The nonlocal theory has been developed with the flavor of accounting for the effects of the long-range interatomic interactions on the material mechanics. Some nonlocal models were early developed [10–12]. The currently used model of the nonlocal theory is the one that has been developed by Eringen [10,11]. In this section, the nonlocal mechanics of particles and the nonlocal mechanics of linear elastic continua are reformulated in the framework of the nonlocal theory. The dispersion relations are then derived on the basis of Eringen’s nonlocal theory.
2.1 Nonlocal Mechanics of Particles
Let us consider a system of particles as shown in Fig.1. According to the conventional mechanics of particles, a particle Aonly interacts with its neighbor particles within the local region. Using Newton’s second law of motion, the equations of motion of the particleAcan be written as:
Fl =mAu¨A (1) xA×Fl =xA×mAu¨A (2)
whereFl denotes the local force vector,u¨Ais the particle’s acceleration, andxA is the particle’s position vector. In addition,mAdenotes the mass of the particle.
Fig. 1 Mechanics of particles in the context of a the classical mechanics andb the nonlocal mechanics
Some cases would require modeling the particle mechanics considering both neighbor and non-neighbor interactions. Accounting for the non-neighbor interac- tions, the dynamic equilibrium of particle Ais conditional by its balance with all surrounding particles. A non-neighbor interaction depends on the distance between the particle and a non-neighbor particle. A weight function, wxA−x, can be introduced to define the intensity of the non-neighbor interaction between particle Aand a particle at pointx, wherexAis the position of particleA. Accordingly, the equations of motion for particle Acan be rewritten as [18]:
w0
Fnl0 +w1
Fnl1 +w2
Fnl2 + · · · =mAu¨A (3) w0
xA×Fnl0 +w1
xA×Fnl1 +w2
xA×Fnl2 + · · · =xA×mAu¨A (4)
where Fnl0 denotes the local force vector and,Fnl1 andFnl2 are the nonlocal force vectors due to the interactions between particle Aand the particles in the first and the second non-neighboring regions, respectively (Fig.1b). In addition,wi are the values of the weight function which satisfy the condition, i.e.,
wi =1 (5)
According to Eqs. (3) and (4), the force and moment equilibriums are globally satisfied in the context of the nonlocal mechanics of particles. This means that the local balance is violated when the interactions of the particles are nonlocal [10,21].
2.2 Nonlocal Continuum Mechanics
Consider an elastic material with volume V and surface S. In the extreme limit, the material is a composition of an infinite number of infinitesimal particles. The physical properties of these particles and all the forces of interactions between them are described by smooth functions of space and time in the context of the theory of continuum mechanics [10]. Thus, the dynamic equilibrium of the material can be obtained based on the nonlocal theory integrating Eqs. (3) and (4) over the material volume, i.e.,
V
[
V
w(x−x)F(x)d V(x)]d V(x)+
S
T(n)(x)d S(x)=
V
ρu(x)d V¨ (x) (6)
V
[
V
w(x−x)(X×F(x))d V(x)]d V(x)+
S
(X×T
(n)
(x))d S(x)
=
V
ρ(X× ¨u(x))d V(x) (7)
whereρis the material’s mass density, andT(n)denotes the surface tractions, such that
Ti(n)=tj inj (8)
where ti j is a nonlocal stress tensor, which is formed accounting for the nonlocal interactions between the different material particles.nj are components of the unit normal vector. Using the Green-Gauss theorem, we have
tj i,j+ fi =ρu¨i (9) where fidenotes the body forces anduiare the Cartesian components of the infinites- imal displacement.
2.3 Eringen’s Constitutive Equations
According to Eringen’s nonlocal theory, equations of motion of a local particle are obtained from the global interaction of the particles with all other particles of the elastic body. These global interactions are considered in the constitutive equations, which gave the stress-based equations of motion identical to the one of the classical elasticity (Eq. (9)). To collect global interactions, the constitutive equations were expressed as follows (for homogeneous-isotropic linear elastic materials) [11,16]:
ti j(x,t)=
V
α(x−x)
λεqq(x,t)δi j +2μεi j(x,t)
d V(x) (10)
whereεi j is the infinitesimal strain tensor, εi j(x)= 1
2[ui,j(x)+uj,i(x)] (11) andλandμare the Lamé constants.δi j is the Kronecker delta andα(x−x)is a nonlocal modulus. The basic features of the nonlocal kernel were discussed in Refs.
[22,23].
