translational degrees of freedom. Obviously, any linear combinationof (3.58) and (3.58′) will also satisfy the governing equation (3.46).

Note that only two parameters of the standing wave solution, the normal frequency andpolarization vector, depend on the intrinsic lattice properties. The remaining parameters, thenormal amplitudes and phases, are only determined by initial conditions, or general provisos,such as the lattice system temperature (see Section 4.2.6). Thus, the knowledge of onlynormal frequencies and polarization vectors provides the homogeneous lattice solutionsin a general form (3.58). In this section, we consider the dispersion functions ω(p) andvectors dp,s for the three lattice structures discussed earlier in Section 3.2.3.

3.4.2 Examples

In this section, we review dispersion properties, that is, frequency spectra and polarizationvectors of three common types of lattice structures: the beamlike monoatomic and diatomiclattices, and the planer monoatomic hexagonal lattice. These structures are depicted inFigures 3.2(a), 3.2(b) and 3.3(b), respectively.

Monoatomic chain lattice

Evaluation of the normal frequencies as functions of the wave number is accomplishedin accordance with the characteristic equation (3.49), which contains the discrete Fouriertransform of the lattice stiffness matrices. Applying the DFT (3.40) to the monoatomiclattice matrices (3.13) gives

K(p) = K−1eip + K0 + K1e

−ip = 2k(cos p − 1) = −4k sin2 p/2 (3.60)

Then, the lattice characteristic equation gives a single acoustic branch:

ω2M − 4k sin2 p

2= 0 ⇒ ω2

p = 4k

Msin2 p

2(3.61)

The corresponding polarization vectors (3.51) are equal to unity for all p.

Diatomic polymer molecule

DFT applied to the K -matrices of a diatomic polymer chain, equation (3.17), yields

K(p) = k

( −2 1 + e−ip

1 + eip −2

)(3.62)

This gives one acoustic (s = 1) and one optical (s = 2) branch,

det

[ω2

(M1 00 M2

)+ k

( −2 1 + e−ip

1 + eip −2

)]= 0 ⇒

ω2p,s = k

µ+ (−1)s

√k2

µ2− 4k2

M1M2sin2 p

2, µ = M1M2

M1 + M2(3.63)


√k/M

p

−p

wp

−p/2 p/2 p

2

(a)

p

2k /M2

−p

wp,s

−p/2 p/2 p

2k /M1

√2k /m

√

√

(b)

Figure 3.14 Dispersion branches of one-dimensional lattices: (a) monatomic and (b)diatomic. The reduced mass µ is defined in equation (3.63).

The relevant polarization vectors are found from (3.51),

dp,s =(

2k − M2ω2p,s

k(1 + eip)

), s = 1, 2 (3.64)

Components of these vectors become dimensionless after normalization.General shapes of the dispersion curves (3.61) and (3.63) are displayed in Figure 3.14.

Hexagonal lattice

The normal frequencies of a two-dimensional hexagonal lattice are derived similarly byutilizing the general equation (3.49) and the lattice stiffness matrices (3.19). There arein total two acoustic branches (s = 1, 2) and there are no optical branches; the normalfrequencies are determined by two wave numbers, p and q:

M

kω2

p,q,s = 3 − cos 2p − 2 cos p cos q+

(−1)s√

(cos 2p − cos p cos q)2 + 3 sin2 p sin2 q (3.65)

Here, M is the atomic mass. Polarization vectors of this lattice are given by

dp,q,s =(

3k(cos p cos q − 1) + Mω2p,q,s√

3 k sin p sin q

), s = 1, 2 (3.66)

3.5 Green’s Function Methods

A direct solution of the original equation of motion (3.9) with a general external load fextn (t)

is difficult, and in most cases it requires numerical time integration, similar to classical


particle dynamics. In this section, we discuss one special technique, known as the Green’sfunction method, which allows a semianalytical solution of the general inhomogeneousequation of motion (3.9). The term semianalytical, or near-analytical, is used because thismethod usually involves numerical methods in obtaining some special preliminary solutions,the Green’s functions, while the final solution to the problem of interest is written in closedform in terms of the Green’s functions.

The Green’s function method has important advantages over direct numerical solu-tions. The first is universality; the Green’s function is a unique structural characteristic.Once obtained for a given lattice structure, it can be used in solving many specific prob-lems involving this lattice. The second is computational effectiveness; as a semianalyticalapproach, the Green’s function method can provide up to several orders of magnitudereduction in processing time and computer memory requirement. Finally, Green’s functiontechniques are important in developing the modern numerical methods that couple classicalparticle dynamics and continuum simulations, Karpov et al. (2005b), Liu et al. (2004c) andPark et al. (2005c).

3.5.1 Solution for a Unit Pulse

Before we proceed to the solution of a general inhomogeneous equation (3.46), at firstconsider the following subsidiary equation:

M gn,n′′(t) −∑

n′Kn−n′ gn′,n′′(t) = I δn,n′′ δ(t) (3.67)

where I gives an S × S identity matrix, δn,n′′ = δn1,n′′1δn2,n′′

2δn3,n′′

3, δi,j is the Kronecker

delta (3.35), and δ(t) is the Dirac’s delta function (3.32). This equation describes theresponse of the lattice structure due to localized pulses of a unit momentum given toeach of the S degrees of freedom of an arbitrary unit cell n′′. This response is representedby the S × S matrix function gn,n′′(t), which is called the lattice dynamics Green’s function,and its physical meaning is explained in Figure 3.15.

