Temperature-dependent physical characteristics and varying heat effects on nonlocal rotating nanobeams due to dynamic load

A theoretical nonlocal thermoelastic model for studying the effects of the thermal conductivity variability on a rotating nanobeam has been described in the present article. The theory of thermal stress is employed using the Euler–Bernoulli beam model and generalized heat conduction with phase lags. It is believed that the thermal conductivity of the current model varies linearly according to temperature. Due to variable harmonic heat, the considered nanobeam excited and was subjected to a time‐varying exponential decay load. Using the Laplace transform process, the analytical solutions for displacement, deflection, thermodynamic temperature and bending moment of rotating nanobeams are provided in final forms and a numerical example has been taken to address the problem. A comparison of the stated results was displayed and additionally, the influences of non‐ local parameter and varying load were analyzed and examined. We also investigate how the linear changes in the temperature of physical properties can influence both the static and dynamic responses to the rotating nanobeam.

based on the law of Fourier, takes an instant reaction to the temperature gradient and tends to a distinctive parabolic equation for temperature development.
To solve the infinite speed thermal propagation phenomena predicted by the classical thermoelasticity theory, modified generalized theories of thermoelasticity have been established. Lord and Shulman [1] suggested one of the modified generalized thermoelasticity theories that contained one relaxation time by introducing a novel law of thermal conductivity to exchange the classic Fourier law. This revised law includes the flux of heat and its partial derivative with respect to time. Among the models that have recently been very popular, the Tzou [2][3][4] model. In this model, Tzou introduced the dual-delay thermal heat conduction model (DPL) to include the effects of microscopic reactions into the rapid transit of the heat transfer mechanism into a microscopic formula. In the constitutive relationship between the heat flux and the temperature gradient, two different phase-lags have been introduced. There are many other suggestions that have been made to overcome the problem of unlimited speeds of heat waves predicted by the classical thermoelasticity theory can be found in [5][6][7][8].
Thermal conductivity is significant, especially when high operating temperatures are reached, in materials sciences, research and electronics, building insulation, and associated fields. The temperature has a distinctive impact on the thermal conductivity for metals and non-metals materials. The thermal conductivity of metal materials is the electric conductivity compared to absolute temperature times. With the expanding temperature, the electrical conductivity in pure metals decreases. The result, the thermal conductivity, is approximately constant along these lines.
In alloys, the electric conductance change is generally smaller, so the temperature rises thermal conductivity, always relative to the temperature [9].
The smallest Electromechanical Systems area (MEMS) is quickly divided into different resistors and applications. In order to satisfy the industry requirements, new technologies have been used to produce a range of MEMS products. MEMS structures include mechanical components, e.g. small cantilevers, extensions and films that have been frequently filled with a variety of geometrical estimates and arrangements [10]. The influences of very small-scale interactions between the neighboring material particles or constituents must be taken into account for these micro / nanostructures, such as actuators, sensors, microscopes, micro / nano-electronic systems (MEMS / NEMS).
It is important for MEMS planners to understand the mechanical properties of adaptable smaller scales in order to remember the true aim of providing for the measure of diversion from associated loads to forestall cracks, improve the execution and produce the lifetime of MEMS devices [11].
From the discussion above, the principle of vibration is well-known and well-studied in dual beam systems. Nevertheless, the scale-dependent vibration of beam systems makes few contribution. The structures of scale-dependent nanobeams are constructed from nano-materials. These nanomaterials have special properties because of their dimensions of the nanoscale. Nanoparticles, nanowires and nanotubes are common examples of the materials with attractive features on the nanoscale [12].
Nanomaterials are technological products for the next century and have intensified the interest of the scientific community in physics, chemistry, biomedicine and technology. In the mechanical properties, the scale of these structures was very small, and both research and atomic reproduction measurements showed a considerable impact. In this sense, the effect of size plays a vital role in the dynamic and static conduct of micro and nanoscale structures and cannot be undermined. It is Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 August 2020 doi:10.20944/preprints202008.0405.v1 understood that the conventional process of the continuum in small and nanoscale systems does not have these dimensional influences. Nanomaterials are used as complex electromechanical nanostructures (NEMS) and complex nanocomposites due to some attractive properties. Nanoscale beams are referred to as structural beams constructed from nanomaterials and nanometer measurements.
The small-scale impact in the non-local elastics theory is defined by the assumption that stress at one point depends at all points on strains in the field (Eringen, [13]). It is distinct from the classical theory of elasticity. Non-local theory takes into account interatomic interaction over a long time and findings are dependent on the body size. The non-local theory includes knowledge about the longrange forces between iota along these lines, and the internal length scale is essentially used as a material parameter to detect the small scale effect. Any inconveniences of classical continuum theory can be easily eschew and the non-local elasticity theory may describe the size-dependent phenomena fairly. The use of conventional theory in studying nanostructures is inappropriate in this particular situation because classical theories are not responsible for the small-scale effects of the size.
Recent nanotechnology consideration has been given to the application of nonlocal elasticity in smaller scales and nanomaterials, and the written text indicates that the non-local elasticity principle is typically used progressively to analyze the nanostructures in a consistent and rapid manner. The fundamental contrast of conventional local theory of elasticity to Eringen's nonlocal theory of elasticity [13][14][15] depends on the context of the stress field. The non-local elasticity theory, however, has been applied in various material science areas, including elastic wave dispersion, mechanical decomposition, etc. and Timoshenko beams (TBT) [16][17][18][19][20][21][22][23].
Another area of research that is emerging due to its current and future applications of various MEMS and NEMS systems is theoretical research into the behavior of rotating micro/nanobeams.
Nanostructures undergoing rotation include nanostructures, nano-turbines, molecular bearings, waft and cord, and multiplicity of gear systems.
Taking into account the number of published studies of the viability of rotating nanoscale structures, this idea is expected to be given significant attention in the near future; existing samples include the analysis of molecular carbon nanotubes and gears [24,25]. The rotational dynamics of carbon nanotubes and gears have been explained by Srivastava [26] under a single imposed laser beam. The mathematical modeling for the rotating nanomotor was carried out by Lohrasebi and Rafii-Tabar [27]. Nanoelectromechanical (NEMS) devices emerge as the next technology, which is capable of making a significant difference in people's lives.
The purpose of this work is to investigate the vibration behavior of rotating isotropic nanobeams in a mathematical model using Eringen's nonlocal theory of elasticity. Since the rotating devices and