2.4 Nonlocal Field Equation
The nonlocal field equation can be obtained by substituting Eqs. (10) and (11) into Eq. (9), as follows:
⎛
⎝
V
α(x−x)
λuq,q(x,t)δi j+μ
ui,j(x,t)+uj,i(x,t) d V
x⎞
⎠
,j
+fi(x,t)=ρu¨i(x,t) (12)
Equation (12) is an integro-partial differential equation. Complications of solving the nonlocal field Eq. (12) have been early addressed in Ref. [13]. To avoid the dif- ficulties of solving the integro-partial differential field equation, Eringen developed an equivalent model that gives the field equation in a differential form, as follows [13]:
(λ+μ)uj,i j +μui,j j+(fi−ρu¨i)−2(fi,j j−ρu¨i,j j)=0 (13) where is a nonlocal parameter. The differential form of the nonlocal theory is equivalent to the nonlocal integral form only over unbounded elastic domains.
2.5 Dispersion Relations
Phonon dispersion is one of the fundamental properties of materials. Here, the propa- gation of a harmonic wave along [100] direction is modeled based on Eringen’s non- local theory. By assuming a high-symmetric direction, the following one-dimensional harmonic functions can be considered:
ur(x,t)=Urexp(i(kx−ωt)) (14) whereur is the vibration of a lattice site about its equilibrium position,kandωare wave number and wave frequency, respectively. Substituting Eq. (14) into Eq. (13) and setting fi =0, we obtain the frequency-wavenumber relations as
ω2L= (λ+2μ)k2
ρ(1+k22) (15)
ω2T = μk2
ρ(1+k22) (16)
whereωLandωTare the frequencies of longitudinal and transverse acoustic waves.
2.6 Limitations of Eringen’s Nonlocal Theory
In this subsection, limitations of Eringen’s nonlocal theory are addressed. It follows from Eq. (12) that Eringen’s nonlocal theory assumes the same attenuation function for all the material coefficients. According to this theory, the nonlocal-dependence of the normal strain is identical to the nonlocal-dependence of the shear strain. How- ever, this is not true for many materials. For example, the experimental dispersion curves of Si, Au, and Pt indicated that the decay of the longitudinal frequencies with the wavenumber is different from the decay of the transverse frequencies with the
Fig. 2 Dispersion curves of Ag [100] based on Eringen’s nonlocal theory (ENT)
wavenumber. For these materials, Eringen’s nonlocal theory is inapplicable where it cannot properly fit their longitudinal and transverse dispersion curves [18,19].
Figure2shows the applicability of Eringen’s nonlocal theory to model the acoustic dispersion phonons of a material with nearly similar nonlocal characters in both the longitudinal and transverse directions. In this figure, the dispersion curves of Ag [100] predicted by Eringen’s nonlocal model are plotted and compared with experimental results reported by Kamitakahara and Brcckhouse [24]. It is observed that the frequencies of both longitudinal and transverse acoustic waves are perfectly predicted by the nonlocal model for2=0.028a2. Figure3shows the limitation of Eringen’s nonlocal theory to model the acoustic dispersion phonons of a material with different nonlocal characters in the longitudinal and transverse directions. This figure shows the dispersion curves of Au [100] obtained from the nonlocal model and the experimental data given in Ref. [25]. These results indicate that Eringen’s nonlocal theory has limitations to model Au. For Au [100], it can only predict one of the acoustic dispersion curves for specific value of the nonlocal parameter. For a detailed discussion about this topic, we refer the interested readers to Ref. [19].
The physical and material parameters used in the generation of these results based on Eringen’s nonlocal theory are taken from Ref. [18] and listed in Table1.
In addition to its limitations to model the dispersion curves of some materials, Erin- gen’s nonlocal theory has limitations to reveal the nonlocal effects on the material’s Poisson’s ratio. It has been demonstrated that, due to nonlocal interatomic interac- tions, the Poisson’s ratioνof the material is generally non-constant and depend on the material dispersion [25], i.e.,
ν= 1−2R2 2
1−R2 (17)
Fig. 3 Dispersion curves of Au [100] based on Eringen’s nonlocal theory (ENT)
Table 1 Physical and material properties of Ag and Au
Material Elastic moduli Lattice constant Density
λ(GPa) μ(GPa) a(nm) ρ(kg/m3)
Silver (Ag) 18 45 0.4079 10490
Gold (Au) 154 48 0.4065 19300
whereR=ωT(k) /ωL(k). The substitution of the frequency-wavenumber relations (Eqs. (15) and (16)) into Eq. (17) gives Poisson’s ratio with the form:
ν = λ
2(λ+μ) (18)
which is the same Poisson’s ratio as defined based on the classical theory of linear elasticity. This indicates that Poisson’s ratio is independent of the nonlocal field in the context of Eringen’s nonlocal theory. This, however, contradicts with the recent observations on the size-dependence of Poisson’s ratio [26–29] and the microstruc- tural topology-dependence of Poisson’s ratio of metamaterials [26,30–32].