We can simplify the equation (3.67) by utilizing the second symmetry principle ofSection 3.1.2. Particularly, owing to the spatial invariance of the K -matrices that follows

s th column

gn,n″(t) =

un (t) n

un (t)

fn″ (t)

fn″ (t) = d(t)n″ s th unitcomponent0

01

Figure 3.15 The sth column of the Green’s function matrix shows the vector of displace-ments for the current cell n, due to the sth unit component of load vector applied to asource cell n′′.


from the definition (3.10), as well as the formal invariance of the load function,

I δn+a,n′′+a δ(t) = I δn−n′′,0 δ(t) = I δn,n′′ δ(t) (3.68)

where a = (a1, a2, a3) is an arbitrary triplet of integer numbers, the lattice Green’s functionmust feature the same translation symmetry,

gn+a,n′′+a = gn,n′′ = gn−n′′ (3.69)

Thus, the Green’s function depends only on the distance between the loaded cell n′′ and thecurrent cell n. This implies independence of the solutions of equation (3.67) with respectto positioning of the load. Therefore, we can assume an arbitrary vector n′′, for example,corresponding to the origin cell (0, 0, 0) = 0, and simplify the equation (3.67) to give

M gn(t) −∑

n′Kn−n′ gn′(t) = I δn,0 δ(t) (3.70)

Similar to the case of K -matrices, a single subscript n at g is used for brevity only; such anotation implies n ≡ n − n′′, or a similar difference.

We assume all trivial initial conditions for the equation (3.70),

gn(0−) = gn(0−) = 0 (3.71)

where 0 is a zero matrix, that is, all lattice particles are at rest right before the unit pulseis applied.

The solution of equation (3.70) is straightforward with the application of functionaltransforms (see Figure (3.13)). We first apply the discrete Fourier transform (3.40) over theequation (3.70) for all components of the lattice site vector n, and utilize the convolutiontheorem (3.42) to obtain

M ˆg(t, p) − K(p) g(t, p) = I δ(t) (3.72)

We next apply the Laplace transform to (3.72) and check Table 3.2 for the transform of aDirac’s delta; this gives (

s2M − K(p))

G(s, p) = I (3.73)

Thus, the Green’s function solution in the transform domain reads

G(s, p) = (s2M − K(p)

)−1(3.74)

Application of the inverse transforms gives the real-space function sought:

gn(t) = L−1s→t F−1

p→n{G(s, p)} (3.75)

Here, the inverse Laplace transform can be accomplished numerically, for example, accord-ing to the Weeks algorithm (3.34).


3.5.2 Free Lattice with Initial Perturbations

A result identical to (3.75) can be obtained by considering the equation of a free lat-tice (3.46) with unit initial velocities of particles in one cell,

un(0−) = 0, un(0−) = I δn,0 (3.76)

Application of the inverse Laplace and discrete Fourier transforms, along with the secondtime-derivative rule (3.30) to equation (3.46) with the initial conditions (3.76) yields

U(s, p) = (s2M − K(p)

)−1 = G(s, p) (3.77)

Thus, we can writeun(t) = gn(t), t > 0 (3.78)

Another possibility is a free lattice with unit initial displacements,

un(0−) = I δn,0, un(0−) = 0 (3.79)

The transform solution to (3.46) and (3.79) is given by

U(s, p) = s G(s, p) (3.80)

Employing the first time-derivative rule (3.30) and the fact that gn(0−) = 0, we obtain

un(t) = gn(t), t > 0 (3.81)

Thus, the Green’s function also solves a free lattice with localized initial perturba-tions. This observation is important in understanding initial conditions for atomic thermalvibrations in lattices, which is discussed in the next chapter.

3.5.3 Solution for Arbitrary Dynamic Loads

As a subsequent complication, consider the solution to the lattice equation (3.9) with ageneral distributed dynamic loading fext

n (t) and with the trivial initial conditions (3.71).Application of the discrete Fourier and Laplace transform then yields(

s2 M − K(p))

U(s, p) = Fext(s, p) (3.82)

where U and Fext are the displacement and force vectors in the transform domain. Accordingto (3.74) and (3.82), the transform solution is simply

U(s, p) = G(s, p)Fext(s, p) (3.83)

The real-space solution is obtained by applying the inverse discrete Fourier and inverseLaplace transforms and taking into account the convolution theorems (3.28) and (3.42),

un(t) =∑

n′

∫ t

0gn−n′(t − τ ) fext

n′ (τ ) dτ (3.84)


3.5.4 General Inhomogeneous Solution

The most general situation with the lattice equation of motion involves arbitrary dynamicloading and arbitrary initial conditions. We write the general initial-value problem con-cisely as

M un(t) − ∑n′ Kn−n′ un′(t) = fext

n (t)

un(0−), un(0−)(3.85)

The solution of this problem is straightforward in the Fourier/Laplace domain,

U(s, p) = G(s, p)(F(s, p) + s M u(0, p) + M ˆu(0, p)

)(3.86)

Thus, in the original domain, we obtain the following solution:

un(t) =∑

n′

∫ t


n′ (τ ) dτ

+ M∑

n′gn−n′(t) un′(0−) + M

∑n′

gn−n′(t) un′(0−) (3.87)

In some special cases, particularly for external loadings fext acting during a short periodof time, the following alternative approach to solving the problem (3.85) can be moreeffective computationally. According to the general theory of ordinary differential equations,a general solution of the inhomogeneous equation (3.9) can be given by a particular solutionto (3.9), augmented with a general solution of the homogeneous equation (3.46). In otherwords, the solution (3.88) can be replaced by

un(t) =∑

n′

∫ t


n′ (τ ) dτ + u+n (t) + u−

n (t) (3.88)

where the displacement vectors u±n (t) are given by (3.58). The normal amplitudes a and

phases φ can be found in (3.58) by employing initial conditions for the problem (3.85), orby using other arguments.

