Vibration analysis of nanobeams subjected to gradient-type heating due to a static magnetic field under the theory of nonlocal elasticity

Nanoelectromechanical systems (NEMS) have received great interest from researchers around the world since the advent of nanotechnology and nanoengineering. This can be attributed due to the unique characteristics of NEMS devices and their wide range of applications. Among these applications, nanobeams and nanotubes now have an important role in the design of a variety of NEMS engineering devices. In the current research, the thermoelastic vibration analysis of Euler–Bernoulli nanobeams has been investigated using the theory of non-local elasticity proposed by Eringen. Also to study the effect of temperature change, the generalized thermoelastic model with dual phase-lag (DPL) is applied. The studied nanobeam is subjected to an axial thermal excitation load and surrounded by a magnetic field of constant strength. The Laplace transform technique has been used to solve the system differential equations and to find an approximate analytical solution for the different physical fields of the nanobeam. The numerical results obtained for the studied variables have been graphically clarified and discussed analytically. The effects of various influencing factors such as magnetic field strength, temperature change, non-local parameter as well as ramp type parameter have been examined and studied in detail.


Governing basic equations
As is evident from Eringen's non-local elasticity theory, stress at any point in the body depends on current stresses as well as strains elsewhere in the body. The dynamics and some experimental results of acoustic scattering are used in atomic network theory to explain Eringen's non-local elasticity. The integral space, which describes the weighted averages of the pressure tensor of a related body point, is one of the basic equations in this theory. The theory introduces the effect of small scale in the integral constitutive relationship of space.
For homogeneous and isotropic elastic solids, the non-local elasticity equation are 22,23 where V is the volume of the elastic body and σ kl denotes the nonlocal stresses tensor, r − r ′ indicates the Euclidean distance. The effect of the strain at point r ′ on the stress at point r in the elastic body is described by the nonlocal kernel α r − r ′ . The value of χ depends on the ratio e 0 a/l which is material constant. The internal characteristic length is a , the external characteristic length is l , and the constant e 0 is specific to each material. The local stress tensor τ kl r ′ may be written 22 In the above relation, θ = T − T 0 is the excess temperature, in which T 0 is the environmental temperature, and µ being Lamé's constants, γ = α t (3 + 2µ) = α t E/(1 − 2ν) , α t symbolizes the thermal expansion coefficient, E denotes Young modulus,ν designates Poisson's ratio,δ ij is Kronecker delta function, ε mm x ′ describes the strain tensor which is given by where u l (r ′ ) is the vector of displacement at the point x ′ in the body. The following are some of the intriguing properties of the nonlocal modulus α r − r ′ 23 : • It reaches its peak at r = r ′ attenuating with r − r ′ .
When the kernel α is chosen as 23,37 : where K 0 is the modified Bessel function, Eq. (1) may be simplified as Tzou in 5-7 provides a simplified version of the classical thermoelastic model, in which the Fourier law is replaced by an approximation of the equation where q indicates the heat flow, K symbolizes the thermal conductivity, τ θ is the phase lag of the heat flow (PLH), and τ q is the phase lag of gradient of temperature (PLT).
According to Eq. (6), the temperature gradient ∇θ generated across a material volume at position r at time t + τ θ results in a heat flux q to flow at a different instant of time t + τ q . This delay is frequently explained in terms of the material's microstructure. We direct the reader to Tzou's [5][6][7] work to clarify the applicability of this type of theory.
We can make use of truncated Taylor expansions to replace Eq. (6) in Tzou's theorem or its extension 6 . The following equation can be used to approximate the previous equation: The heat flux q r, t + τ q has been enlarged to second order, and the temperature gradient ∇θ(r, t + τ θ ) has been expanded to first order. It is also possible to propose a higher-order thermoelasticity theory. Equation (7) describes the Cattaneo heat transfer model for τ θ , and simplifies to the classical Fourier law when τ q = τ θ = 0.