Nonlocal model of thermoelasticity
Eringen [13] proposed a non-local continuum mechanics theory to study the structural problems at a micro-scale. For nano-scale beams the general equations of nonlocal constituent relations can be described as follows [14].
The constitutive equation is expressed as Where and are the Lame' constants and refers to the function of Kronecker's delta. If = − is the temperature increment over the uniform reference temperature and = div ⃗ is the volumetric strain.
The modified heat conduction equation that includes two phase delay will be follows [2][3][4] 1 + , , = 1 + Where is the heat source = /(1 − 2 ), where is the thermal expansion coefficient, is the module of Young and is the ratio of Poisson, implies the thermal conductivity, denotes the specific heat for each unit mass and is the material density.
The theory of classical thermoelasticity, the Lord and Shulman model [1], and dual-phase-delay model [2][3][4] can be derived from Eq. (3) for the selection of different phase lag parameters and .


The generalized theory of nonlocal thermoelasticity (LS theory) can be gotten after placing = 0 and = ( is the relaxation time).


The nonlocal dual-phase-lag model (DPL) can be obtained by setting 0 < ≤ .
When one of the thermo-physical characteristics ( , and ) is based on temperature, Eq. (3) is converted to a nonlinear partial differential equation.

Problem formulation
where denotes the nanobeam deflection. Equation (5) can be used to simplify the one dimension constitutive relationship (1) as where indicates the nonlocal axial thermal stress, = /(1 − 2 ) and = ( ) . The bending moment of the nanobeam may be found from the following integration: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 August 2020 doi:10.20944/preprints202008.0405.v1 We can get the following partial differential equation satisfied by the bending moment after substituting Eq. (6) in Eq. (7) where and = ℎ /12 respectively represent the bending rigidity of the nanobeam and the inertia moment of the cross-section. In Eq. (8) indicates the thermal moment of nanobeam which is given by It is assumed that the nanoscale beam distributed by a transversely varying load ( , ), then the transverse motion equation can be written as [37] = − ( , ) + ℎ , We suppose that the nanobeam rotates about an axis parallel to the -axis with an angular velocity Ω centered at a small distance from the first edge of the nanobeam. The centrifugal tensional force ( ) is introduced as a result of rotation. In this case, the equation of transverse motion (10) can be expressed as [37] If Ω = 0; i.e., there is no rotation and therefore the centrifugal tension force disappears.
The axial force ( ) due to centrifugal stiffening at a distance from the origin (Fig. 1) is is given as [37,38] ( ) = ∫ Ω ( + ) The constant is the distance from the rotation center to the first edge of the nanobeam (a hub radius) ( see Figure 1). After integration, then Eq. (19) can be simplified as For calculating the bending moment ( , ), Eqs. (16) and (18) can be used If the moment is removed from Eqs. (18) and (21), the equation of motion (66) may be written as The modified heat conduction equation (11) is given in the absence of heat sources ( = 0) as

