Free Vibrations of Stepped Nano-Beams

In the recent decade there has been considerable progress in the use of nano-plates and nano-beams due to the need of micro- and nano-electromechanical systems. It is known that the behaviour of nano-structures can be modelled with non-local theories of elasticity (see Eringen [1, 2], Reddy [3]). Lim [4, 5] has developed non-local bending theories and applied these for quasistatically loaded nano-beams. Buckling of nano-beams was studied with the help of non-linear non-local models by Reddy [3], Emam [6], Challamel et al. [7]. Analytical solutions for the transverse vibration of simply supported and clamped at both ends nano-beams with axial force are obtained by Li et al. [8], Lu et al. [9]. In Ref. [10] the Ritz method is accommodated for buckling and vibration of non-local beams. The vibrations of nano-beams with cracks are studied in Ref. [11] making use of the non-local theory of elasticity. In the present paper the free vibrations of nano-beams clamped at both ends are investigated. It is assumed that the nano-beams have stepped cross sections and that the beams are weakened with cracks located at the corners of steps.

Let us consider natural vibrations of a nano-beam of length l. The edges of the beam at x = 0 and x = l are fully clamped. The coordinate axis Ox coincides with the axis of corresponding straight beam; the onset of coordinates is located at the centre of the left-hand end of the beam. It is assumed that the nano-beam has rectangular cross sections with the width b and the height

In eqn. (1) the quantities h_{0 ,} h₁ and a are considered as given numbers. The nano-beam is weakened with a crack of length c at x = a. The crack length is expected to be constant. Thus the crack area is

The aim of the study is to determine the eigenfrequencies of natural vibrations of the nanobeam and to clarify the sensitivity of eigenfrequencies on the geometrical parameters of the beam and on the physical parameters of the material. The material of the nano-beam is assumed to be an elastic material obeying the constitutional equations of a non-local theory of elasticity (see Eringen [2], Lellep & Lenbaum [12]).

It was shown in the previous study by the authors [12] that the constitutional equations of non-local elasticity lead to the equation

where M stands for the bending moment and M_c is the moment calculated by the rules of the classical bending theory of thin-walled beams. Evidently (see Ref. [12]) the bending moment M_c and the deflection w are coupled as

where E is the Young’s modulus and I stands for the moment of inertia of the cross section of the beam. In eqn (3) η is a material constant (it is connected with the dimensions of the lattice of the nano-material). Combining eqns (3) and (4) one can write

On the other hand, it follows from the equilibrium equations (see Ref. [8]), that

In eqn (6) µ = ρbh stands for the mass per unit length of the nano-beam and ρ being the density of the material. Here t denotes time and N is the axial tension applied at the edges of the beam. Substituting eqn (6) into eqn (5) results in

where the notation

is used. Combining eqns (6) and (7) one easily obtains

The latter can be presented in the form

This is the equation of motion for nano-beams. Taking η = 0 in eqn (10), one obtains

The latter is the equation of free vibrations of beams in the classical beam theory. The eqn (10) can be solved with the method of separation of variables making use of appropriate boundary conditions. In the case of beams clamped at both ends the boundary conditions are

and

For solution of the linear fourth order differential equation with partial derivatives let us assume that

where j = 0 for x ∈(0,a) and j = 1 if x ∈(a,l). In eqn (14) the function W_j(x) depends only on x and the function T(t) is the function of time t. Differentiating eqn (14) with respect to x and t one can easily recheck that

In eqns (15) on has to take j = 0, if x ∈(0,a) and j = 1, if x ∈(a,l ) . Substituting the derivatives in eqns (15) into eqn (10), one obtains

The separation of variables in eqn (16) leads to the equation

where ω stands for the frequency of natural vibrations. It immediately follows from eqn (17) that

and

It is reasonable to assume that at the initial time instant $w(x, 0)=0 ; \dot{w}\left(x_*, 0\right)=v_0$. Taking eqn (14) into account, one can assume that

Evidently, the particular solution of eqn (18) satisfying the initial conditions in eqn (20) can be presented as

The eqn (19) is a linear fourth order differential equation with constant coefficients. The characteristic equation corresponding to eqn (19) has the form

The roots of the characteristic equation (22) are

where i is the imaginary unit,

and

In eqns (24) and (25) for the briefness sake the notation

is used. Taking eqns (22)–(26) into account, one can present the general solution of eqn (19) as

for $x \in S_j$, provided $S_0=[0, a], S_1=[a, l]$. In order to define particular solutions for regions $S_0$ and $S_1$, one has to satisfy the boundary conditions in eqns (12) and (13), also the boundary conditions for $M(x, t)$ and the appropriate continuity requirements for the deflection and bending moment.

