# Natural vibrations of curved nano-beams and nano-arches

Jaan Lellep and Shahid Mubasshar  Institute of Mathematics and Statistics,
University of Tartu, Estonia<EMAIL_ADDRESS>Institute of Mathematics and
Statistics, University of Tartu, Estonia<EMAIL_ADDRESS>
###### Abstract

The natural vibrations of curved nano-beams and nano-arches are studied. The
nano-arches under consideration have piece wise constant thickness; these are
weakened with stable cracks located at re-entrant corners of the steps. A
method of determination of natural frequencies is developed making use of the
method of weightless rotating spring. The aim of the paper is to assess the
sensitivity of the eigenfrequencies on the geometrical and physical parameters
of the nano-arch. The results of the calculations favourably compare with
similar works of other researchers.

##  1 Introduction

The main ideas of the non-local theory of elasticity were initially formulated
by Eringen [1] and Eringen and Edelen [2] several decades ago. However, the
rapid progress in the non-local theories got its start with the wide use of
nanomaterials (see Thai [3], Thai et al [4], Reddy [5]). Comprehensive reviews
of papers dedicated to the mechanical behaviour of nano-beams, nano-plates and
nano-arches can be found in the review papers by Faghidian [6, 7] Farajpour,
Ghoyesh and Farokhi [8], also in Wang and Arash [9]. It is interesting to
remark that the non-local theory was initially formulated in the integral form
and later it was reformulated by Eringen in the differential form making use
of a specific kernel function. Since the differential form is more simple it
is widely used in the analysis of nanostructures.

A general approach to the application of the non-local theories of the
elasticity in the bending, buckling and vibration of nano-beams was developed
by Aydogdu [10]. Resorting to the results obtained by Nayfeh and Emam [11] a
comparative study on the buckling and post-buckling analysis of nano-beams was
undertaken by Emam [12]. The formulations corresponding to the classical
Euler-Bernoulli approach, the first order Timoshenko theory and higher order
shear deformation Reddy theories are analysed in detail. The comparison of
classical bending theory and the shear deformation theories is carried out by
Reddy [5, 13], as well.

Vibrations of nano-beams containing cracks and other defects are studied by
Roostai and Haghpanahi [14], Loya [15], also by Lellep and Lenbaum[16],
Loghmani and Yazdi [17], Hossain and Lellep [18, 19]. In the paper [19] the
effect of the temperature is taken into account.

In the paper by Jiang and Wang [20] analytical solutions are developed for
nano-beams subjected to axial and thermal stresses.

The vibrations of curved beams or arches and the detection of damages in these
structures is studied by Cerri and Ruta [21], Viola, Hrtioli and Dilena [22]
with the help of vibrational analysis. Viola and Tournabene [23] undertake the
analysis of circular arches of variable cross-section. The effects of cracks
are taken into account in the paper by Karaagae et al [24].

In the present study an attempt is made to define the natural frequencies of
circular nano-beams and to study the sensitivity of eigenfrequencies on the
geometrical and material parameters of the nano-arch.

##  2 Formulation of the problem

Let us consider the dynamic behaviour of a nano-arch of radius $R$. For
determination of the positions of central points of the arch the polar
coordinates are introduced (Fig 1).

Figure 1: Elastic stepped nano-arch.

Assume that the angles $\varphi=0$ and $\varphi=\beta$ are correspond to the
edges of the nano-arch, respectively.

It is assumed that the thickness of the nano-arch is defined as

$h=\begin{cases}\begin{aligned} &h_{0}\quad&,\quad&\varphi\in[0,\alpha)\\\
&h_{1}\quad&,\quad&\varphi\in(\alpha,\beta]\\\ \end{aligned}\end{cases}$ (1)

In (1), $h_{0},h_{1}$ are given numbers and $\alpha\in(0,\beta)$. The width
$b$ of the nano-arch is assumed to be a constant. The aim of the study is to
determine the eigenfrequencies of stepped nano-beams weakened with defects at
the re-entrant corners of steps. It is assumed that at the cross section
$\varphi=\alpha$ a crack with length $c$ is located. The defect is treated as
a stable crack; no attention will be paid to the extension of the crack during
vibrations.

