Nonlocal Fractional Thermoelasticity Theory for Nanobeams with Variable Thermal Conductivity
Ashraf M Zenkour1,2 and Ahmed E Abouelrega3,4*
1Department of Mathematics, Faculty of Science, King Abdulaziz University, Saudi Arabia
2Department of Mathematics, Faculty of Science, Kafrelsheikh University, Egypt
3Department of Mathematics, Faculty of Science, Mansoura University, Egypt
4Department of Mathematics, College of Science and Arts, University of Aljouf, Saudi Arabia
Submission: December 22, 2017; Published: January 17, 2018
*Corresponding author: Ahmed E Abouelrega, Department of Mathematics, Faculty of Science, Mansoura University, Egypt, Email: firstname.lastname@example.org
How to cite this article: Ashraf M Z, Ahmed E A. Nonlocal Fractional Thermoelasticity Theory for Nanobeams with Variable Thermal Conductivity . JOJ Material Sci. 2018; 3(5): 555622. DOI: 10.19080/JOJMS.2018.03.555632
In this work, a new theory of nonlocal thermoelasticity is constructed in the context of a new consideration of heat conduction with fractional orders. The closed form expressions for the transverse vibrations of a homogenous isotropic, thermoelastic thin beam with variable thermal conductivity, based on Euler-Bernoulli theory have been derived. We consider the nanobeam is subjected to a harmonically varying heat. The effects of the variable thermal conductivity parameter, the nonlocal parameter, the fractional order, the angular frequency of thermal vibration on the lateral vibration, the temperature, the displacement, and the bending moment of the nanobeam are presented in graphical and discussed.
Nanotechnology has wide applications, which include the fabrication of engineering structures at nanoscale enabling production of materials with revolution properties and devices with enhanced functionality. One of these structures is nanobeams, which have been used widely in systems and devices such as atomic force microscope, nanowires, nanoactuators and nano-probes . The thermoelastic solutions may be completely incorrect with the ignoring of small-scale effects in sensitive nano-designing fields. Since mechanical behavior of nanostructures could not be predicted accurately by classical continuum theories. So, nonlocal continuum theories have been proposed to include the small scale effect in vibration and buckling of nanobeams.
In 1972, Eringen proposed a theory, called nonlocal continuum mechanics, in an effort to deal with the small-scale structure problems. In the classical (local) continuum elasticity, the material particles are assumed to be continuously distributed and the stress tensor at a reference point is uniquely determined by the strain mechanics is based on the constitutive functionals being functional of the past deformation histories of all material points of the body concerned. The small length scale effects are counted by incorporating the internal characteristic length such as the length of the C-C bond into the constitutive relationship. Solutions from various problems support this theory [2-4]. For examples, the dispersion curves by the nonlocal model are in excellent agreement with those by the Born-Karman theory of lattice dynamics; the dislocation core and cohesive stress predicted by the nonlocal theory are close to those known in the physics of solids [3,4]. Some authors determined the values of scaling effect parameter experimentally and they used these values in vibration and buckling problems of nanostructures [5,6].
The current manuscript is an attempt to study the induced temperature and stress fields in an elastic half space. The governing equations are written in the context of two- temperature generalized and fractional order thermoelasticity theories. The half space is considered to be made of an isotropic homogeneous thermoelastic material. The bounding plane surface is heated by a non-Gaussian laser beam. The present paper investigates the coupling effect between temperature and strain rate in nanobeams with variable thermal conductivity based on the nonlocal dual-phase-lag model [7-10]. The solution for the generalized thermoelastic vibration of a nanobeam subjected to time-dependent temperature is developed. The Laplace transform method is used to determine the lateral vibration, temperature, displacement, and bending moment of the nanobeam. The effects of nonlocal parameter, phase- lags, angular frequency of thermal vibration and the variability thermal conductivity parameter are investigated.