Maxwell's relations
Due to the application of an initial magnetic field H , both the magnetic field h and the electric field E l are induced. Below are the basic linear electrodynamic equations of a slow-moving medium for the ideal uniform electrical conductivity of an elastic solid (neglecting the load density) 32 where ∇ is the Hamilton arithmetic operator (nabla) and µ 0 denotes the magnetic permeability. Let the displacement vector U = (u, ν, w) , and a longitudinal magnetic field vector H = (H x , 0, 0) , then we have The Lorentz force F = f x , f y , f z is created from the length of the magnetic field and is given by:

Formulation of problem
Consider slight bending thin nanobeam with sizes L(0 ≤ x ≤ L) , thickness h(−h/2 ≤ z ≤ +h/2) and width b −b/2 ≤ y ≤ +b/2 (see Fig. 1). We set the x-axis along the beam axis and the y and z-axes match width and thickness. The nanobeam has small amplitude bending motions along the x-axis so that the deflection is consistent with the linear principle of Euler-Bernoulli. his implies that every plane cross-section is perpendicular to the beam axis, flat, and perpendicular to the neutral surface during curvature. The displacements are therefore are given by where w is the lateral deflection. The Lorentz force in z direction is written as The non-local constitutive Eq. (5) can be written as 22 : where σ x is the nonlocal axial stress, α T = α t /(1 − 2ν) and ξ = (e 0 a) 2 is the nonlocal parameter. The equilibrium conditions of the Euler-Bernoulli beam can be described as the transverse nanoscale vibrations described by the equation. The equilibrium conditions of the Euler-Bernoulli beam can be described as the transverse nanoscale vibrations which can be described by the equation where f (x) is the force per length application of an initial magnetic field and A = bh. .

Analytical solution
The upper and lower sides of the beam have no heat flow (thermally insulated), That's how it is ∂θ ∂z = 0 at z = ±h 2 . For the very thin beam, assuming that the temperature increases vary according to the sin pz function along the thickness direction (sinusoidal anisotropy), where, Thus, Eqs. (20) and (21) become Multiplying Eq. (22) by means of 12z/h 3 and integrating it with respect to z through the thickness of the nanobeam from −h/2 to +h/2 , we get In order to simplicity the basic equation, the following non-dimensional quantities are considered: The governing equations are therefore simplified as (dropping the primes)

Initial and boundary conditions
It is important to emphasize that both initial conditions and boundary conditions must be considered in order to solve the problem. Suppose that our problem is homogeneous in the initial conditions. First, the initial requirements are considered The following boundary conditions can be taken into account. We will assume that the nanobeam is simply supported at the two ends We also consider, that the first end x = 0 of the nanobeam is thermally loaded by ramp-type varying heat. In this case, then we have following boundary condition where the ramp-type parameter is denoted by t 0 and 0 is a constant.
Furthermore, at the end side x = L of the nanobeam is thermally insulated. Then we have

Solution of the problem
Applying the Laplace transform to Eqs.  . From Eqs. (23) and (47), the temperature in the transformed domain is given by Using the above equations, the bending moment M represented in Eq. (37) can be given by In addition, after using Eq. (12), the axial displacement u will be in the form In addition, the strain e will be The boundary conditions after using Laplace transform can be written as Using the above conditions, six linear equations are given as follows in matrix format

Inversion of the Laplace transforms
The Riemann-sum approximation technique is employed to get numerical results for the purposes of determining the temperature, the displacement, bending moment and the stress distributions in the time domain. Any functions in the Laplace domain can be converted to the time domain using the following relation 38 where Re and i are the real part and the imaginary number unit. For faster convergence, numerical experiments have shown that the value that satisfies the above relation is ζ ≃ 4.7/t 7 .
Sample transform functions. To show the degree of precision, the findings of our numerical inversion approach are compared with the accurate estimates.
We will check two examples to illustrate the applicability of the suggested numerical inversion technique. We take the transformed function g 1 (s) = 1 1+s+s 2 and g 2 (s) = exp(−1/s) which are the transforms of the functions g 1 (t) = 2/ √ 3 exp(−t/2) sin √ 3t/2 and g 2 (t) = sin 2 √ t √ π . To invert the transform g 1 (s) , the parameter values of 6 were used, namely ζ = 0.421 , N = 19 and t 1 = 7.5 39,40 . Table 1  The enhanced approach can quickly achieve great accuracy without requiring large calculations. In addition, as compared to other algorithms with comparable high accuracy, the programming effort for the approach is modest 39,40 . More importantly, by increasing processing effort by a linear amount, our inversion approach may make large improvements in accuracy. Our technique is thought to work well for a broader variety of functions.