3.5.5 Boundary Value Problems and the Time History Kernel

In the previous sections, we considered solutions of the initial value problem over periodiclattices, including several special cases. Another class of problems in lattice mechanics isthe dynamic boundary value problem, where the solution to the lattice equation of motionis determined by known dynamic displacements of lattice edges.

A straightforward solution to this problem can be obtained with the assumption thatthe lattice boundary is given by a straight line/strip or plane/slab for the two- and three-dimensional lattices respectively. Meanwhile, the domain of interest on one side of thisboundary is large enough so that the effects of any peripheral boundaries are negligible.These assumptions are justified in most cases when we are looking for the solution only inthe direct vicinity of the boundary for a considerably short period of time.

This problem statement is particularly important in modern multiscale methods that cou-ple molecular dynamics and continuum simulations of solids. Since the continuum descrip-tion can only provide an approximate description of the atomic displacements through the


interpolation by finite element shape functions, it is important to know accurate displace-ments of lattice particles in the finite element domain in close vicinity of the interface. Aprojection of the accurate atomic displacements onto the finite element displacement fieldcan provide necessary updates to the coupled model; this comprises a key idea of the bridg-ing multiscale method, for example, Karpov et al. (2005b), Park et al. (2005c), Wagner andLiu (2003) and Wagner et al. (2004). We discuss this application in greater detail in thelater chapters of this book.

A straight line or planelike boundary can be defined by fixing one of the lattice siteindices ni in the triplet n = (n1, n2, n3). Owing to translational symmetry, we can employany specific value of this index; the final results will be identical. For example, let n1 = 0;then we write the boundary value problem of interest as follows:

M un(t) − ∑n′ Kn−n′ un′(t) = 0

n = (n1 > 0, n2, n3), un(0−) = un(0−) = 0, u0,n2,n3(t)(3.89)

where the displacements u0,n2,n3 are enforced and the value n1 is small.An important detail regarding the solution of the problem (3.89) consists in realizing

that the motion of the boundary atoms can be caused either by the displacements of theatoms to be kept or by an external force acting upon the boundary atoms in a hypotheticallarge lattice that extends also for n1 < 0. Therefore, we assume that the motion of theboundary atoms is due to unknown external forces that act only at n1 = 0,

M un(t) −∑

n′Kn−n′ un′(t) = δn1,0 f0,n2,n3 (3.90)

Applying the DFT (3.40) and Laplace transform (3.27) to the equation of motion (3.90),we get

U(p, s) = G(p, s) Fext(p, s) + R(p, s) (3.91)

R(p, s) = M G(p, s)(s u(p, 0−) + ˆu(p, 0−)

)(3.92)

where the transform domain force is Fext = F0(p2, p3). Substitution of this force into (3.91)and further application of the inverse DFT over p1 lead to

Un1 − Rn1 = Gn1 F0 (3.93)

Here, the tilde notation stands for the mixed (transform/real space) quantities that dependon n1 and also p2, p3, s. The force vector F0 can be eliminated by writing two equa-tions (3.93) for a general n1 and for n1 = 0 and substituting F0 from the second of theseequations into the first; this gives

Un1 − Rn1 = �n1

(U0 − R0

)(3.94)

�n1 = Gn1G−10 (3.95)

Finally, application of the inverse DFT transform over p1 and p2 as well the inverseLaplace transform onto (3.94) results in a solution of the form

un(t) =∑

n′δn′

1,0

∫ t

0θn−n′(t − τ )(un′(τ ) − Rn′(τ )) dτ + Rn(t) (3.96)


un′(t)un(t)

un′(t) = d(t)

un(t)

qn-n′(t) =

10

sth unit

sth column

component

Figure 3.16 The sth column of kernel matrix θ shows the vector of displacements for thecurrent cell n, due to the sth unit component of a boundary displacement vector at n′, withthe rest of boundary being fixed.

whereθn(t) = L−1

s→t F−1p2→n2

F−1p3→n3

{Gn1 G−10 } (3.97)

and the vector R depends on the initial conditions (the capital letter R is used for bothLaplace and time domain functions),

Rn(t) = M∑

n′(gn−n′(t) un′(0−) + gn−n′(t) un′(0−)) (3.98)

The time history kernel matrix θ in equation (3.96) represents another class of latticedynamics Green’s functions. While the function (3.75) relates displacements of the lat-tice particles with the localized external forcing, the kernel (3.97) relates the displacementsolution with the localized boundary perturbations. The physical meaning of the kernel θ

is explained in Figure 3.16.Let the external force in (3.90) formally represent all nonrandom, nonequilibrium effects

at the lattice boundary. Then, the initial conditions in (3.89) and (3.92) can be associatedwith the chaotic thermal motion only. Therefore, the random thermal part of the displace-ment solution, vector R, can be replaced by the general homogeneous solution (3.58),

Rn(t) = u+n (t) + u−

n (t) (3.99)

Here, the normal amplitudes and phases for u±n can be found by employing the methods of

statistical mechanics in application to periodic lattices in thermal equilibrium. This techniqueis discussed later in this book. As compared to (3.98), utilizing the random term of theform (3.99) is more effective computationally. For lattices initially at rest, we have Rn = 0.

