3 Applications for One-Dimensional Problems
3.3 Dynamical Problems
3.3.1 Free Vibration Problem of a Beam
The model of the modified kernel presents a flexible behavior in comparison with the other investigated models. When e0a/L >0.03, however, the aforementioned model is affected by the locality more than the corresponding model in case of a concentrated load.
In addition, the deflections of the TPNI stress model monotonically evolve with respect tok1-parameter and when the nonlocal parameterer increases in Fig.16c, d.
The nonlocal deflection also tends to that of the classic-local model because of the augmentation ofk1-parameter.
N n=1
⎡
⎣k14
j=1
cnj xn
xn−1
d2φin
d x2 E Id2φnj
d x2 d x
+k2
xn xn−1
d2φni
d x2 E I M m=1
4 j=1
cmj sm
sm−1
A(|x−s|,e0a)d2φmj
ds2 dsd x
−ω2 4
j=1
cnj xn
xn−1
m0φni(x)φnj(x)d x
⎤
⎦
= −φiN(L)Q(L)˜ +φ1i(0)Q(0)˜ +dφiN
d x (L)M˜(L)−dφ1i
d x (0)M(0)˜ (64) where the arbitrary finite elements of the beam are denoted by Vn=(xn−1,xn) andVm =(sm−1,sm)and the numbers of FEs by N andM, respectively. Hermite shape functions are also denoted byφni(x)(i=1, . . . ,4), the approximate transverse deflection byψ(x)=4
j=1cnjφnj(x)as well as the generalized displacements bycj, respectively. Gauss quadrature rule is used for the integrals’ calculation of Eqs. (63) and (64).ω2(or|ω|) is then calculated by det
KN L−ω2M
=0.N=M =100 is also selected for all the below investigated problems.
(a) Cantilever beam
The essential and the natural BCs of a cantilever nanobeam are given by Eqs. (55) and (56). In Fig.17, the eigenfrequencies of the nonlocal models appear to have a softening response in comparison with those of the classic-local model, apart from the fundamental eigenfrequency of the nonlocal differential model. To be more specific, the fundamental eigenfrequency of the differential model increases as the nonlocal parameter augments (Fig. 17a). This remark has been obviously presented in the literature [35] and it is regarded as the paradox of the fundamental eigenfrequency of the nonlocal differential model of a cantilever beam.
What is more, the fundamental eigenfrequency of the model of the modified kernel takes larger values than those of the TPNI stress model for eachk1-parameter (Fig.
17a). However, that is not the case for the model of the modified kernel when the eigenfrequencies ω¯2,ω¯3,ω¯4 are studied. In those cases, its eigenfrequencies take smaller values than the eigenfrequencies of all the other investigated models (Fig.
17b–d).
Furthermore, the fundamental eigenfrequency of the TPNI stress model appears a non-monotonic behavior with respect tok1-parameter. Perhaps an explanation of this behavior centers on the fact that the TPNI stress model is unsuitable to suc- cessfully handle the boundary (Fig. 17a). On the other hand, the three remaining eigenfrequencies of the aforementioned model present a totally different behavior.
To be more specific, the corresponding eigenfrequencies of the TPNI stress model evolve in a monotonic way and they tend to those of the classic-local model ask1- parameter increases. Moreover, all the eigenfrequencies of the TPNI stress model exhibit a monotonic behavior with respect to the nonlocal parameter (Fig.17b, c).
Fig. 17 The first four normalized eigenfrequencies of a cantilever nanobeam with respect to the nonlocal parameter.aω¯1,bω¯2,cω¯3,dω¯4
The third and the fourth eigenfrequencies of the nonlocal differential model demonstrate a softening response in comparison with those of the TPNI stress model fork1 = {0.1,0.5}as well (Fig.17c, d).
To recap, the responses of the nonlocal integral models present a softening behav- ior compared to those of the classic-local model. The last-mentioned results are in accordance with the results of a structural element in the context of a lattice atomic model [29].
