Size-dependent thermoelastic initially stressed micro-beam due to a varying temperature in the light of the modified couple stress theory

The bending of the Euler-Bernoulli micro-beam has been extensively modeled based on the modified couple stress (MCS) theory. Although many models have been incorporated into the literature, there is still room for introducing an improved model in this context. In this work, we investigate the thermoelastic vibration of a micro-beam exposed to a varying temperature due to the application of the initial stress employing the MCS theory and generalized thermoelasticity. The MCS theory is used to investigate the material length scale effects. Using the Laplace transform, the temperature, deflection, displacement, flexure moment, and stress field variables of the micro-beam are derived. The effects of the temperature pulse and couple stress on the field distributions of the micro-beam are obtained numerically and graphically introduced. The numerical results indicate that the temperature pulse and couple stress have a significant effect on all field variables.


Introduction
More recently, small scale and nano-scale structures have been used in several design devices such as nuclear drive and switches. A few models are developed for small scale and nanostructures, including strain gradient and couple stress. The modified couple stress (MCS) theory can be seen as an exceptional example of the strain gradient theory. The MCS theory considers the revolution as a variable in determining curvature, while the strain inclination theory considers the strain to be variable in curvature pronounce [1][2][3][4] .
The classical elasticity theory is not suitable to capture the effect of the size of the microstructure. In addition, it is appropriate to study the material behavior on a large scale. As

Basic equations
For simplicity and readability propose, all the mathematical notations appearing in this paper are defined in nomenclature. The basic equations of the isotropic homogenous thermoelastic medium (in the Oxyz coordinates) at an initial uniform temperature T 0 based on the MSC theory are given as follows [17,23,32] : where λ and μ denote Lame's constants, σ ij are components of the stress tensor, e kl are components of the strain tensor, u k is the displacement vector, θ = T − T 0 is the excess temperature distribution, ω i is the rotation vector, m ij is the couple stress tensor, α = l 2 μ is a higherorder modulus and is considered as the rotational modulus indicating the resistance of the material versus the gradient of its element rotation, l is the material length scale parameter, γ = α t E/(1 − 2ν), E is Young's modulus, α t is the thermal expansion coefficient, ν is Poisson's ratio, and δ ij is the Kronecker delta function. The parameter α is indeed a higher-order modulus which can be regarded as the rotational modulus representing the resistance of the material against the gradient of the rotation of its elements.
In this paper, we take the equation of heat conduction corresponding to that conducted in Ref. [33]. This equation can be written as follows: where K is the thermal conductivity, k = K/(ρC E ), C E is the specific heat per unit mass at constant strain, τ 0 indicates the thermal relaxation time, which guarantees that Eq. (6) will expect a limited speed of heat spread. The values of λ and μ are defined as [34]

Mathematical model description
In this model, we assume that the thermoelastic micro-beam properties are unstrained, unstressed, and clamped-clamped. Let the constants ρ, σ 0 , and A denote the density, the initial tension, and the area cross-sectional of the micro-beam, respectively. As shown in Fig. 1, the micro-beam length, width, and uniform thickness are denoted as L, b, and h, respectively.
In order to formulate the governing equations of the thermoelastic micro-beam, the following assumptions are considered: (i) the cross-sectional area along the x-direction does not change; (ii) the Euler-Bernoulli beam theory is employed [35][36] ; (iii) the only transverse motion is taken into consideration. As explained in Refs. [36]- [39], the axial motion can be very important and in some cases cannot be ignored. Based on the Euler-Bernoulli beam theory, the displacements may be written as follows: in which u is the axial displacement and w is the lateral deflection. By substituting Eq. (8) into Eq. (4), the rotation vector can be written as The components of the curvature tensor are derived by substituting Eq. (9) into Eq. (3) as Using Eq. (8), the axial thermal stress σ x given in Eq. (1) is reduced to The bending moment M resultant of the micro-beam can be defined as follows: Substituting Eqs. (2), (10), and (11) into Eq. (12), the moment M is given by where The transversely governing equation of motion due to the initial compression stress σ 0 can be formulated as [30,40] Note that a negative estimation of σ 0 suggests the initial tensile stress. By substituting Eq. (13) into Eq. (15), the equation of motion can be expressed as It can be noticed from Eq. (16) that the motion equation of the Euler-Bernoulli beam is composed of some parts: some of them are associated with ρA, σ 0 A, EI, and αA, and the others are relevant to α T EI. The first part is the same as the one in the classical model, while the second part is added due to the existence of the MCS theory. The third part is due to the initial stress, and the last part is due to the temperature field. When the parameter of the couple stress of the material vanishes (α = 0), the basic equation reduces to the classical model.
Substituting Eq. (8) into Eq. (6), the heat conduction will be in the form The temperature θ(x, z, t) can be assumed to be in the form of sinusoidal function [41] , i.e., By substituting Eq. (18) into Eqs. (16) and (13), we get Integrating Eq. (17) along the z-axis, the following heat equation is attained: We use the non-dimensional quantities as follows: The non-dimensional basic equations can be rewritten as (dropping primes for convenience) where