However, the sensitivity of the eigenfrequency with respect to the position of
the crack and other parameters will be clarified.

##  3 Governing equations and main hypotheses

The system of governing equations includes the equilibrium equations and
constitutive and geometrical relations with boundary conditions. The
equilibrium equations for an element of the arch or curved beam can be
presented as (see Soedel [25]),

$\displaystyle\frac{\partial{N}}{\partial s}{}+{}\frac{Q}{R}$
$\displaystyle+p_{u}$ $\displaystyle=$ $\displaystyle\bar{\rho}h\ddot{U},$ (2)
$\displaystyle\frac{\partial{Q}}{\partial s}{}-{}\frac{N}{R}$
$\displaystyle+p$ $\displaystyle=$ $\displaystyle\bar{\rho}h\ddot{W},$
$\displaystyle\frac{\partial{M}}{\partial{s}}$ $\displaystyle-Q$
$\displaystyle=$ $\displaystyle 0$

In (2), $N$, $M$ and $Q$ denote the membrane force, bending moment, and shear
force, respectively. Here $s$ is curvilinear coordinate related to the current
angle as

$s=R\varphi.$ (3)

In (2), $U$, $W$ stand for the displacements in the circumferential and normal
direction, respectively and $p_{u}$ and $p$ are the intensities of the
distributed loading in these directions. Here $h$ stands for the thickness of
the arch and $\bar{\rho}$ is the mass per unit length of the arch. Let $b$ be
the width of the arch. In (2), dots denote the differentiation with respect to
time $t$. This means that

$\ddot{U}=\frac{\partial^{2}{U}}{\partial
t^{2}},\quad\ddot{W}=\frac{\partial^{2}{W}}{\partial t^{2}}.$ (4)

The strain-displacement equations can be taken according to Soedel [25] as

$\varepsilon=\frac{1}{R}W+\frac{\partial{U}}{\partial{s}}$ (5)
$\varkappa=-\frac{\partial^{2}{U}}{\partial
t^{2}}+\frac{\partial{U}}{\partial{s}.R}$ (6)

In (5), (6) $\varepsilon$ stands for the relative extension of the curved
element and $\varkappa$ is the curvature of the middle line of the arch.

It is assumed herein that the vibrations of the arch are in-place motions
only.

According to the Hook’s law in the classical theory of the elasticity one has

$M_{c}=EI\varkappa,$ (7)

where $M_{c}$ denotes the bending moment in the classical theory Here $E$ is
the Young modulus and $I$—the moment of inertia of the cross section of the
arch. Evidently, in the case of the rectangular cross section

$I=\frac{1}{12}bh^{3}.$ (8)

As $h$ is a piecewise continuous function of the current angle the second
moment (8) has the values $I_{0}$ and $I_{1}$ in the regions ($0,\alpha$) and
($\alpha,\beta$), respectively. It is assumed that the material of the arch
obeys the constitutive equations of the non-local theory of elasticity (see
Eringen [1], Reddy [5]). In the non-local theory the stresses at a given point
of the body depend on the strains at all points of the body. One of the
simplest physical relations of this type can be presented as (see Reddy [5],
Emam [12])

$\sigma_{ij}-\eta\nabla^{2}\sigma_{ij}=\sigma^{c}_{ij},$ (9)

where $\sigma_{ij}$ denotes the components of non-local stress tensor and
$\sigma^{c}_{ij}$ stands for the classical elastic stress tensor. Here
$\eta=(e_{0}a)^{2}$ stands for a material constant, $a$ being the dimension of
the lattice of the material and $e_{0}$—a physical constant to be defined
experimentally.
Instead of stress components $\sigma_{ij}$ in (9) one can use the generalized
stresses used in the equilibrium equations (2).
However, it is reasonable to adapt the system (2) to the current problem. Let
us assume that $p_{u}=0$, $U(Q,t)$ is small and that $\ddot{U}$ is also
negligible. In this case the system (2) can be rewritten as