Thermal properties of materials
The identification and resolution of nonlinear problems will be carried out if the material is temperature-dependent and then the specific heat and the thermal conductivity depends upon the temperature distribution [24]. In this work, it can be assumed that other physical parameters are constant and not dependent on temperatures, such as the Poisson's ratio and the thermal expansion coefficient [39]. The material's thermal conductivity will be assumed to be a linear function of the variation of the temperature [13] as Where the parameter denotes the thermal conductivity at = and is a factor that characterizes thermal conductivity variety.
By substituting from equation (13) in equation (20), we obtain a partial nonlinear differential equation in the form The previous equation can be converted to a linear equation by defining the mapping [40] = ∫ ( )d , After inserting Eq. (13) in Eq. (14) and integration, then we have [40] = 1 + .
Through differentiating relation 5 once with regard to distances and also in terms of time, the following relationships can be deduced By using Eqs. (16) and (17) then, the heat conduction equation (23) may be reduced to where 1/ = / is the thermal diffusivity.

Sinusoidal solution
To solve the problem, we take the temperature change solution as (sinusoidal solution).
After introducing dimensionless quantities (24) in Eqs. (21)- (23), we can obtain In Eqs. (29)-(31), the primes are omitted for convenience Special external transverse load type is now taken into account. We consider exponential time of decay to vary vertically load operating towards the thickness of the beam [41].
where is the magnitude of the dimensionless point load and is the dimensionless decaying parameter of the applied load, respectively ( = 0 for the uniformly distributed load).
In this paper, we assume that the nanobeam rotates at a constant angular speed, and consider the centrifugal tension force ( ) to be the maximum [37] value. The maximum axial force ( ) due to centrifugal stiffening at the root ( = 0) takes the form [37] = ∫ Ω ( + ) = Ω (2 + ) The motion equation (29) is therefore can be described as The bending moment in Eq. (31) may also be defined as

Initial and boundary conditions
The initial conditions are supposed to be ii) The nanobeam is harmonically heated as Where Θ is constant and is the thermal vibration angular frequency. The problem will thermal shock if by taking = 0.
The following differential equation can be accomplished once the function Θ has been extracted from the Eqs. (41) and (42) − We achieve the general solution for by solving the differential equation (44) From the given boundary conditions the undetermined parameters , ( = 1,2. . ,6) , may be calculated. The parameters , and also satisfy the following equation The bending moment is determined from (43) using the solutions (46) and (49) ( , ) = ∑ e + e + The axial displacement u can be calculated by using Eq. (46) The boundary conditions (37) = 0.
When the above conditions are applied to Eqs. (46) and (49) The undetermined parameters , ( = 1,2. . ,6) can be calculated in the resolution of the above system equations.

Laplace transform Inversion
With the aim of getting solutions into the physical domain, at last, we invert the transformation of Laplace to the functions that govern. We now follow a numerical overlay strategy based on an extension of the Fourier series [42]. Any functions in the domain Laplace ̅ ( , ) can be changed to the time field ( , ) in this procedure by using the relation.
Various numerical studies have shown that the parameter satisfies the relation ≈ 4.7 for speedier convergence [43].