For one-dimensional lattices, such as the polymer chains depicted in Figure 3.2, thesolution (3.96) reduces to

un(t) =∫ t

0θn(t − τ )(u0(τ ) − R0(τ )) dτ + Rn(t) (3.100)


3.5.6 Examples

In this section, we derive the Green’s function solution for the monoatomic chain latticediscussed earlier in sections 3.2.3 and 3.4.2. The simplest monoatomic lattice, Figure 3.10,with a single longitudinal degree of freedom per unit cell and the lattice stiffness matri-ces (3.13) is unique, which allows all the Green’s function solutions to be found in closedform. Considering a more general lattice structure, numerical inversion of the Fourier or/andLaplace transforms is typically required.

Dynamic Green’s function

The lattice dynamics Green’s function in the transform domain is derived by substitutingthe Fourier transform of K -matrices (3.60) into equation (3.74),

G(s, p) = 1

s2M − 2k(cos p − 1)(3.101)

The inverse Fourier integral (3.41) of this function gives

Gn(s) = 1

s

(√

s2 + ω2 − s)2n

Mω2n√

s2 + ω2, ω = 2

√k

M(3.102)

where the characteristic frequency ω is equal to the maximum normal frequency of themonoatomic lattice (see Figure 3.14).

The inverse Laplace transform of the above function can be accomplished by utilizingentries 4 and 17 of Table 3.2,

gn(t) = 1

M

∫ t

0J2n(ωτ)dτ, gn(t) = 1

MJ2n(ωt) (3.103)

where J is the Bessel function (3.33). In Figure 3.17, we plot this function for variousspatial order parameters n.

0 5 10 15 20 25 300

0.5

1

1.5

t × ω

g n ×

Mω

n = 0n = 1n = 2

(a)

0 5 10 15 20 25 30−0.5

0

0.5

1

t × ω

∂gn

/ ∂t ×

M

n = 0 n = 1 n = 2

(b)

Figure 3.17 Dynamic Green’s function of a monoatomic chain. Dispersion branches ofone-dimensional lattices: (a) displacement and (b) velocity.


0 5 10 15t × w

20 25 30

0.4

0.3

0.2

0.1

0

−0.1

−0.2

Θn

/ ω

n = 1n = 2n = 3

Figure 3.18 Time-history kernel of a monoatomic chain lattice.

Time history kernel

The derivation of the lattice time history kernel, a solution to the boundary value prob-lem (3.89), starts from substitution of the transform function into equation (3.95):

�n = GnG−10 = (

√s2 + ω2 − s)2n

ω2n(3.104)

Note that the DFT is not relevant for the one-dimensional version of (3.95). The timedomain function is found by employing the last entry of Table 3.2:

θn = 2n

tJ2n(ωt), n = 1, 2, . . . (3.105)

A plot of this function is provided in Figure 3.18.Qualitatively, the behavior of the functions (3.103) and (3.105) is typical for most

monoatomic lattices, including two- and three-dimensional systems.

3.6 Quasi-Static Approximation

The quasi-static (time-independent) approximation is used in various applications of thelattice model, where any noticeable change of the external forcing in (3.9) occurs duringa period of time that is much longer than the characteristic time of atomic vibrations,ω−1

p,s . Typical examples include mechanical failure, indentation and scratching of crystallattices, where the peripheral elastic region can be considered within the harmonic approx-imation (3.7).

3.6.1 Equilibrium State Equation

The mathematical formulation for the quasi-static model is obtained by assuming that theexternal loading is equilibrated by the elastic response of the lattice structure at all times,


that is, by ignoring the inertial force term in equation (3.9). This leads to

−∑

n′Kn−n′ un′ = fext

n , n = (n1, n2, n3) (3.106)

Thus, the static lattice deformation is described with the displacement vectors un = rn − n,where the vectors n and rn determine equilibrium positions of the lattice particles beforeand after application of the external forcing. The nonloaded configuration represents aperiodic lattice structure. As before, the subscript vector n gives a triplet of lattice siteindices (n1, n2, n3).

3.6.2 Quasi-Static Green’s Function

The solution of the static equation is straightforward with the use of DFTs (Section 3.3.3).In the Fourier domain, we obtain

−K(p) u(p) = fext(p) (3.107)

This leads to the transform solution

u(p) = g(p) fext(p), p = (p1, p2, p3) (3.108)

whereg(p) = −K−1(p) (3.109)

The real-space solution is obtained by applying the inverse DFT over (3.107) andaccounting for the convolution theorem (3.42),

un =∑

n′gn−n′ fext

n′ (3.110)

Here, the matrix g is the lattice static Green’s function,

gn = F−1p→n{−K−1(p)} (3.111)

In order to avoid possible singularity of the matrix K(p) at p = 0 in numerical calcula-tions, the DFT procedures over the quasi-static equations should utilize the half-integer wavenumber pi , as in equation (3.44). This singularity is due to the fact that the solution (3.110)can be augmented with an arbitrary constant vector to describe a rigid-body displacementof the lattice structure. If required, a rigid-body indeterminacy of the quasi-static solutioncan be eliminated by applying displacement boundary conditions as in engineering structuralanalysis.