$\displaystyle N^{{}^{\prime}}$ $\displaystyle=-Q,$ (10) $\displaystyle
Q^{{}^{\prime}}$ $\displaystyle=N+R(\rho h\ddot{W}-p),$ $\displaystyle
M^{{}^{\prime}}$ $\displaystyle=RQ,$

where prim denotes the differentiation with respect to the current angle
$\varphi$.
It can be seen from (10) that in the case when $M(\varphi)$ and $N(\varphi)$
vanish at the same cross section one has

$M=-RN.$ (11)

Equations (10) and (11) result in

$M^{{}^{\prime\prime}}+M+R^{2}(p-\rho h\ddot{W})=0.$ (12)

Taking in (9) instead of $\sigma_{ij}$ the moment $M$ yields

$M-\eta M^{{}^{\prime\prime}}=M_{c}.$ (13)

Combining (13) and (7) with (12) one obtains

$M=\frac{1}{1+\eta}[-EIW^{{}^{\prime\prime}}+h\eta\rho\ddot{W}].$ (14)

Substituting the bending moment from (14) into (12) one obtains the equation

$EI(W^{IV}+W^{{}^{\prime\prime}})+\eta\rho hR^{2}\ddot{W}-\eta\rho
hR^{2}(1+\eta)\ddot{W}^{{}^{\prime\prime}}=0.$ (15)

Equation (15) is a fourth order equation. The boundary conditions for (15) are
in the case of simply supported arches

$\displaystyle W(0,t)=0,$ (16) $\displaystyle W(\beta,t)=0$

and

$\displaystyle M(0,t)=0,$ (17) $\displaystyle M(\beta,t)=0.$

##  4 Solution of the governing equations

Making use of the method of separation of variables it is assumed that

$W(\varphi,t)=X(\varphi).T(t),$ (18)

where $X(\varphi)$ a function of the coordinates $\varphi$ and $T(t)$—a
function depending on time $t$ only.
Differentiating (18) one can find easily that

$\displaystyle\ddot{W}$ $\displaystyle=X(\varphi).\ddot{T}(t),$ (19)
$\displaystyle W^{{}^{\prime\prime}}$
$\displaystyle=X^{{}^{\prime\prime}}(\varphi).T(t),$ $\displaystyle W^{IV}$
$\displaystyle=X^{IV}(\varphi).T(t),$
$\displaystyle\ddot{W}^{{}^{\prime\prime}}$
$\displaystyle=X^{{}^{\prime\prime}}(\varphi).\ddot{T}(t).$

The substitution of (19) in (15) leads to the differential equations

$\ddot{T}+\omega^{2}T=0$ (20)

and

$EI\frac{X^{IV}+X^{{}^{\prime\prime}}}{X}-\omega^{2}\eta\rho
hR^{2}(1-\frac{X^{{}^{\prime\prime}}}{X})=0$ (21)

In (20) $\omega$ stands for the frequency of natural vibrations of the arch.
The equation (21) must be solved separately in regions $(0,\alpha)$ and
$(\alpha,\beta)$ respectively, since $h$ is defined by (1) so that it has
different values in these sections. Evidently, the equation (20) has
periodical solutions. One of these can be taken as

$T=\bar{A}\sin(\omega t)$ (22)

Equation (21) can be presented in the form

$X^{{}^{\prime\prime\prime\prime}}+(1+K_{j})X^{{}^{\prime\prime}}-K_{j}X=0,$
(23)

where

$K_{j}=\begin{cases}\begin{aligned}
&K_{0}\quad&,\quad&\varphi\in(0,\alpha),\\\
&K_{1}\quad&,\quad&\varphi\in(\alpha,\beta).\\\ \end{aligned}\end{cases}$ (24)

In (23), (24)

$K_{j}=\frac{\omega^{2}\eta\rho h_{j}R^{2}}{EI_{j}}$ (25)

where $j=0,1$. Here

$I_{0}=\frac{bh_{0}^{3}}{12}\quad,\quad I_{1}=\frac{bh_{1}^{3}}{12}$ (26)

In order to solve the linear fourth order equation (23) one has to solve the
characteristic equations

$\lambda_{j}^{4}+(1+K_{j})\lambda_{j}^{2}-K_{j}=0$ (27)

