Vibration of Gold Nano Beam in Context of Two-Temperature Generalized Thermoelasticity Subjected to Laser Pulse

In the present work, the model of vibration of gold nanobeam induced by laser pulse heating is developed in the context of two-temperature generalized thermoelasticity and non-Fourier heat conduction. The analytic solution has been derived in the Laplace transform domain. The inverse Laplace transform has been calculated numerically and the numerical results have been presented graphically in two and three dimensions figures with some comparisons to stand on the effects of the twotemperature parameter and the laser pulse parameters on all the studying fields and which one of that parameters plays a vital role in the damping of the energy which has been generated inside the beam.

Gold Nano-Beams based nano-carriers have great potential in biomedical fields, such as biological monitoring, imaging, thermotherapy and multifunctional nano-complex diagnose, and all these bring ideas and hopes to the development of biomedicine.In connection with structure, property and bio-effects of Gold Nano-Beams, developing real-time, sensitive, high through put detection, and analysis methods could be the important consults to rational design of Gold Nano-Beams based nano-carriers.Ultimately, through surface modification and functionalization, it could improve targeting of Gold Nano-Beams, reduce immune response and other negative effects, these Gold Nano-Beams-based multifunctional nano-carriers will play crucial role in future biocatalysis, disease diagnosis, imaging and therapy Wang et al. (2013).
Many works and articles have been made recently to investigate the elastic properties of nanostructured materials by atomistic simulations.Diao et. al. (2004) studied the effect of free surfaces on the structure and elastic properties of gold nanowires by atomistic simulations.Although the atomistic simulation is a terrific way to calculate the elastic constants of nanostructured materials, it is only applicable to homogeneous nanostructured materials, for example, nanoplates, nanobeams and nanowires with a limited number of atoms.Moreover, it is difficult to obtain the elastic properties of the heterogeneous nanostructured materials using atomistic simulations.For these and other reasons, it is prudent to seek a more practical approach.One such approach would be to extend the classical theory of elasticity down to the nanoscale by including in it the hitherto neglected surface/interface effect.For this, it is necessary first to cast the latter within the framework of continuum elasticity.
Nano-mechanical beams have attracted considerable attention recently due to their many significant technological applications.Accurate analysis of various effects on the characteristics of beams, such as resonant frequencies and quality factors, is crucial for designing high-performance components.Many authors have studied the vibration and heat transfer process of beams.Kidawa (2003) has studied the problem of transverse vibrations of a beam induced by a mobile heat source.The analytical solution to the problem was obtained using the Green's functions method.However, Kidawa did not consider the thermoelastic coupling effect.Boley (1972) analyzed the vibrations of a simply supported rectangular beam subjected to a suddenly applied heat input distributed along its span.Manolis and Beskos (1980) examined the thermally induced vibration of structures consisting of beams, exposed to rapid surface heating.They have also studied the effects of damping and axial loads on the structural response.Al-Huniti et. al. (2001) investigated the thermal induced displacements and stresses of a rod using the Laplace transformation technique.Ai Kah Soh et al. (2008) studied the vibration of micro/nanoscale beams induced by ultra-short-pulsed laser by considering the thermoelastic coupling term.Sun et. al. (2008Sun et. al. ( , 2006) ) constructed a model of thermoelastic damping in micro-beams, and Fang et. al. (2006) got the analysis of the frequency spectrum of laser, induced vibration of microbeams.Eringen (1983) reduced Integro-partial differential equations of the linear theory of nonlocal elasticity to singular partial differential equations for a special class of physically admissible kernels.Civalek and Akgöz (2010) presented free vibration analysis of microtubules (MTs) based on the Euler-Bernoulli beam theory.Liew et al. (2008) simulated the flexural wave propagation in a single-walled carbon nanotube (SWCNT) by using molecular dynamics (MD) based on a second-generation reactive empirical bond order (REBO) potential.Civalek and Demir (2011) formulated the equations of motion and bending of Euler-Bernoulli beam using the nonlocal elasticity theory for cantilever microtubules (MTs).Youssef and Elsibai (2010) investigated the vibration of gold nano-beam induced by many types of thermal loading.Youssef (2006) investigated two-temperature generalized thermoelasticity theory together with a general uniqueness theorem and solved many applications in the context of this theory with Al-Lehaibi (2007), Harby (2007) and Bassiouny (2008).In this work, the model of vibration of gold nano-beam induced by laser pulse heating will be developed in the context of two-temperature generalized thermoelasticity and non-Fourier heat conduction.The analytic solution will be derived in the Laplace transform domain.The inverse Laplace transform will be calculated numerically and the numerical results will be presented graphically in two and three dimensions figures with some comparisons to stand on the effects of the twotemperature parameter and the laser pulse parameters on all the studying fields and which one of that parameters plays a vital role in the damping of the energy which will be generated inside the beam.

