2 Discrete and Nonlocal Rods 2.1 Axial Lattices
2.6 Lattice with Direct and Indirect Interactions
βm,n=k∗+ (mπ)2 1+e0a
L
2
(mπ)2 =k∗+(mπ)2
1−(mπ)2e0a L
2 +O
1 n4
(45) The small length scale coefficient can be easily identified for low frequencies and a sufficient number of elementsn, by comparing Eqs. (44) and (45):
e0= 1 2√
3 ≈0.289 (46)
For the highest frequency, one obtains form=n:
βm,n=k∗+4n2=k∗+ (nπ)2
1+π2e20 ⇒ e0= 1 π2
π2 4 −1
≈0.386 (47)
The valuee0=0.39 is reported by Eringen [3] who equated the nonlocal wave dispersion frequency and the corresponding lattice dynamic results at the end of the first Brillouin zone (in the absence of elastic medium interaction-see also Challamel et al. [37]). It is worth mentioning that the stiffnesskof the elastic medium does not affect this length scale calibration in this case.
Fig. 3 Axial lattice composed of concentrated masses withp-neighbour elastic interaction
the wave dispersive properties of an infinite generalized lattice (lattice with short and long range interaction). Rosenau [64] studied the specific problem of a general- ized lattice with an interaction that includes two closest neighbours (Rosenau [64]
also derived the results for more neighbours). Pipes [59], Chen [60] and Chen [61]
studied a specific generalized lattice with N-neighbour interaction, characterized by equal stiffness for each interaction. The exact eigenfrequencies of this specific gen- eralized lattice have been obtained. The problem studied by Pipes [59], Chen [60]
and Chen [61] is very specific in the sense that short- and long-range interactions are equal. For many physical systems, the effect of long range interaction decreases with respect to the distance so that a differentiation between short- and long-range interactions is necessary. When considering the N-neighbour interaction problem with generalized interactions, Eaton and Peddieson [62], calculated the exact eigen- frequencies of a generalized lattice with N-interaction for fixed-free, fixed-fixed and free-free boundary conditions. They also used a continualization procedure based on a Taylor-based asymptotic expansion of the pseudo-differential operator involved in the difference formulation of the generalized lattice (associated with a difference operator of order 2N) (this methodology was also followed by Rosenau [64]). Both Eaton and Peddieson [62] and Rosenau [64] derived with such a continualization procedure a higher-order gradient approach associated with the generalized lattice.
The wave dispersion relation of this generalized lattice, was studied by Brillouin [63], Eaton and Peddieson [62] and Rosenau [64]. Rosenau [64] followed the same methodology used for the direct interaction problem, and presented a lower order spatial differential wave equation (with spatio-temporal derivatives), for capturing the inherent scale effects in the generalized lattice. We shall follow this approach by considering a nonlocal equivalent rod model associated with the finite generalized lattice. Triantafyllidis and Bardenhagen [26] also investigated the static behaviour of a generalized lattice and formulated an equivalent gradient elasticity approach at the
macroscopic scale. More generally, generalized and complex interactions in lattice mechanics are still important to be studied, for a better understanding of the macro- scopic properties of nonlocal and nonlinear generalized wave propagation properties (see Maugin [42]).
Nonlocal medium and generalized lattices with generalized interactions have been already related in the literature. Eringen and Kim [2] studied an N-neighbour inter- action model as a discrete formulation of a strain-based nonlocal continuum model.
The N-neighbour interaction lattice model can be also considered as the discrete for- mulation of a peridynamic model (nonlocal relative displacement-based model) [65].
Carcaterra et al. [66] revisited this problem with generalized interactions, and built a microscopic generalized lattice model (with direct and indirect neighbour interac- tions) based on a macroscopic higher-order gradient continuum media. Tarasov [67]
studied the generalized elastic lattice with two closest neighbours. Tarasov [67] also continualized the higher-order difference operators to derive a higher-order constitu- tive law which is similar to the one derived by Eaton and Peddieson [62] or Rosenau [64]. Tarasov [67] also commented the possible loss of definite positiveness of the macroscopic elastic energy (issued of the continualization procedure) from the struc- ture of the generalized interactions at the lattice level. The definite positivity of the generalized interaction is not necessarily associated with the definite positivity of the equivalent gradient elasticity approach. Michelitsch et al. [68] investigated the properties of generalized lattices with both short range and long-range interactions governed by a power law. They showed the link between such generalized lattice and some equivalent nonlocal media based on fractional nonlocal mechanics. Ghavanloo et al. [69] studied the wave propagation of diatomic lattices, with both a discrete and a nonlocal lattice-based approach formulated from a continualization procedure.