It is easy to recheck that

$\lambda_{j}^{2}=\frac{1}{2}(-1-K_{j}\pm\sqrt{1+6K_{j}+K_{j}^{2}})$ (28)

and

$\lambda_{j}=\pm\\{\frac{-1-K_{j}}{2}\pm\frac{1}{2}\sqrt{1+6K_{j}+K_{j}^{2}}\\}^{\frac{1}{2}}$
(29)

Thus the general solution of the (21) has the form

$X_{j}=C_{1}\cosh\mu_{j}\varphi+C_{2}\sinh\mu_{j}\varphi+C_{3}\cos\nu_{j}\varphi+C_{4}\sin\nu_{j}\varphi$
(30)

where

$\displaystyle\mu_{j}$
$\displaystyle=-\frac{1}{2}(1+K_{j})+\frac{1}{2}\sqrt{1+6K_{j}+K_{j}^{2}}$
(31) $\displaystyle\nu_{j}$
$\displaystyle=\frac{1+K_{j}}{2}+\frac{1}{2}\sqrt{1+6K_{j}+K_{j}^{2}}$

It is expected in (27)- (31) that $j=0$, if $\varphi\in(0,\alpha)$ and $j=1$
if $\varphi\in(\alpha,\beta)$.
The boundary conditions (16) and (17) together with (18) admit to assert that
the boundary requirements for $X_{j}(t)$ are

$\displaystyle X_{j}(0)$ $\displaystyle=0,$ (32) $\displaystyle
X^{{}^{\prime\prime}}_{j}(0)$ $\displaystyle=0,$

and

$\displaystyle X_{j}(\beta)$ $\displaystyle=0,$ (33) $\displaystyle
X^{{}^{\prime\prime}}_{j}(\beta)$ $\displaystyle=0.$

Note that (32) and (33) hold good in the case of the arch simply supported at
both edges.

##  5 Local flexibility due to the crack

It is recognized long time ago that cracks and other defects are the sources
of additional structural compliance. Dimarogonas [26] Chondros et al [27] also
Kukla [28], suggested to treat the slope of deflection $W^{{}^{\prime}}$ as a
discontinuous variable and to define a new variable

$\theta=W^{{}^{\prime}}(\alpha+,t)-W^{{}^{\prime}}(\alpha-0,t)$ (34)

so that

$\theta=cM(\alpha,t)$ (35)

where $c$ can be considered as the additional compliance. In (35) $M$ stands
for the moment of internal forces at $\varphi=\alpha$. Evidently, $c$ can be
an appropriate matrix, as well. In this case instead of $M$ one has the vector
of corresponding internal forces applied at the same cross section.
It is known in the linear elastic fracture mechanics that the energy release
rate due to the crack can be calculated as

$G=\frac{M^{2}}{2b}\frac{dC}{c}.$ (36)

In (36) $b$ is the width of the crack of rectangular cross section and $M$ is
the moment applied at this cross section and $c$ stands for the length of the
crack.
On the other hand, the stress intensity factor can be defined as (see Anderson
[29]),

$K=\sigma\sqrt{\pi c}F(s)$ (37)

where $s=\frac{c}{h}$ and

$\sigma=\frac{6M}{bh^{2}}$ (38)

where $F(s)$ is the shape function.
The quantity $K$ is coupled with the energy release rate as

$G=\frac{K^{2}}{E^{{}^{\prime}}}$ (39)

In (39) $E^{{}^{\prime}}=E$ for the plane stress state and
$E^{{}^{\prime}}=\frac{E}{1-\nu^{2}}$ for plane strain state. Following
Dimarogonas [26] , Rizos et al [30] one can use the function $F(s)$ in the
form

$F(s)=1.93-3.07s+14.53s^{2}-25.11s^{3}+25.8s^{4}$ (40)

Another shape function often used by researchers is (see Tada et al [25],
Dimarogonas [26])

$F(s)=\frac{\sqrt{\tan\psi}}{\psi\cos\psi}[0.923+0.199(1-\sin\psi)^{4}]$ (41)

where $\psi=\frac{\pi s}{2}$.