Initial and boundary conditions
The initial conditions are assumed to be homogeneous, i.e., We consider the case that the two ends of the micro-beam are clamped. Then, the following boundary conditions are expressed as: Also, we suppose that the surface x = 0 of a micro-beam is subject to a temperature pulse in the form where Θ 0 is the amplitude of the thermal load, and ω is the temperature pulse. Furthermore, the temperature change at the end boundary x = L should satisfy the condition

Solution in the transformed domain
The Laplace transform is characterized by the integral The governing equations in the Laplace domain are given by where Combining Eq. (33) and Eq. (34) gives the differential equation for w or Θ as where the coefficients A, B, and C are given, respectively, by Introducing m i into Eq. (38) yields where D = d/dx, and m 2 1 , m 2 2 , and m 2 3 are the characteristic roots of the equation The roots of Eq. (41) achieve the notable relations The analytical solution to Eq. (40) can be calculated as where L i and M i are constant coefficients depending on s, and The bending moment M , given in Eq. (36) in the Laplace space using Eqs. (43) and (44), can be gained as where γ i = A 6 m 2 i + A 7 β i . Also, the axial displacement after using Eq. (43) takes the form Furthermore, the strain can be calculated as In the domain of the Laplace transform, the boundary conditions (29)-(31) take the forms Substituting Eqs. (50)-(52) into Eqs. (43) and (44) yields To complete the analytical solution within the Laplace transform domain, it remains for us to find the values of the coefficients L i and M i . These values can be easily derived by solving the system of equations (53)-(58).
To obtain the field variables in the physical domain, the inversion of the Laplace transform will be derived. Due to the complexity of obtaining the inversion of the Laplace transform analytically, instead, the Riemann-sum approximation technique will be employed. To accomplish this task, the following well-known Riemann-sum approximation formula defined will be used [42] : For convergence of the solution reasons, the Riemann-sum approximation technique [43] insists that υ ≈ 4.7/t.

Numerical results
In this section, the effects of the temperature pulse, the absence and presence of the couple stress, and the relaxation time on the field amounts are numerically discussed. All the numerical calculations are conducted using MATHEMATICA platform. The mechanical properties of the Nickel micro-beam can be introduced as [44] E = 210 GPa, ρ = 8 900 kg/m 3 , C E = 438 J/(kg · K), We assume that the values of the micro-beam non-dimensional operational parameters are assigned as h = 10, L/h = 10, b/h = 0.5, t = 0.12 s, and Θ 0 = 1. Using the approximation method defined in Eq. (59), the values of the deflection w, temperature θ, thermal moment M , displacement u, and stress σ x are computed. The effects of the presence of the couple stress, initial stress, and temperature pulse on the micro-beam will be introduced.