(b) Clamped-clamped beam
The essential BCs of a clamped-clamped nanobeam are given by Eq. (58). The eigen- frequencies of the nonlocal models appear to have a softening response in comparison with those of the classic-local model in Fig.18. Apart from the fundamental eigen- frequency, all the eigenfrequencies of the model of the modified kernel take smaller values than those of the other investigated models. As regards the fundamental eigen- frequency, the responses of the nonlocal integral models clearly differ from that of the nonlocal differential model (Fig.18a).
Fig. 18 The first four normalized eigenfrequencies of a clamped-clamped nanobeam with respect to the nonlocal parameter.aω¯1,bω¯2,cω¯3,dω¯4
What is more, the eigenfrequencies of the TPNI stress model monotonically evolve with respect tok1-parameter. In other words, the aforementioned eigenfrequencies tend to those of the classic-local model ask1-parameter increases. Besides, these eigenfrequencies decrease further when the nonlocal parameter augments.
Regarding the model of the modified kernel, its fundamental eigenfrequency diminishes up to a specific value of the nonlocal parameter (extremum), and when the nonlocal parameter exceeds this value, the eigenfrequency’s behavior changes and starts to augment (Fig.18a).
For the other three investigated eigenfrequencies, the model of the modified ker- nel and the TPNI stress model withk1=0.01 present a competitive behavior with each other (Fig.18b). The modified kernel’s model also demonstrates a monotoni- cally decreasing behavior with respect to the nonlocal parameter (Fig.18b–d). The responses of the nonlocal differential model demonstrate a monotonically decreasing behavior as well.
(c) Clamped-pinned beam
The essential and the natural BCs of a clamped-pinned nanobeam are given by Eqs.
(59) and (60). The responses of a clamped-pinned beam appear to have a similar
Fig. 19 The first four normalized eigenfrequencies of a clamped-pinned nanobeam with respect to the nonlocal parameter.aω¯1,bω¯2,cω¯3,dω¯4
behavior with those of a clamped-clamped beam in Fig.19. Unlike the fundamental eigenfrequencies of a clamped-clamped beam, the fundamental eigenfrequencies of the nonlocal differential and the nonlocal integral models of a clamped-pinned beam do not present such a significant difference (Fig.19a).
(d) Simply supported beam
The essential and the natural BCs of a simply supported nanobeam are given by Eqs. (61) and (62). The eigenfrequencies of the nonlocal models appear to have a softening response in comparison with those of the classic-local model in Fig.20.
Apart from the fundamental eigenfrequency whene0a/L>0.03 (Fig.20a), all the eigenfrequencies of the model of the modified kernel take smaller values than those of the other investigated models.
What is more, the eigenfrequencies of the TPNI stress model monotonically evolve with respect tok1-parameter. In other words, the above eigenfrequencies tend to those of the classic-local model ask1-parameter increases. Besides, these eigenfrequencies decrease further when the nonlocal parameter augments.
As regards the model of the modified kernel, the decrease rate of its fundamental eigenfrequency reduces when the nonlocal parameter increases (Fig.20a). It is crit- ical to mention that the transverse deflections of the corresponding static problem present a similar behavior with respect to the nonlocal parameter. To elucidate, the increase of the contribution of the neighbouring particles to the nonlocal stresses at the interior points of the body is triggered by the augmentation of the nonlocal parameter. Nevertheless, the locality’s contribution in stresses at an area close to the boundary is significant. The model of the modified kernel therefore suggests a behavior according to which the local attribute is of importance when the nonlocal parameter increases. Furthermore, the rest of the eigenfrequencies of the model of the modified kernel appear to have a linear, decreasing behavior with respect to the nonlocal parameter (Fig.20b–d).
The responses of the nonlocal differential model demonstrate a monotonically decreasing behavior as well.
Fig. 20 The first four normalized eigenfrequencies of a simply supported nanobeam with respect to the nonlocal parameter.aω¯1,bω¯2,cω¯3,dω¯4