Results and Discussion
Throughout this section, we provide some discussions and numerical results to illustrate the general solution behavior of the theoretical outcomes. In addition, numerical results are provided to determine the effects on the physical fields analyzed with three appropriate parameters. We have taken specific physical parameters from various current literature works into account for computational purposes. In this analysis, we use silicon data as the physical material discussed: We consider a nano-beam with dimensionless parameters as given in equation (24). The ratios of the nanobeam are set in the current calculation, i.e., /ℎ = 10 and /ℎ = 0.5 . The dimensionless nanobeam length is taken as = 1 and take = ℎ/3 and time = 0.1. Three cases for discussion and analysis are considered using the nonlocal theory of Eringen.
As mentioned previous, a linear function of the variation in temperature is assumed to be the thermal conductivity of the material (see Eq. (13)). That is the main goal of the analysis. From the figures we have noted that the variability parameter has a considerable impact on all fields that make our consideration of the variable thermal conductivity more seriously. The nanostructures can also be observed to based physically on temperatures and to increase external temperatures, results of small-scale nonlocal theories increase.
The deflection with varying values of thermal conductivity variability parameter is shown in Figure 2. As can be shown, with the increased value of the parameter , the lateral vibration decreases. Figure 2 illustrates that the deflection distribution that starts and ends at zero values (i.e. disappears) and meets the limit conditions of the rotating nanobeam at = 0 and = . The temperature for variability parameter values can be seen in Figure 3. The temperature decreases with increasing distance , in order to drive towards wave propagation, as shown in Figure   3. Figure 3 indicates that a decrease in the parameter increases the temperature distribution . We also note that the rise in in Figure 5 is intended to increase the bending moment distribution. The figures show that the effect of a change in thermal conductivity should not be disregarded [44]. The mechanical distributions of the nano beam show that the wave spreads in medium as a wave with a finite speed [45].  In second case, an effect on dimensionless field quantities has been performed on the angular rotation velocity Ω. It is supposed to be consistent both the non-local parameter , angular frequency of the harmonic thermal load , and phase lags and . The calculations has been determined using the fixed values ̈ = 0.1, = 5, = 0.02 and = 0.01 have been taken into account.
The variation in the fields considered for three different values angular velocity (Ω = 0, 0.1, 0.3) is shown in Figures 10-13. In the absence of rotation, the coefficient of rotation is equal to zero (Ω = 0), and this is a special case in current procedure.
The angular rotation s Ω that influences the deflection of the nanoscale beam w, is shown in Figure   10. This parameter has been found to have a substantial effect on the distribution of deflection and variation in outcomes in the case of presence and lack of rotation. Through increasing the angular velocity Ω, the deflection decreases. These results are consistent with those reported [46].  that at certain ranges, with a rising rotation, the distribution of the displacement decreases and increases at other intervals. Figure 13 shows the variation of varying angular velocity Ω values in the distribution of bending moment of the rotating nanobeam. It is observed from the figure that the rotation of the moment has a great effect on the curves, and that the amplitude of the moment increases with the moment .
One of the objectives of this analysis is to explain the approach to the temperature field and to the angular velocity of certain nano-device blades such as nano-turbines [50], which provide valuable insights. Via the previous observations, we can also deduce the major effect of rotation on the different physical distributions. The findings are in line with the earlier results of [49][50][51].
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 August 2020 doi:10.20944/preprints202008.0405.v1 For various values of the non-local scaling coefficient , Figure 4 shows the curves for non-rotating nanobeam deflection. It can be seen that with an increase in the parameter of the nonlocal scaling, the deflection becomes very small and the dispersed nature turns into a non-dispersed form. The temperature distributions are also shown for the various non-local scaling parameter in the axial direction in Figure 3. The Figure shows that the temperature convergence can be accomplished by increasing the nonlocal parameter across the space. Figure 6 shows that the displacement amplitude increases with the non-local parameter due to the inclusion of the nonlocality in the given model. The influence of the non-local coefficient on the distribution of the non-dimensional moment is shown in Figure 7. As the non-local coefficient increases, the bending moment will decrease.
The reported findings and conclusions are consistent with those of various literature researchers [52].
The results shown typically show that the non-local parameter has a major effect on all observed physical variables. The distinction between local thermoelasticity models and non-local thermoelasticity models in thermal fields is clarified [53]. In nanoscale systems and devices the impact of this parameter should also be taken into account.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 August 2020 doi:10.20944/preprints202008.0405.v1   Figure 6 indicates that there is a greater disparity between physical quantities with the point load in the highest points of the figures.
We also note from these figures that the absolute values of the field variables increase with the magnitude of point load if the varying load exponential decay with time [54].   The dependency from the temperature on thermal conductivity has a significant influence on mechanical and thermal interactions.
 The nonlocal parameter effects can be significant on all fields studied.
 Dynamic loads have a major impact on all physical quantities.
 There are significant variations in the fields investigated between the exponential time of decay and the load distributed uniformly. The thermoelastic stresses, and temperature, on the other hand, are highly dependent on the angular rate of the thermal vibrators parameter.
 Nanobeam research is an important subject in nanotechnology as it encompasses the optical and electronic characteristics in nanobeams.
 Current research may be used in applications including resonators, sensors for the voltage surge, frequency filters, accelerometers and relay switches.
 This study will recognize demands and various requirements for the design and production of the environmentally sensitive resonator machines.