FORMULATION OF THE PROBLEM
Since beams with rectangular cross-sections are easy to fabricate, such cross-sections are commonly adopted in the design of NEMS beams.Consider small flexural deflections of a thin elastic beam of length   , for which the x, y and z axes are defined along the longitudinal, width and thickness directions of the beam, respectively.In equilibrium, the beam is unstrained, unstressed, without damping mechanism, and the temperature is T0 everywhere, Soh et al. (2008), Sun et al. (2006), Youssef and Elsibai (2010).
In the present study, the usual Euler-Bernoulli assumption is adopted, i.e., any plane crosssection, initially perpendicular to the axis of the beam remains plane and perpendicular to the neutral surface during bending.Thus, the displacements u , v, w are given by Hence, the differential equation of thermally induced lateral vibration of the beam may be expressed in the form: where E is Young's modulus, I [= bh 3 /12] the inertial moment about x-axis,  the density of the beam, T  the coefficient of linear thermal expansion,   w x,t the lateral deflection, x the dis- tance along the length of the beam, A hb  is the cross section area and t the time and T M is the thermal moment, which is defined as: where is the dynamical temperature increment of the resonator, in which T(x, z, t) is the temperature distribution and T0 the environmental temperature.The Laser where p t is a characteristic time of the laser-pulse, 0 I is the laser intensity (the total energy carried by a laser pulse per unit cross-section of laser beam) Sun et al. (2006).
The heat source where  is the absorption depth of heating energy and R is the surface reflectivity.
According to Youssef model of two-temperature thermoelasticity, the non-Fourier heat conduction equation has the following form: where C  is the specific heat at constant volume, 0  the thermal relaxation time, k the thermal conductivity, in which  is Poisson's ratio,  is the conductive temperature increment it satisfies the following relation Youssef (2008): where  is non negative parameter (two-temperature parameter).
The volumetric strain takes the form Since there is no heat flow across the upper and lower surfaces of the beam, then, For a very thin beam and assuming the temperature varies in terms of a   sin pz function along the thickness direction, where p /h   , gives: and equation ( 6) gives After doing the integrations, equation (13) takes the form In equation ( 14), we multiply the both sides by z and integrating with respect to z from where Now, for simplicity we will use the following non-dimensional variables, Youssef (2008): Then, we have and where (We have dropped the prime for convenience)

FORMULATION THE PROBLEM IN THE LAPLACE TRANSFORM DOMAIN
We will apply the Laplace transform for equations ( 18) -( 20), which is defined by the following formula Hence, we obtain the following system We will consider the function  as follows: Then, we have and where Hence, we have where where and Eliminating 1  from equations ( 33) and ( 34), we get Eliminating  from equations ( 33) and ( 34), we obtain Now, we will consider the first end of the nano-beams x=0 is clamped and has no thermal load, which gives After using Laplace transform, the above conditions take the forms Considering the other end of the beam x   is clamped and remains at zero increment of temper- ature as follows: After using Laplace transform, we have After some simplifications, we get the final solutions in the Laplace transform domain as follows: By using the boundary conditions, we get the following system of linear equations Solving the above system, we get

THE STRESS AND THE STRAIN-ENERGY
The stress on the x-axis, according to Hooke's law is, Fang et al. (2006): By using the non-dimensional variables in (9), we obtain the stress in the form After using Laplace transform, the above equation takes the form: The stress-strain energy which is generated on the beam is given by or, we can write as follows: where   1 L   is the inversion of Laplace transform.

NUMERICAL INVERSION OF THE LAPLACE TRANSFORM
In order 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 Tzou (1996): where Re is the real part and i is imaginary number unit.For faster convergence, numerous numer- ical experiments have shown that the value of  satisfies the relation t 4.7   Tzou Tzou (1996).

Numerical Results and Discussion
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 Youssef and Elsibai (2010): The aspect ratios of the beam are fixed as / h 10   and b / h 1/ 2  , when h is varied,  and b change accordingly with h.
For the nanoscale beam, we will take the range of the beam length    The figures were prepared by using the non-dimensional variables which are defined in (9) for beam length 1.0   , z h /6  and t 5.0  .
The two-dimensional figures 2-7 and the three-dimensional figures 9-14 show the heat conduction distribution, the dynamical heat distribution, the deflection distribution, the stress distribution, the strain distribution and the strain-stress energy distribution respectively for the two cases of one and two temperature models of thermoelasticity at constant time to stand on the effect of the two temperature parameter effect on all the studied fields.We can see that the two temperature parameter has significant effects on the heat conduction distribution, the dynamical heat distribution, the stress distribution and the strain-stress energy distribution while it has a week effects on the strain and the displacement distribution.The two temperature parameter plays a vital role on the damping of the stress-strain energy where when this parameter increases that energy decreases.
Figure 8 shows the deflection distribution with different values length to stand on the effect of the scale of the beam on the deflection, and we found that, the length of the beam has a significant effect on its deflection, where the deflection increases when the length of the beam increases.The figures 15-20 show the heat conduction distribution, the dynamical heat distribution, the deflection distribution, the stress distribution, the strain distribution and the strain-stress energy distribution respectively with constant value of the two temperature parameter and with different values of time to stand on the effect of the time on all the studied fields and we find out that, the time has significant effects on all the studied fields.x t e W x t

CONCLUSION
The two-temperature parameter has significant effects on the heat conduction temperature, the dynamical temperature, the stress and the stress-strain energy.
Increasing the value of the two-temperature parameter causes decreasing on the values of the stress strain energy which gives more damping of that energy.The values of the time have significant effects on all the studied fields.
The length of the beam has a significant effect on its deflection.

Figure 15 :
Figure 15: The heat conduction distribution at a = 0.4.

Figure 20 :
Figure 20: The strain-stress energy distribution at a = 0.4.