For modelling the influence of the crack on the eigenfrequencies the method of distributed line springs suggested by Dimarogonas and his co-workers [13] will be used. According to this method, one has to introduce the additional compliance K , so that

where

Making use of eqns (7) and (14), one can introduce functions m_j(x), so that

where

for x ∈ S_j; j =0, 1. The relations in eqns (14), (30) and (31) admit to present the requirement in eqn (28) as

Since the bending moment M is considered at x = a, at any time instant, one gets m₁ (a+) = m₀ (a−), or according to eqn (31)

here square brackets denote the finite jump of corresponding quantity. This means that

for any variable y = y(x). Due to their physical background, it is clear that the deflection W(x) and the shear force Q = M ′ are continuous as well. Thus one has

and

Following Dimarogonas [13], also Lellep and Kraav [14], Lellep and Liyvapuu [15], the additional compliance caused by the crack is evaluated as

where I = min (I₀, I₁) , h = min (h₀, h₁) and

The function F = F(s) is obtained with the help of experimental data. In the current paper, as in [12]

In order to define the eigenfrequencies of the vibration, one has to specify the constants $C_{i j}(i=1, \ldots, 4 ; j=0,1)$ in eqn (27). Making use of the boundary conditions in eqns (12) and (13) and taking into account eqns (14) and (27), one easily obtains

and

Due to the continuity of the displacement W and the shear force at x = a, one has to take into account the requirements

where according to eqn (30) $M^{\prime}=m_j^{\prime}(x) T(t)$. It is worthwhile to mention, that the total set of boundary and intermediate requirements, which consists of eqns (32), (33) and (40) - (42), is a linear system of algebraic equations with respect to unknown constants $C_{10}, \ldots, C_{40} ; C_{11}, \ldots, C_{41}$. Since it is a linear homogenous system, the non-trivial solution exists under the condition, that the determinant $\Delta$ of this vanishes. Equalizing $\Delta=0$ leads to the equation for determination of natural frequencies.

The obtained set of algebraic equations is solved numerically with the aid of the computer program Mathcad. Numerical results are obtained for the nano-beam with dimensions $l=500 \mathrm{~nm}, b=10 \mathrm{~nm}$ and the material constant $\sqrt{\eta}=2 \mathrm{~nm}$. The results of calculations are presented in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5, where the lowest eigenfrequency $\omega$ is depicted. Here the stepped nano-beam with thicknesses $h_0=50 n m, h_1=\gamma \cdot h_0$ is investigated, where $s=\frac{c}{h_0}$ and $a$ is the point at which the crack occurs. In Figure 1, Figure 2, Figure 3 and Figure 4 the density of the material is $\rho=7850 \frac{\mathrm{~kg}}{\mathrm{~m}^3}$ and Young's modulus $E=200 G P a$. Figures 1 and 3 correspond to the case where $\gamma=1.2$. Different curves in Figure 1 and Figure 3 correspond to the crack extension $s=0.2 ; s=0.3$ and $s=0.4$. It can be seen from Figure 1 and Figure 3 that the shorter is the crack the higher is the natural frequency of vibration. The dependence of the eigenfrequency on the axial force $N$ and the crack location $a$ can be seen in Figure 1 and Figure 3, respectively. In Figure 2 and Figure 4 different curves correspond to the different ratios of thicknesses and a fixed crack extension $s=0.2$. It shows from Figure 2 and Figure 4 that a higher ratio of thickness corresponds to a higher frequency. The eigenfrequency versus the axial force $N$ (see Figure 2) and the step location $a$ (see Figure 4) can be observed here as well. Also an example for the case of a nano-beam with constant thickness is presented in each of Figure 1, Figure 2, Figure 3 and Figure 4. In Figure 5 the dependence of the eigenfrequency on the material of the nanobeam is depicted for a case where $\gamma=1.2, s=0.2$ and the axial force $N=500 n N$. Different curves in Figure 5 correspond to a nano-beam made of iron, copper, zinc or nickel, respectively.

A vibration analysis of nano-beams clamped at both ends was undertaken in the case of stepped beams with stable cracks or crack-like defects. It was shown that the defects have essential influence on the eigenfrequencies of nano-beams. Calculations carried out showed, that the lowest eigenfrequencies correspond to nano-beams with more severe defects. It was revealed that the higher is the tension applied to the nano-beam the higher is the frequency of natural vibrations.