Ghavanloo and Fazelzadeh [70] generalized the lattice problem with long-range interactions with additional coupling with internal mass called metamaterial.
The generalized axial lattice comprises concentrated massesmiconnected by an elastic network as shown in Fig.3 (mi =ρAa except at the border). Fixed-fixed boundary conditions will be assumed for the generalized lattice. The generalized lattice is characterized by a distribution of elastic stiffnesses kj, where kj is the axial stiffness of the j-th spring associated with its neighbouring interaction, for
j ∈ {1,2, ...,p}. This stiffness can be rewritten via a scaling law as:
kj=αj
E A
j2a (48)
whereais the lattice spacing (identical to the lattice problem with direct interactions), αjis a dimensionless influence function which characterizes the contribution of the long range interaction forces. E Ais the equivalent axial rigidity of the continuum rod (local continuum rod). The higher-order mixed differential-difference equation of the generalized lattice is given by (see for instance Challamel et al. [57])
E A p
j=1
αj
ui+j−2ui+ui−j
(j a)2 −ρAd2ui
dt2 =0 (49)
withαj ≥0 is a positive parameter (generally a decreasing function with the inter- action distance) and the new normalization requirementp
j=1αj =1.
By assuming a harmonic motion, the eigenvalue problem is governed by difference equation of order 2p
E A p
j=1
αj
ui+j−2ui+ui−j
(j a)2 +ρAω2ui =0 (50) This difference equation can be equivalently written as:
p j=1
αj
ui+j−2ui+ui−j
j2 + β
n2ui=0 (51)
We consider fixed-fixed boundary conditions with the higher-order anti-periodic boundary conditions:
u−(k−1)= −uk−1, un−(k−1)= −un+k−1 for k∈ {1,2, ...,p} (52) For this generalized lattice with such a generalized boundary condition, the exact vibration mode can be assumed in a sinusoidal form as
ui =u0sin
πi n
(53) The fundamental frequency is obtained from the substitution of Eq. (53) into Eq.
(52):
β = p
j=1
αj
2n j
2 sin2
π j 2n
(54)
Eaton and Peddieson [62] also obtained a similar formulae by using dimensional quantities without scaling factors.
It is possible to develop a nonlocal approach associated with the lattice problem, by using a continualization procedure. The higher-order difference Eq. (50) can be continualized with the introduction of a pseudo-differential operator:
E A
⎡
⎣ p
j=1
4αj
(j a)2sinh2 j a
2 d d x
⎤⎦u+ρAω2u=0 (55)
The pseudo-differential operator is expanded with a Taylor-based asymptotic expansion truncated at order 2:
⎡
⎣p j=1
4αj (j a)2sinh2
j a 2
d d x
⎤⎦u(x)= d2 d x2
⎡
⎣1+a2 12
p j=1
j2αj d2 d x2
⎤
⎦u(x)+O
a4
(56) The Padé approximant of order [2,2] for the pseudo-differential operator can be alternatively used, as proposed by Rosenau [64]:
⎡
⎣p
j=1
4αj
(j a)2sinh2 j a
2 d d x
⎤⎦u(x)=
d2 d x2
1+a122 p
j=1j2αj d2 d x2
u(x)+... (57)
The pseudo-differential equation is then approximated by a linear second-order differential equation which is Eringen’s type nonlocal model defined by
1−βlc
2d2u
d x2 +βu=0 (58)
where the nonlocal length scale ratio calibrated from:
lc2=a2 p
j=1
j2αj
12 (59)
For direct neighbor interaction, this general equation simplifies to:
αj =0 if j ≥2 ⇒ lc2= a2
12 (60)
For fixed-fixed boundary conditions, the dimensionless frequency of the approx- imated nonlocal rod (of Eringen’s type) is then given by
β= π2 1+πL2l2c2
= π2 1+12nπ22
p j=1j2αj
(61)
Equation (61) is an approximation of Eq. (54) which is the exact eigenfrequency of the generalized lattice problem.