3.6.3 Multiscale Boundary Conditions

Over the past decade, there has been considerable effort in finding fundamental descrip-tions of strength and failure properties of nanoscale materials, taking into account theiratomic structures. Numerous experimental observations of material behavior cannot bereadily explained within the framework of continuum mechanics: dislocation patterns infatigue and creep, surface roughening and crack nucleation in fatigue, the statistical nature


of brittle failure, plastic flow localization in shear bands, and so on. The use of classi-cal atomistic simulations has provided useful information of chemical interactions at thenanoscale. However, typical atomistic simulations are still restricted to very small systemsconsisting of several million atoms or less and timescales on the order of picoseconds.Thus, even for nanoscale structures and materials, atomistic modeling would be compu-tationally prohibitive. The limitations of atomistic simulations and continuum mechanics,along with practical needs arising from the heterogeneous nature of engineering materials,have motivated the research on multiscale simulations that bridge atomistic simulations andcontinuum modeling; this topic is discussed in greater detail in Chapter 5.

In order to make the computations tractable, multiscale models generally make use ofa coarse–fine grain decomposition for the physical domain under analysis. An atomisticsimulation method is used in a small subregion, the fine grain, where it is crucial to capturethe individual atomistic behavior accurately. A continuum simulation is used in a largerperipheral region, the coarse grain, where the deformation is considered to be homogeneousand smooth; the macroscopic material behavior and properties are mainly determined by thecollective atomic motion. Since the continuum region is usually chosen to be much largerthan the atomistic region, the overall domain of interest can be considerably large. A purelyatomistic solution is normally not affordable on this domain, though the multiscale solutionis expected to provide the detailed atomistic information only where it is necessary.

The key issue is then the coupling between the coarse and fine scales. Typically, con-current coupling involves the idea of a “handshake” or “pad” region, where pseudoatomsare available on the continuum part of the interface and share the physical space withfinite elements, as shown in Figure 3.19. The purpose of the handshake region is to assure

Atomistic Handshake Continuum

u1 = Nd

u0

Figure 3.19 Structure of the transition region in coupled atomistic/continuum simulations.Positions of pseudoatoms in the handshake area are interpolated with finite element shapefunctions. Reprinted from International Journal of Solids and Structures, E. G. Karpov,H. Yu, H. S. Park et al., Multiscale boundary conditions in crystalline solids: Theory andapplication to nanoindentation, 2005, with permission from Elsevier.


a smoother coupling between the atomistic and continuum regimes. The group of pseu-doatoms serves to eliminate nonphysical surfaces in the atomistic lattice structure, so thatthe real atoms along the interface have a full set of interactive neighbors in the contin-uum domain. However, introduction of a nonphysical handshake region leads to severalconcerns. The first is double counting of the strain energy in the handshake region bythe atomistic and continuum models. The second is ill-conditioning of the finite elementstiffness matrices, due to the need for atomic length scale refinement of the finite elementmesh for the handshake region. Indeed, an extremely fine mesh is required in order to pro-vide accurate positions of the pseudoatoms, since they are found by interpolating the finiteelement nodal positions. The third concern is related to the need, in a nonlocal continuummodel, for the small length scales that are typical for the handshake region.

The FE interpolation procedure for the handshake atoms can be written symbolically as

u1 = Nd (3.112)

where u1 are displacements of the pseudoatoms interacting with the interface atoms, d arefinite element nodal displacements, and the columns of the rectangular matrix N give a setof interpolation basis vectors, computed by utilizing the FE shape functions.

The handshake and continuum domains serve to absorb the fine grain excitations and totransfer effects of the peripheral (coarse grain) boundary into the central atomistic region ofinterest. In principle, this requires evaluation of the displacement vector u1, which dependson the interface displacements u0 and the peripheral boundary displacements ua (a is acoarse grain size parameter) at each step of the solution procedure. This dependence canbe written in the symbolic form,

u1 = �{u0} + �{ua} (3.113)

which represents the multiscale boundary conditions (MSBCs) for the atomistic model onthe fine grain Karpov et al. (2005). The concept behind this relationship is illustrated inFigure 3.20. In equation (3.113), � and � are unknown mathematical operators. In prin-ciple, if both of these operators were known from alternative arguments, for example, byemploying the methods of lattice mechanics, the explicit continuum model as well as thehandshake domain would become redundant. Meanwhile, all the atomistic degrees of free-dom between u1 and ua (Figure 3.20) could be eliminated from the numerical solution.In this section, we discuss a semianalytical quasi-static method to derive the multiscaleboundary conditions for periodic lattice interfaces, expressed by the equation (3.113). Thisapproach involves only lattice mechanics tools; it is based on the intrinsic atomic structuremodel without involving a continuum homogenization procedure. The method is applied tonanoindentation of a crystalline gold substrate.