An exact solution of the problem is first obtained in Laplace transform space. The inversion of the Laplace transform will be computed numerically by using a method based on Fourier expansion technique. The numerical estimates of the conductive temperature, the thermodynamic temperature, the stress and the strain distributions are obtained. The derived expressions are computed numerically for copper and the results are presented in graphical form. The effects of the fractional parameter, the two-temperature parameter, and the laser-pulse and the laser intensity parameters are estimated. Some special cases are also deduced.
During recent years, several interesting models have been developed by using fractional calculus to study the physical processes particularly in the area of heat conduction, diffusion, viscoelasticity, mechanics of solids, control theory, electricity, dielectrics and semiconductors through polymers to fractals, glasses, porous, and random media, porous glasses, polymer chains, and biological systems. There exists many material and physical situations like amorphous media, colloids, glassy and porous materials, manmade and biological materials/polymers, transient loading etc., where the classical thermoelasticity based on Fourier type heat conduction breaks down. In such cases, one needs to use a generalized thermoelasticity theory based on an anomalous heat conduction model involving time fractional derivatives.
The classical thermoelasticity is based on the principles of the classical theory of heat conductivity, specifically on the classical Fourier law, which relates the heat flux vector q to the temperature gradient as follows:
Where q denotes the heat flux vector, K denotes the thermal conductivity, θ = T -T0 is the temperature increment of the resonator, in which T0 is the environmental temperature, T(x, z, t) denotes the temperature distribution.
The modified of classical thermoelasticity model is given by Tzou [11-13] theory in which the Fourier law is replaced by an approximation
Where τθ and τq represent the delay times and
0≤τθ < τq The delay time τθ is said to be the phase-lag of temperature gradient and τq is the phase-lag of heat flux.
Ezzat  constructed a new model of the magneto- thermoelasticity theory in the context of a new consideration of heat conduction equation by using the Taylor series expansion of time fractional order, developed by Jumarie  as Ezzat, M.A.: Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer. Phys. B 406, 30-35 (2011). Jumarie, G.: Derivation and solutions of some fractional Black- Scholes equations in coarsegrained space and time. Application to Merton’s optimal portfolio. Comput. Math. Appl. 59, 11421164 (2010).
Now, taking divergence of both sides of Equation (3), we get
Using Equation (5), we get, the fractional heat conduction equation corresponding to the dual-phase-lag model proposed by Tzou [11-13] in the form
where CE denotes the specific heat per unit mass at constant strain, is the volumetric strain, in which αT is the thermal expansion coefficient, E is Young’s modulus and V is Poisson’s ratio, ρ is the material density of the medium, and Q is the heat source.
The constitutive equation take the form
Where σiJ is the stress tensor, λ and μ are the Lame’ constants, and δij is the Kronecker delta function.
We may consider equation (9) as the fractional ordered generalized heat conduction equation in isotropic, thermoelastic solids in two temperatures.
The general nonlocal constitutive equations for a nanobeam can be written as 
where σ kl denotes the nonlocal stress tensor, Tkl denotes the classical (Cauchy) or local stress tensor and ∇2 is the Laplacian. The parameter a is the internal characteristic length, l is the external characteristic length (e.g., crack length, wavelength), and e0 is a constant appropriate to each material.
Equation (3) describes the non-local coupled dynamical thermoelasticity theory, the generalized thermoelasticity theories proposed by Lord and Shulman , and dual-phase- lag model [11-13] for different sets of values of phase-lags parameters τq and τθ as follows:
a)The nonlocal coupled thermoelasticity (CTE) theory results from Equation (3) by taking τθ = τq = 0 .
b) The nonlocal generalized theory with one relaxation time (LS theory) can be obtained whenτθ =0, τq = τ0 (τ0 is the relaxation time).
c)The nonlocal generalized theory with two dual-phase- lags (DPL) theory is given by setting τq≥τθ>0.
d)It can be seen that the corresponding local thermoelasticity is recovered by putting ae0 = 0 in the above equations.
Let us consider small flexural deflections of a thin elastic nanobeam with dimensions of length L (0 ≤ x ≤ L) , width b (-b / 2≤ y ≤+b / 2) and thickness h (-h /2≤z ≤+h / 2) , as shown in Figure 1. We define the -axis along the axis of the beam and the y- and z-axes correspond to the width and thickness, respectively.