Numerical results
In order to demonstrate the previous analysis and to compare the theoretical results obtained in the previous sections, we now take a numerical example to give computational results. Silicon, as an anthropic substance, is the material chosen for this numerical evaluation. Thus, the physical properties of the problem are expressed in SI units as 15,16 The nanobeam aspect ratios are taken as L/h = 10 and b/h = 0.5 . To study the effect of a non-local parameter ξ , the values ξ = 10 6 ξ is considered. The figures were prepared for L = 1 , t = 0.12 and z = h/3 in the wide range 0 ≤ x ≤ 1 . Through the Laplace inversion, the distributions of temperature, bending moment, deflection (56) f (t) = e ζ t t  Comparison investigation. In this part, a comparison study will be carried out to assess the reliability and validity of the findings, as well as the correctness of the suggested model. Using Laplace transforms, the analytical formulae for all of the distinct physical fields were recovered in the preceding section. The Riemannsum approximation approach is used to get numerical results. Table 1 compares the non-dimensional physical fields (temperature θ and deflection w ) of nanoscale beams with previously published results 15 . In this case ξ = 0.003 , τ q = 0.2 , τ θ = 0.1 and t 0 = 0.1 , and H x = 0 . When these findings were compared to those obtained from literary works 15 , it was observed that the behavior of mechanical and thermal waves of varying magnitudes is relatively comparable. The existence of non-local operator, ramping time and phase lag parameters in the behavior of thermal and mechanical waves is indicated by the data shown in the table. The findings of the current technique and those of Ref. 15 are in good agreement, suggesting that our model is valid, as shown in Table 2.
These numerical calculations are made in the following four cases: Case I: Explore how the values of different non-local parameter ξ values vary with the thermodynamic temperature, dimensionless deflection, displacement and bending moment (see Figs. 2,3,4,5). In this case, we take t 0 = 0.1 and assume that the PLs τ q and τ θ are assumed to be constants τ q = 0.2, τ θ = 0.1 .         Case I: The effect of the nonlocal parameter. The non-local theorem is proved by replacing the variables in the standard continuous medium with nonlocal coefficients and expressing internal forces as nonlocal expressions. When looking for mechanical behavior at the micro/nano-scale, other physical phenomena must also be taken into account. The physical qualities of the structure, as well as the possibilities of deformation, are influenced by physical events. At nanometer scales, the size effect is often prominent and its effect on the mechanical performance of the nanostructures must be explicitly addressed. Thus, magneto-electro-elastic nanostructuring has become an active research project that depends on its size. For this reason, we will study the effect of the nonlocal parameter on the distributions of different fields.
In this case, three values for the non-local parameter ξ = 0 are taken into consideration. For the nonlocal theory, we will take ξ = 0.003 and ξ = 0.005 while in the classical theory of elasticity we put ξ = 0 . The values of the other factors will be fixed as τ q = 0.2 , τ θ = 0.1 and t 0 = 0.1 . The numerical values of the different domains are represented in Figs. 2, 3, 4 and 5.
From the figures, the following important conclusions and notes can be written:  www.nature.com/scientificreports/ • As reported by 41 , small-scale effects play a significant impact in the thermal and mechanical properties and responses of micro/nanostructures, according to the results of experiments. As a result, nonlocal influences should be addressed while analyzing the mechanical behavior of nanostructures. • It is clear from Fig. 2 that the s lateral vibration w satisfies the boundary conditions as it always starts and ends with zero values at both edges of the nanobeam. • The gap between non-local and classical theories increases with the increase of the non-local parameter.
• The non-local parameter has a significant effect on the lateral vibration w change. With the increase of the non-local parameter ξ the amount of deflection w increases and thus the stiffness of the nanobeam decreases. This is the case because the stiffness of the overall beam has been reduced. • The decrease in deflection is more evident at higher levels and should not be disregarded. In summary, the small scale influence makes the nanobeam more flexible because elastic springs connect the atoms in nonlocal theory. • The maximum deflection occurs at the center of the beam ( x = L/2 ) while the ends meet the stipulated boundary criteria. The largest deflection that occurs in the middle of the beam may be due to the simply supported boundary conditions. • The curve representing the shape of the temperature change of the beam for certain non-local parameters is given in Fig. 3. It is clear from Fig. 3 that the non-local coefficient has little effect on the change of temperature θ distribution as reported by 37 . • Figure 3 shows that the temperature θ gradually decreases in the direction of wave propagation away from the first edge as the distance x increases. • With and without nonlocality impact, Fig. 4 shows how the displacement u varies as a function of both time and distance x. The displacement begins with positive values at the first edge and gradually lowers until it becomes negative and inverted at x = 0.5. • The crossing phenomena in displacement were demonstrated to be dependent on the linear coupling stiffness between the layers, as well as the nonlocal parameter. • Fig. 4 shows that this parameter affects the displacement field considerably. The waves achieve the constant state according to the non-local value of parameter ξ . The amplitude of displacement u is increasing with ξ , as shown in Fig. 4. • In Fig. 5, the bending moment M fades out at the first beam boundary x = 0 and after a certain distance, it reaches a local maximum value. It decreases finally to zero. From Fig. 5 it is clear that the bending moment is reduced by the parameter ξ . • Figure 5 shows that tiny size has a big impact on the bending moment M but fades away as the beam gets longer. • The moment M reduces as the coefficient increases, implying that the nonlocal beam moment is smaller than the local equivalents. • The nonlocal parameter may thus be set appropriately to minimize the bending moment values and ensure that the nanobeam behavior is in the elastic region. • Because in nonlocal theory, elastic springs connect the atoms, the small scale effect makes the beam more flexible 25,42 . • Since the classical continuum theory does not account for nonlocality effects when modeling materials, nonlocality-dependent continuity theories must be used. • The figures indicate that the tiny scale has a major influence on short nanobeams, and that as the beam length increases, the influence gradually decreases. As a result, the tiny size will be reduced to produce a long skinny nanobeam.
Case II: The effect of the ramping time parameter. The time it takes the heated material to achieve the desired temperature is known as the ramp-up time. In other words, the time it takes for the heated medium to reach the target temperature is known as the ramp-up time. Once the material heating system has reached the required temperature, it will just need to cycle on and off to keep it there, consuming far less energy than during the initial ramp-up period. For aerospace constructions, the temperature ramp rate is important for modeling real-world temperature changes. The second case studies how the non-dimensional deflection, thermodynamic temperature, displacement and bending moment of the nanobeam varies with the time gradient parameter (ramping time) t 0 . In the calculations, the phase lags (PLs) τ q and τ θ and the nonlocal ξ parameter persist constantly. Figures 6, 7, 8 and 9 are shown to examine the effect of parameter t 0 on the different distributions of the studied field variables.
From the figures, we can see the major influence on all the fields studied of the ramping time t 0 parameter. The high value of the parameter t 0 decreases in the magnitudes of all studied field variables, which can be seen very clearly at the peaks of the curves. The effect of ramping time heating on the deflection may be seen clearly in Fig. 6. The magnitudes of the deflections decrease with the increase of the heating coefficient of the ramping time t 0 . Figure 7 displays the temperature changes for the different values of the heating coefficient for the ramping time t 0 . We note from Fig. 7 that the temperature changes are very sensitive to the change of the heating coefficient t 0 . The temperature jumps with a large decrease with the change and increase of the heating factor for the time of ramping t 0 . This corresponds to the boundary condition of the problem since the temperature is inversely proportional to this parameter. The effect of the time of ramping t 0 on the displacement is shown in Fig. 8. We can see from the figure that the displacement increases with the increase of the parameter t 0 in some periods and decreases in other periods along the axis of the beam. Moreover, as shown in Fig. 9, we observed the distribution of the bending moment M decreasing with the increase of the parameter t 0 . From these figures, we show that the classical thermoelasticity theory CTE is near to the DPL model of thermoelasticity. However, The LS model distributions differ from the DPL model distributions. We observed moreover as presented in Fig. 10 the distribution of the flexure moment M in the case of the CTE model is small compared to the other models. It is noticed in Fig. 6 that the temperature distribution in the case of the classical theory is higher than the distributions in the case of the other generalized models. In other words, heat waves propagate at lower speeds in the case of generalized models than in the traditional theory.