Boundary condition operators

Similar to the dynamic boundary value problems (Section 3.5.5), we consider a planelikeinterface determined by fixing one of the lattice site indices; in this section, we use the(n, m, l) notation for the components of lattice vector n, instead of (n1, n2, n3). The corre-sponding components of the vector p will be referred to as (p, q, r). The task of derivingthe 3D multiscale boundary conditions reduces to expressing, for example, the atomic


Fine grain Coarse grain

uau0

u1 = Θ{u0} + Ξ{ua}

Figure 3.20 Multiscale boundary conditions: atomic positions within the coarse grain areevaluated through the displacements of atoms at the interface and the outer boundary ofthe coarse grain. Continuum modeling is not used. Reprinted from International Journal ofSolids and Structures, E. G. Karpov, H. Yu, H. S. Park et al., Multiscale boundary condi-tions in crystalline solids: Theory and application to nanoindentation, 2005, with permissionfrom Elsevier.

displacements un,m,1 in terms of un,m,0 and un,m,a ,

un,m,1 = �{un,m,0} + �{un,m,a} (3.114)

where the slab l = 0 represents the fine/coarse grain interface, and l = a assigns the coarsegrain boundary. A more general (polyhedron) shape of the fine grain can be obtained bycombining several planelike interfaces, where each of them is treated independently usingan equation similar to (3.114).

The deformation of the coarse grain domain can be viewed as occurring owing to theexternal boundary forces applied to the atoms (n, m, 0) and (n, m, a):

fextn ≡ fn,m,l = δl,0fn,m,0 + δl,afn,m,a (3.115)

We then apply the DFT to the static equation (3.106) for all three spatial indices, utilizingthe external force vector (3.115) and the convolution theorem (3.42) to obtain the Fourierdomain solution

u(p, q, r) = g(p, q, r)(f0(p, q) + e−ira fa(p, q)

)(3.116)

g(p, q, r) = −K−1(p, q, r) (3.117)

Further use of the shift theorem (3.45) with the inverse (r → l) DFT of equation (3.116)yields

ul (p, q) = gl (p, q) f0(p, q) + gl−a(p, q) fa(p, q) (3.118)

gl (p, q) = F−1r→l {g(p, q, r)} (3.119)


We then write three equations (3.118) by substituting l = 0, l = 1 and l = a, and rear-range them to express the vector u1 through u0, ua , and to eliminate the vectors f0, fa:

u1(p, q) = �(p, q) u0(p, q) + �(p, q) ua(p, q) (3.120)

where

(�(p, q) �(p, q)

) = (g1(p, q) g1−a(p, q)

) (g0(p, q) g−a(p, q)

ga(p, q) g0(p, q)

)−1

(3.121)

By applying the inverse DFT for p and q to equation (3.120), and taking into accountthe convolution theorem (3.42), we obtain the final form of the 3D multiscale boundaryconditions:

un,m,1 =∑n′,m′

�n−n′,m−m′ un′,m′,0 +∑n′,m′

�n−n′,m−m′ un′,m′,a (3.122)

�n,m= F−1p→n F−1

q→m

{�(p, q)

}, �n,m= F−1

p→n F−1q→m

{�(p, q)

}(3.123)

Thus, the operators of MSBCs act as discrete convolution sums over the interfaceand coarse grain boundary displacements. These operators are represented by compactmatrix kernels � and �, depending on the lattice geometry and structure of the interatomicpotential.

In some applications, including the nanoindentation simulations to be discussed later inthis section, the coarse scale boundary can be assumed to be rigidly fixed, so that un,m,a = 0for all n and m. This assumption simplifies equation (3.122) to the form

un,m,1 =∑n′,m′

�n−n′,m−m′ un′,m′, 0 (3.124)

Note that the kernel matrices � and � are dimensionless quantities, because they relate“displacements with displacements” (see equations (3.122) and (3.124)). This indicates that,for a given interatomic potential, their values should depend on the geometry of the crystallattice and the choice of the elementary cell, rather than on scaling parameters of thepotential. This feature is attractive because it implies the existence of unique kernel matricesfor various materials with identical or similar crystalline structures.

Application: nanoindentation of crystalline gold

We verify correctness of the MSBCs on a benchmark nanoindentation problem with acrystalline metallic substrate. Consider a parallelepipedic fine grain with periodic boundaryconditions on the side faces and the multiscale boundary conditions on the bottom face(Figure 3.21). The top face of the fine grain domain is originally traction free and is thensubjected to a solid nondeformable indenter.

A substantial part of the substrate is considered as a bulk coarse scale domain, whoseatomistic degrees of freedom will be eliminated from the multiscale model. The resultantmultiscale solution will be compared with the benchmark data, obtained by molecularmechanics simulations on the original full domain, that is, by preserving the completeatomistic resolution and all the relevant degrees of freedom at both fine grains and coarse


Indenter

Finegrain

Coarsegrain

Multiscale BC

un,m,a = 0

un,m,1

un,m,0 1/4

Figure 3.21 Nanoindentation problem with the multiscale boundary conditions applied tothe bottom face of the fine grain; periodic boundary conditions are applied to the lat-eral faces. Reprinted from International Journal of Solids and Structures, E. G. Karpov,H. Yu, H. S. Park et al., Multiscale boundary conditions in crystalline solids: Theory andapplication to nanoindentation, 2005, with permission from Elsevier.

grains. The coarse scale boundary displacements in both problems are set to be zero;therefore, equation (3.124) will be utilized.