In equilibrium, the beam is unstrained, unstressed and at temperature T0 everywhere. The displacements according to Euler-Bernoulli beam theory are given by
where w is the lateral deflection. It is to be noted that, the nonlocal theory assumes that stress at a point depends not only on the strain at that point but also on strains at all other points of the body. So, the one-dimensional constitutive equation gives the uniaxial tensile stress only, according to the differential form of the nonlocal constitutive relation proposed by Eringen , as
where EI is the flexural rigidity of the beam in which
I = bh3 /12 denotes the inertia moment of the cross-section and MT is the thermal moment,
The equation of motion is given by
where A = bh denoes the cross-section area. Substituting Equation (7) into Equation (9), the equation of motion becomes
In case of the nonlocal theory, the bending moment can be written as
It is exactly seen that when the parameter ξ is set identically to zero, the local Euler-Bernoulli beam theory will be obtained
Generally, the assumption that the solid body is thermo sensitivity (the thermal properties of the material vary with temperature) leads to a nonlinear heat conduction problem. The exact solution of such problem can be found by assuming the material to be (simply nonlinear) meaning that the thermal conductivity K and specific heat CE depend on the temperature, but thermal diffusivity k is assumed is constant.
The thermal conductivity K is assumed to vary linearly with temperature according to 
Where K0 is the thermal conductivity at ambient temperature T0 and K1 is the slope of the thermal conductivity- temperature curve divided by the intercept K0 . Now, we will consider the Kirchhoff transformation 
After substituting Equations (15) and (16) into Equation (3), the new form of the general heat equation of solids with temperature-dependent thermal conductivity is obtained by the form
where ρCE = K/k. Substituting the Euler-Bernoulli assumption given in Equation (5) into Equation (17) gives the thermal conduction equation for the beam without the heat source (Q = 0), as
For a nanobeam, assuming that the temperature increment varies sinusoidally along the thickness direction. That is
Using Equations (8), (12) and (15), Equation (10) will be in the form
The preceding governing equations can be put in non- dimensional forms by using the following dimensionless parameters:
So, the governing equations, and the bending and constitutive equations in non-dimensional forms are simplified as (dropping the primes for convenience)
The closed form solution of the governing and constitutive equations may be possible by adapting the Laplace transformation method. Applying the Laplace transform to Equations (24)-(26), one gets the field equations in the Laplace transform space as
where an over bar symbol denotes its Laplace transform, denotes the Laplace transform parameter and
Substituting Equations (44) into the above boundary conditions, one obtains six linear equations in the matrix form as
After solving the above equations, we have the values of the constants Cii =1,2,...,6 whose solution complete the solution of the problem in the Laplace transform domain. Hence, we obtain the expressions for the non-dimensional lateral vibration, displacement, the stress and other physical quantities of the medium.
In order to invert the Laplace transform in the above equations, we adopt a numerical inversion method based on a Fourier series expansion. Durbin  derived the approximation formula
Where being the Laplace transform. It is known that Re and Im tend to 0 when n tends to infinity. To apply efficiently each of the above mentioned procedures, one would have first to find, in each case, the value n of after which decreases monotonically to 0 . This is virtually impossible for the very complicated . So, the real and imaginary part of are evaluated together through a complex, single precision arithmetic subroutine, but are converted into double precision constants for the summation up to NSUM; the results are then turned back to single precision expressions. Thus, one avoids time and storage consuming systematic double precision computation; NSUM can be determined by the convergence criterion:
It should be noted that a good choice of the free parameters and is not only important for the accuracy of the results, but also for the application of the Korrecktur method and the methods for the acceleration of convergence. We found that st1 = 5 to 10 give good results for NSUM ranging from 50 to 5000 . The values of all parameters in Equation (39) are defined as t =120 ,s = 0.25 and N = 1000 in this paper .