Scientific Reports
We note that due to the presence of relaxation times the speed of thermal and mechanical waves is reduced. We also note that in the case of the Tzou model 5 , there are two relaxation times (phase lags), so the reduction is greater than in the case of Lord and Shulman model 2 .
Case IV: The effect of the initial magnetic field. In the last case, we have analyzed the influence of the initial magnetic field parameter H x on all studied field variables ( θ , w, u, M ) in the wide range 0 ≤ x ≤ 1 . We take two values of the initial magnetic field H x = 5, 10 in the presence of the magnetic field. When H x = 0 , the absence of a magnetic field was considered. When the parameters τ q = 0.2, τ θ = 0.1, ξ = 0.003 and t 0 = 0.1 ,  The effect of the magnetic field on the responses of the nanobeam can be seen clearly in this situation. The amplitude of vibrations is significantly altered by raising the magnetic field value. The amplitude of deflection field vibration responses rises as the magnetic field increases (see Fig. 14), but the temperature does not change as the magnetic field changes (see Fig. 15). From Fig. 16, we can see that the magnitudes of the displacement u rise with the rising magnetic field. As seen in Fig. 17, the amplitude of the bending moment decreases when the magnetic field is increased, demonstrating the effect of the magneto-thermoelastic field on bending moment vibration responses. When the magnetic force characteristics change from tension to compression, the bending moment falls as predicted.

