^{1}

^{*}

^{2}

^{*}

In this paper, we will study the most important effects in the nano-scale resonator: the coupling effect of temperature and strain rate, and the non-Fourier effect in heat conduction. A solution for the generalized thermoelastic vibration of nano-resonator induced by thermal loading has been developed. The Young’s modulus is taken as a linear function of the reference temperature. The effects of the thermal loading and the reference temperature in all the studied fields have been studied and represented in graphs with some comparisons. The Young’s modulus makes significant effects on all the studied fields where the values of the temperature, the vibration of the deflection, stress, displacement, strain, stress-strain energy increase when the Young’s modulus has taken to be variable.

Diao et al. [

Recently, nano mechanical resonators have attracted considerable attention due to their many applications on technology. The analysis of various effects on the characteristics of resonators, such as the resonant frequencies and the quality factorsis crucial for designing high-performance components. Many authors have studied the vibration and the heat transfer process of nano-beams [

The temperature dependence of the Young’s modulus for some materials was measured in the range of 293K and 973 K, using the impulse excitation method and compared with literature data reported. The data could be fitted with [

The values of parameters E_{0} and T_{0} are related to the temperature and the parameter B to the harmonic character of the medium.

Farraro and Rex found that no departure from linearity was detected when they studied the dependency of the Young’s modulus on the temperature, and the get the linear relation [

where

Now, we will consider the Young’s modulus depends on the temperature by the following function

where

In this paper, the non-Fourier effect on heat conduction, and the coupling effect between temperature and strain rate in the nano-scale beam will be studied when Young’s modulus is variable as a function of temperature. A general solution for the generalized thermoelastic vibration of gold nano-beam resonator induced by thermal shock will be developed. Laplace transforms and direct method will be used to get the lateral vibration, the temperature, the displacement, the stress-strain energy of the beam. The effects of Young’s modulus will be studied and represented graphically.

Since nano-beams with rectangular cross-sections are easier to fabricate, such cross-sections are commonly adopted in the design of NEMS resonators. Consider small flexural deflections of a thin elastic beam of length

along the longitudinal, width and thickness directions of the beam, respectively (_{0} everywhere [

In the present work, the Euler-Bernoulli equation is considered, and then, any plane cross-section, initially perpendicular to the axis of the beam remains plane and perpendicular to the neutral surface during bending. Thus, the displacements are given by [

Thus, the differential equation of thermally induced lateral vibration of the beam may be expressed in the form [

where ^{3}/12] the inertial moment about x-axis,

where _{0} the room temperature.

According to Lord-Shulman model (L-S), the non-Fourier heat conduction equation has the following form [

Where

and assuming the temperature varies in terms of a

Hence, Equation (6) gives

Moreover, Equation (8) gives

After doing the integrations, Equation (10) takes the form

In Equation (11), we multiply the both sides by z and integrating with respect to z from

we obtain

where

For simplicity, we will use the following dimensionless variables [

Then, we have

and

where

For convenience, we dropped the prime.

Applying the Laplace transform for Equations (14) and (15), this is defined by the following formula

Hence, we obtain the following system

and

We will consider the function

Then, we have

and

where

Consider the first end of the beam x = 0 is clamped and loaded thermally, which gives [

and

where

By using Laplace transform, the conditions will take the forms

and

Consider the other end of the beam

After using Laplace transform, we have

After some simplifications by using MAPLE programme, we get the final solutions in the Laplace transform domain as follows:

The lateral deflection

The temperature

The displacement

The Strain

where

and

The stress on the x-axis, according to Hooke’s law is:

By using the non-dimensional variables in (13), we obtain the stress in the form

By using Laplace transform, the above equation takes the form:

The stress-strain energy, which is generated by the beam, is given by

We can re-write Equation (36) to be in the form

To complete the solution in the Laplace transform domain, we have to determine the type of heating which we have used to load the boundary of the medium thermally.

We have applied harmonic thermal loading as follows [

after using Laplace transform, we obtain

To determine the solutions in the time domain, the Riemann-sum approximation method is used to obtain the numerical results. In this method, any function in Laplace domain can be inverted to the time domain as

where Re is the real part and i is imaginary number unit. For faster convergence, numerous numerical experiments have shown that the value of

Now, we will consider a numerical example for which computational results are given. For this purpose, Gold (Au) is taken as the thermoelastic material for which we take the following values of the different physical constants [

The aspect ratios of the beam are fixed as

For the nano-scale beam, we will take the range of the beam length

The figures (

The Young’s modulus has significant effects on all the studied fields. The values of the temperature, the vibra-

tion of the deflection, stress, displacement, strain, stress-strain energy increase when the Young’s modulus is variable. The peak points of all the distributions increase when the Young’s modulus is variable with large differences in the case of Young’s modulus is constant.

Eman A. N.Al-Lehaibi,Hamdy M.Youssef, (2015) Vibration of Gold Nano-Beam with Variable Young’s Modulus Due to Thermal Shock. World Journal of Nano Science and Engineering,05,194-203. doi: 10.4236/wjnse.2015.54020