We introduce the three-dimensional numbering of the atomic locations in the fcc lattice,so that the atoms occupy the even staggered locations (n, m, l) as illustrated in Figure 3.22.For this lattice, we utilize the pair-wise Morse potential (2.63) with parameters of crystalline

y,m

x,n

z,l

(2,0,0)(1,1,0)

(0,0,0) (0,2,0)

(2,1,1)(1,2,1)(1,0,1)

(0,1,1)

(2,2,0)

Figure 3.22 Face-centered cubic crystal: Bravais lattice (left); numbering of equilibriumatomic positions (n, m, l) in two adjacent (001) planes, corresponding to l = 0 and l = 1(right). The interplanar distance is exaggerated. Reprinted from International Journal ofSolids and Structures, E. G. Karpov, H. Yu, H. S. Park et al., Multiscale boundary condi-tions in crystalline solids: Theory and application to nanoindentation, 2005, with permissionfrom Elsevier.


gold, for example, Harrison (1988),

ε = 0.560eV, β = 1.637A−1

, ρ = 2.922A (3.125)

Accounting only for nearest-neighbor interactions in equations (2.46) and (3.10), we obtainthe following set of K -matrices in the Cartesian coordinate system shown in Figure 3.22:

K1,1,0 = k

1 1 0

1 1 00 0 0

, K1,−1,0 = k

1 −1 0

−1 1 00 0 0

, K1,0,1 = k

1 0 1

0 0 01 0 1

,

K1,0,−1 = k

1 0 −1

0 0 0−1 0 1

, K0,1,1 = k

0 0 0

0 1 10 1 1

, K0,1,−1 = k

0 0 0

0 1 −10 −1 1

,

K0,0,0 = −8k

1 0 0

0 1 00 0 1

, k = εβ2

(3.126)Six additional matrices are obtained from (3.126) by employing K−n,−m,−l = Kn,m,l , whileany other subscript triplet (n, m, l) yields a zero matrix. The Fourier transform (3.40) ofthese matrices can be shown in closed form:

K(p,q,r)=−8kI+4k

(cosq+cosr)cosp − sinp sinq − sinp sinr

− sinp sinq (cosp+cosr)cosq − sinq sinr

− sinp sinr − sinq sinr (cosp+cosq)cosr

(3.127)

where p, q and r are real-valued Fourier parameters on the interval [−π, π ] and I is theidentity matrix.

Further steps in computing the kernel matrix � are accomplished numerically, accordingto the procedures (3.117), (3.119), (3.121), (3.123) and the DFT inversion formula (3.44).The coarse scale parameter a is chosen to be 31.

Elements of the matrix � decay quickly with increasing absolute values of the spatialorder parameters n and m. We illustrate this property in Figure 3.23 by plotting element(1,1) of this matrix as a function of n at various m. The overall trends shown are typical forall other elements of �. In accordance with the staggered atomic arrangements in the fcccrystal, the matrices � for even n+m appear trivial, and they are ignored in Figure 3.23.The value of the element (1,1) is seen to decay by a factor of 100 with the growth of |n|or |m| from 0 to 4. This decay is a valuable property of the kernel matrix. Taking this intoaccount, we can truncate the summation in equation (3.124) at some critical differencesn−n′ and m − m′ without a considerable loss of the accuracy. By denoting these criticalvalues nc and mc, we can update equation (3.124) to give the final form,

un,m,1 =n+nc∑

n′=n−nc

m+mc∑m′=m−mc

�n−n′,m−m′ un′,m′,0 (3.128)

which is used in the computer simulation. In most applications of the fcc lattice model, it isproper to truncate this convolution sum at nc, mc = 4∼6. This truncation can dramaticallyreduce the computational cost of the MSBCs.


−10 −5 0 5 1010−4

10−3

10−2

10−1

100

n

m = 0m = 4m = 8 Θn,m (1,1)

Figure 3.23 Typical dependence of the components in the kernel matrix on the valuesof spatial order parameters. Reprinted from International Journal of Solids and Structures,E. G. Karpov, H. Yu, H. S. Park et al., Multiscale boundary conditions in crystalline solids:Theory and application to nanoindentation, 2005, with permission from Elsevier.

Another important feature of this kernel matrix is that it is invariant with respect tothe value of the interaction coefficient k in matrices (3.126), and therefore with respect toparameters of pair-wise potentials. Thus, the present kernel matrix � for a given value a isa general characteristic of all monoatomic fcc crystals with nearest neighbor interactions.

In the course of the simulation, the atomic displacements along the deformable boundarylayer l = 1 in the reduced domain ( see Figure 3.21) are related to the displacements ofatoms in the adjacent layer l = 0 through the discrete convolution operator, accordingto equation (3.128). The original size of the full domain is 17 × 17 × 40 atomic layers,and the fine grain domain (the reduced domain) comprises 1/4 of the total volume, that is,17 × 17 × 10 layers. This reduction corresponds to the value a = 31. According to (3.125),the interlayer distance along the coordinate axes is 2.066A, and the total height of the finegrain is 18.6A. The curvature radius of the indenter is 10A.