Conclusion
In this paper, thermal vibration analysis of nanobeams by considering the magneto-thermo-elastic effect was studied using the non-local elasticity of Eringen and the Euler-Bernoulli beam principle. Also, based on the two-phase delay thermoelastic model, the problem was also studied. The effect of a permanent magnetic field on the vibrations of nanobeams was modeled using the concept of the Lorentz force. The Laplace transformation method was used to calculate analytical findings for dimensionless temperature, deflection and thermal moment. Although the illustrations are self-explaining in showing the various peculiarities that occur during the wave spread, the following observations can be added.  www.nature.com/scientificreports/ • A significant effect was observed for the non-local response compared to the absence of the non-local effect. The non-local coefficient has a prominent role in changing the studied physical fields except for temperature. • The maximum deflection that occurs at the center of the beam while the ends meet the stipulated boundary criteria. The largest deflection occurs in the middle of the beam may be due to the simply supported boundary conditions. • The crossing phenomena in displacement were demonstrated to be dependent on the linear coupling stiffness between the layers, as well as the nonlocal parameter. • The wave propagation characteristics of nanobeams are affected by nonlocal scale parameter, temperature change, thermal load, and external magnetic potential, according to numerical results. • All physical fields within the body are strongly affected by changing ramping time parameter. The temperature ramp rate can be used to model real-world temperature variations in aerospace structures. • The variation of the field quantities is shown to have a significant effect on thermal-phases lag parameters.
It is shown that in conventional thermoelasticity the profile of the physical field quantity is larger than in generalized models. • These results obtained in this paper are important for the design of integrated functional devices and smart devices. • The current work only analyses the nanostructure exposed to longitudinal magnetic field; in the future, it would be helpful to observe the nanostructure exposed to magnetic field in 3-dimensional directions.

Data availability
Data will be provided on request.