In terms of the Riemann-sum approximation defined in Equation (52), numerical Laplace inversion is performed to obtain the non-dimensional lateral vibration, temperature, displacement and bending moment in the nanobeam. In the present work, the thermoelastic coupling effect is analyzed by considering a beam made of silicon. The material parameters are given as:
The aspect ratios of the beam are fixed as L /h = 10 and b/ h =. The figures were prepared by using the non- dimensional variables which are defined in Equation (14) for a wide range of beam length when L = 1 , z = h/6 and t = 0.12 sec .
Numerical calculations are carried out for four cases. In the first case the non-dimensional lateral vibration, temperature, displacement and bending moment with different phase-lags are investigated when the pulse-width and nonlocal parameter remain constants The graphs in Figures 2 (a-d) represent the curves predicted by three different theories of thermoelasticity obtained as special cases of the present dual- phase-lag model. The computations are performed for various values of the parameters τq and τθ to obtain the coupled theory (CTE)(τq =τθ=0),the Lord and Shulman (LS) theory ( τθ=0 and τq =0.2) and the generalized theory of thermoelasticity proposed by Tzou (DPL) (τq =0.2 and τθ=0.1 ).
Figure 2a depicts the distribution of the lateral vibration which always begins and ends at the zero values (i.e. vanishes) and satisfies the boundary condition at x = 0, L. In the context of three theories, the values of starts increasing to maximum amplitudes in the range 0 ≤ x < 0.3 , thereafter it moves in the direction of wave propagation. The behavior of DPL model may be differ than those of other two theories. It is observed from Figure 2 b that the temperature θ decreases as the distance x increases to move in the direction of wave propagation. The temperature of DPL model may be similar than those of other models. Figure 2 c shows that the values of the displacement u<start decreasing in the range 0 ≤ x ≤ 0.1 , thereafter increasing to maximum amplitudes in the range 0.1 ≤ x ≤ 0.4. The displacement moves directly in the direction of wave propagation in the range 0.4≤ x ≤ 1 . In Figure 2d, the bending moment is zero at x = L at the boundary of the beam and it attains a stationary maximum value at some positions.
It can be observed that the phase-lags parameters have great effects on the distribution of field quantities. The mechanical distributions indicate that the wave propagates as a wave with finite velocity in medium. The fact that in thermoelasticity theories (DPL, LS, CTE), the waves propagate with finite speeds is evident in all these figures. The behavior of the three theories is generally quite similar. In second case, we consider three different values the variability thermal conductivity parameter K1 to discuss the thermo sensitivity (the thermal properties of the material vary with temperature). We take K1 =-1, - 0.5 for variable thermal conductivity and K1 = 0 when thermal conductivity is independent of temperature. In this case the angular frequency of thermal vibration ω= 5 and phase-lag of the heat flux τq = 0.02 and the phase-lag of temperature gradient τθ= 0.01 . From Figure 3(a-d), the parameter K1 has significant effects on all the fields. From Figures 3a, 3c, 3d, the lateral vibration w , the temperature θ and bending moment M decreases as the K1 decreases. From Figure 3b, it can be found that the displacement U decreases as the parameter K1 value decreases in the range 0 ≤ x ≤ 0.2 and increases in the range 0.2 ≤ x ≤ 1.
The medium close to the beam surface suffers from tensile stress, which becomes larger with the time passing. We have noticed from the figures that, the variability thermal conductivity parameter K1 has a significant effect on all the fields, which add an importance to our consideration about the thermal conductivity to be variable. It is clear that the maximum values of occur near the surface x = 0 of the beam and its magnitude decreases with the increase of x. In the third case, three different values of the angular frequency of thermal vibration were considered. For thermal shock problem, we put ω = 0 and for harmonically heat ω is set to ω = 5,10. In this case the variability thermal conductivity parameter K1 is fixed to -0.5. From Figure 4(a-d), the values of the lateral vibration, temperature, displacement and bending moment decreases ω as decreases. These figures illustrate that, the angular frequency parameter ω has a significant effects on all the fields.