The indentation process is simulated in a series of iteration steps, where the maximumindentation depth of 10A is achieved at the 100th step. The result of this simulation, theload/indentation depth curve, is plotted in Figure 3.24 in comparison with the benchmarksimulation that preserves the complete atomistic resolution for both the fine grain and thecoarse grain. The plot shows a good agreement between the benchmark and multiscalesimulations at all three indentation regimes: elastic load, plasticity (the top part of theindentation curve) and unload. Meanwhile, the multiscale approach reduces the computa-tional effort by a factor of 10. More details on the MSBCs in solids can be found in Karpovet al. (2005) and Medyanik et al. (2005).


0 20 40 60 80 100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Indentation step

Nor

mal

ized

load

Full domainMultiscale, a = 31

Loading Unloading

Figure 3.24 Load versus indentation depth: comparison of full domain atomistic and mul-tiscale solutions for nanoindentation problem described in Figure 3.21.

Application: deformation of graphene monolayers

Application of the MSBC method to the mechanics of carbon nanostructures was demon-strated by Medyanik et al. (2005). In this work, a spherical indenter of radius R is appliedin a displacement controlled manner to the center of a graphene sheet of hexagonal shape ina direction perpendicular to the plane of the sheet (see Figure 3.25). During each step, theindenter is lowered by 0.5 A followed by energy minimization using the conjugate gradientmethod. The indenter is modeled using a repulsive potential so that a repulsive force actson every atom located in the vicinity of the center of the indenter closer than the indenterradius:

Fi = α(R − ri)2 (3.129)

where α is a parameter governing indenter stiffness and ri is a distance between thecenters of the indenter and current atom. In the numerical simulations, the value of A

is taken as 100 eV/A corresponding to a very stiff (nearly rigid) indenter compared totransverse stiffness of the graphene structure; the radius of the indenter is R = 10 A.

The outer boundary of the full domain, represented by a solid line in Figure 3.25, isfixed; the first term in equation (3.113) is then trivial. MSBCs are applied at the boundary ofthe reduced domain, which is shown with a dashed line. The size of the reduced domain isdetermined by the domain reduction parameter a, which represents the number of unit celllayers between the boundaries of the reduced domain and the full domain. The characteristicdimension of the coarse scale region being eliminated from the simulations, La , is equal tothe domain reduction parameter times the size of the representative unit cell. The MSBCsare applied to each of the six zigzag sides of the reduced domain.


Spherical indenter,radius R = 10Å

MSBCs Fixed

Side view

Top view

Fine scale region(reduced domain)

Coarse scale region(eliminated part)

Y

X

L≈220Å

La≈65Å

ZX

Figure 3.25 Application of multiscale boundary conditions to nanoindentation of a singlegraphene sheet: schematic problem statement. The dark gray domain of width La has beeneliminated from the molecular mechanics simulation.

Molecular mechanics simulations were carried out for reduced domains of differentsizes, using the MSBCs of the type

um,1 =m+mc∑

m′=m−mc

�m−m′ um′,0 (3.130)

Note that this equation is a reduced form of (3.128) that is valid for planar structures.The following domain reduction and cutoff parameters were utilized: a = 30 and mc =4. The full domain and the reduced domain were composed of 2646 and 15,000 atoms,respectively.

The results were compared with the corresponding full domain (benchmark) solutions.Figure 3.26 shows the reduced domain along with the fixed boundary of the full domain atfour stages of the indentation process starting from the initial configuration, when all theatoms were in the same plane, and then after the indenter penetrates to the depths of 10,20, and 30 A. The pictures in Fig 3.26 represent the true aspect ratio, that is, they use thesame scale in all dimensions.

The deformed configuration of the reduced domain at the final stage (after indenting tothe depth of 30 A) can be seen in more detail in Figure 3.27. This figure shows a comparisonof the vertical displacements of the reduced domain to the full domain solution for the atomsat the cross section of the sheet along the vertical plane y = 0. Initially, the graphene sheet


Figure 3.26 Multiscale simulations of the indentation of a graphene layer; only the reduceddomain and the fixed boundary of the full domain are shown. The snapshots correspond tofour different indentation depths: 0, 10, 20 and 30 A.

−100 −80 −60 −40 −20 0 20

10

0

−10

−20

−30

−4040 60 80 100

Figure 3.27 Vertical (z-axis) displacements of the atoms lying on the middle line crosssection: reduced domain multiscale simulations (crosses) versus full domain solution (cir-cles). Domain reduction parameter, a = 30.

was positioned at z = 0. It can be seen that owing to the applied multiscale boundaryconditions (MSBC) the whole boundary of the reduced domain has moved from its initialposition downward in the direction of indentation, representing a deformable boundary. Thecomparison of the benchmark and multiscale displacements shown in Figure 3.27 showsthe high degree of accuracy of the MSBC method. Note that Figures 3.26 and 3.27 showthe actual scale of deformation, proving that the method works well even for moderatedeformations. Figure 3.28 shows very good agreement between multiscale and full domain


40

35

30

15

10

5

00 5 10 15

Depth20 25 30

25

Forc

e

20

Full domainMulti–scale

Figure 3.28 Force (eV/A) versus indentation depth (A).

results for the force versus depth of indentation curve. The force is given here in unitsof eV/A.

Copyright

Material of pages 67–74, including Figures 3.19–3.23, is reproduced from Karpov et al.(2005), Copyright 2005, with permission from Elsevier.

nano mechanics and materials (theory, multiscale methods and applications) || lattice mechanics

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