##  6 Numerical results

Numerical results are presented for nano-arches simply supported at both ends
in fig 2-6. Here the nano-arches with a single step whereas $R=30nm$,
$h_{0}=10nm$, $h_{1}=20nm$, the material constant are $E=7\times 10^{11}Pa$,
$\nu=0.3$

Figure 2: Natural frequency Vs thickness of the nano-arch

Figure 3: Natural frequency Vs step location

Figure 4: Natural frequency Vs step location for different values of $\beta$

Figure 5: Natural frequency Vs radius of the nano-arch.

Figure 6: Natural frequency Vs material constant of the nano-arch.

Figure 7: Natural frequency Vs central angle of the nano-arch. Mode | $\eta$ | Present | Thai[3]
---|---|---|---
1 | 0 | 9.75821445 | 9.2745
| 1 | 7.05584295 | 8.8482
| 2 | 5.80188130 | 8.4757
| 3 | 5.04192108 | 8.1466
| 4 | 4.51883759 | 7.8530
Table 1: Natural vibration of the nano-arches of the constant thickness,
without any crack and with central angle very very small

In figure (2) the natural frequency of the nano-arch versus the thickness is
depicted for different values of the radius $R$. It can be seen from the
figure (2) that the natural frequency increases if the thickness of the arch
increases. However, if the radius of the arch increases then the natural
frequency decreases.
In the next figures (3)-(7) corresponding data for one-stepped arches is
presented. Here $h_{0}=10nm$, $h_{1}=20nm$.
In figures (3) and (4) the natural frequency as a function of the step
location $\alpha$ is presented. In figure (3) the nano-arch with the central
angle $\beta=1rad$ is treated. Different curves in figure (3) are correspond
to the crack extensions $s=0$, $s=0.1$. $s=0.2$, $s=0.3$,
$s=0.4$,$s=0.5$,$s=0.6$,$s=0.7$ respectively. It can be seen from figure (3)
that the lowest values of the natural frequency correspond to the arch without
any cracks. In figure (4) similar results are presented for different angle
values of the central angle $\beta$ (here $s=0.7$). It reveals from figure (4)
that in the case of smaller values of the angle $\alpha$ the lowest
eigenfrequency is achieved in the case of the largest value of the central
angle $\beta$. However, in the case when $\beta>2\alpha$ this relationship is
more complicated.
The relationship between the natural frequency and the radius $R$ of the nano-
arch is shown in figure (5) in the case $\beta=1$ and $s=0.7$ (here
$\eta=1nm$).
It reveals from figure (5) that the smaller is $\alpha$ the larger is the the
natural frequency of the nano-arch.
On the other hand, the larger is $\alpha$ the lower is the eigenfrequency.
The dependence of the eigenfrequency on the material parameter
$\eta=(e_{0}a)^{2}$ is demonstrated in figure (6) for different value of the
radius $R$. Generally speaking, the eigenfrequency decreases if the material
parameter increases. Thus, in the case $\eta=0$ the natural frequency has its
maximal value.
The sensitivity of the eigenfrequency on the central angle $\beta$ of the
nano-arch is shown in figure (7) for radius $R=30nm$. Different curves in
figure (7) correspond to nano-arches having the step in different cross
sections (here $\eta=1nm$). One can see from figure (7) that in the case of
larger values of the angle $\beta$ curves corresponding to different values of
the step angle are quite close to each other. However, if $\beta$ is around
the value $0.5$ the discrepancies between these curves are large.
The results obtained in the current study are compared with those obtained by
Thai [3] in the case when $\alpha=0$, $\beta=0.5^{\circ}$. Table (1) shows
that the results are in reasonable correspondence.

##  7 Concluding remarks

Natural vibrations of curved nano-beams are treated in the current study. The
nano-beams have piecewise constant thickness with stable cracks at the re-
entrant corners of steps. A method for determination of natural frequencies
based on the methods of the linear elastic fracture mechanics is developed.
Calculations carried out showed that the cracks effect essentially the values
of natural frequencies.
It is shown that the lowest values of the natural frequency correspond to
nano-arches without any defect. The results of current study are compared with
results obtained by Thai[3]. The results compare favourably in the case of
small values of the parameter $\eta$.

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