Next, the DPL model is used to investigate the non- dimensional lateral vibration, temperature and displacement for different values of the nonlocal parameter The numerical results are obtained and presented graphically in Figure 5(a-d). It is seen that the effect of the nonlocal parameter on all the studied fields is highly significant. The last case is to investigate the non-dimensional lateral vibration, temperature and displacement for various values of the nonlocal parameter when the pulse width remains constant It can be seen that the corresponding local thermoelasticity model is recovered by putting The variations of the field variables with various values of the nonlocal parameter are depicted in Figure 5(a-d). Generally, the field variables are very sensitive to the variation of the nonlocal parameter.
In Figures 1-5 the temperature, the stress and the displacement distributions and perturbed magnetic fields both in the vacuum and in the half-space are plotted against x respectively for various values of the order of fractional derivative of Caputo derivative. The different values of the parameter α with wide range (0<α≤ 1) cover the two cases of the conductivity; (0<α< 1) for weak conductivity and α=1 for normal conductivity (ordinary heat conduction equation).
It should be pointed that, the increasing of the value of the parameter α causes increasing in the speed of the waves propagation of the stresses, the displacement and perturbed magnetic field in the medium distributions, whereas the distribution of the temperature and perturbed magnetic field in the vacuum decreasing. We have also noticed that, the value of α has a significant effect on all distributions.
In this paper, a new model of nonlocal generalized thermoelasticity with dual-phase-lags for the Euler-Bernoulli nanobeam under variable thermal conductivity is constructed. The vibration characteristics of the deflection, temperature, the displacement and the bending moment of nanobeam subjected to harmincally varying heat heating are investigated. The effects of the nonlocal parameter and the angular frequency of thermal vibration ω on the field variables are investigated.
From the numerical results, concluded that:
a) The variability thermal conductivity parameter has significant effects on the speed of the wave propagation of all the studied fields.
b) The dependence of the thermal conductivity on the temperature has a significant effect on thermal and mechanical interactions.
c) According to the results shown in all figures, we find that the nonlocal parameter has significant effects on all the field quantities. On the other hand the thermoelastic stresses, displacement and temperature have a strong dependency on the angular frequency of thermal vibration parameter.
d) The dual-phase-lag model of thermoelasticity predicts a finite speed of wave propagation that made the generalized theorem of thermoelasticity more consistent with the physical properties of the material. Also, it was found that the phase lags of the heat flux and temperature gradient τq and τ&&952; play a significant role on the behavior of all field variables.
e) The Lord and Shulman (LS) theory and the classical thermoelasticity theory (CTE) are obtained as special cases of the present model.
f) The paper also concludes the governing equation of motion for a nonlocal nanobeam can be formed by replacing the bending moment term in the classical equation of motion with an effective nonlocal bending moment as presented herewith. It is found that the effect of the nonlocal parameter is very pronounced. Finally, this work is expected to be useful to design and analyze the wave propagation properties of nanoscale devices.
A new model of thermoelasticity theory was investigated in the context of a new consideration of heat conduction with fractional derivatives. This model based on the heat conduction equation with the Caputo fractional derivative of order a . The solution is obtained by applying the Laplace integral transform. The numerical results for temperature, displacement, and stresses are computed and illustrated graphically. The results are graphically described for the medium of copper
The analysis of the results can be summarized as follows:
a) The dependence of the fractional parameter has a significant effect on the thermal and mechanical interactions, and plays a significant role in all the physical quantities.
b) At any point the distributions of displacement, stress, and perturbed magnetic field in the medium are increased with an increase in &3945; but the effect of fractional parameters is to decrease the values of the temperature field and perturbed magnetic field in the vacuum with a wide range (0 < α ≤ 1).
c) It is clear from Figures that the different times play a significant role in all the physical quantities.
d) All the physical quantities satisfy the boundary conditions and initial conditions.
e) It is also observed that the theories of coupled thermoelasticity and generalized thermoelasticity with one relaxation time can be obtained as limited cases.
f) The values of the distributions of all the physical quantities converge to zero with increasing the distance X .
g) The phenomenon of finite speeds of propagation is manifested in all these figures.
h) As a final remarks, the results presented in this paper should prove useful for researchers in material science, designers of new materials, low temperature physicists as well as for those working on the development of a theory of hyperbolic thermoelasticity with fractional order.