Couple stress effect
This subsection indicates the effects of the small length scale α (couple stress) on microbeams in terms of the deflection w, temperature θ, thermal moment M , displacement u, and stress σ x in two cases. Namely, the presence and absence of couple stress are indicated as α = 2.5 and 0.0, respectively. Figures 2-6 illustrate the two cases, where the parameters σ 0 , ω, and τ 0 are fixed to certain values.
In order to verify the results obtained in this subsection, they are compared with the results of the classical Euler-Bernoulli beam. It is worth noting that the values obtained by the MCS theory in the present study are always greater than the calculated values based on the classical theory. This important observation is consistent with the results obtained by Babaei et al. [45] .
As displayed in Fig. 2, compared with the classical theory considering the effect of the material length scale parameter on the MCS theory, it leads to a decrease in the lateral deflection of the micro-beam. Therefore, taking into account the parameter of the material length scale, the maximum and minimum of the lateral deflection decrease. Also, as time goes by, the deflection starts from zero and reaches a maximum value at x = 0.2 based on the couple stress theory, and finally quickly tends to zero, satisfying the mechanical boundary condition.  Thus, we can conclude that the thermal vibrations of the micro-beams obtained with the help of the MCS are higher than those predicted by the classical theory of Euler-Bernoulli beams. Comparing the results with Kong et al. [46] , we find that there is a convergence of the results.
As can be seen in Fig. 4, the MCS model predicts lower values of natural axial stress σ x in comparison with the classical theory. The parameter α has a significant effect on the distribution of the bending moment M as shown in Fig. 5. It is clearly observed in Fig. 5 that the value of the bending moment M decreases with a decrease in the value of the small length scale parameter α, and this corresponds to Refs. [47]- [49]. Figure 6 illustrates that the value of the axial displacement u goes down in the range 0 x 0.1 and then goes up to the maximum amplitudes in the range 0.1 x 0.3. In addition, the small length scale parameter α has a great impact on the displacement distribution u. From Figs. 2-6, we note that the values of distribution fields increase in the presence of the couple stress term and decreases in the absence of that term.

The initial stress effect on solid materials
This case depicts the variety of the dimensionless physical fields of the micro-beam with three different values of the dimensionless initial stress σ 0 (see . In this case, we have assumed that the pulse of temperature ω = 0.2 and the relaxation time τ 0 = 0.1. Note that, the value of σ 0 = 0 indicates the absence of the initial stress, while other values indicate that the micro-beam is subject to compressive strength. Figure 7 shows the deflection of the micro-beam for various values of the non-dimensional initial stress parameter σ 0 . Obviously, by raising the value of the parameter σ 0 , the deflection increases. Figure 8 displays the temperature response of the micro-beam for various values of the initial stress parameter versus the distance x. In this figure, the increase in the initial stress force σ 0 causes a weak effect on the temperature of the micro-beam. Figures 9-11 indicate the variations of the thermal stress σ x , the displacement u, and the bending moment M for various values of the dimensionless initial stress σ 0 . As can be seen, the initial pressure increase leads to an increase in the field variables. Figures 12-16 display the reactions of a clamped-clamped micro-beam resonator in the direction of the axial x with different temperature pulses ω, σ 0 = 1, and α = 2.5. From these figures, we note that the pulse temperature constant has a significant effect on the mechanical behaviors of the Euler-Bernoulli micro-beam. The inclusion of the temperature pulse effect in the beam theory leads to an increase in the thermodynamic interactions in the micro-beam. As a result, this study gives a physical realization that can be practical to design and analyze the vibrations for micro-/nano-structures.

Comparable results in Lord-Shulman (LS) and coupled thermoelasticity (CTE) theories
In the current case, the responses of variable fields in the cases of the theories of coupled thermoelasticity and the Lord-Shulman theory are analyzed. The calculations are conducted based on the values of the relaxation parameter τ 0 . If τ 0 = 0, the responses of variable fields are applied to the CTE theory. If τ 0 > 0, the responses of variable fields are applied to the LS theory. The other parameters are assumed to be constants. From Figs. 17-21, we conclude that: (i) the magnitudes of the considered field variables in the CTE model are greater compared with those in the LS model; (ii) the parameter τ 0 has a great effect on the propagation of all field quantities; (iii) the micro-beam exhibits more deflection in the case of the coupled thermoelastic (CT) beam than that for the generalized thermoelastic LS model; (iv) the mechanical distributions indicate that the wave propagates with a finite velocity in the medium; (v) although the thermal wave spreads with a finite speed in the coupled theory thermoelasticity as seen in Figs. 17-21, there are great differences between the coupled theory and the generalized theory.

Conclusions
In this work, the effects of the initial stress, temperature pulse heating, and MCS term on the thermoelastic response of a micro-beam are mathematically analyzed. A governing equation that governs the studied problem dependent on the Euler-Bernoulli theory is proposed by the use of Hamiltonian's principle. Using the Laplace transform, the expressions of the field variables are derived. The proposed model is numerically analyzed using MATHEMATICA.
The numerical results indicate that the parameter of the couple stress has significant effects on all the distributions of the studied fields. Increasing the effect of couple stress leads to decreasing the values of the basic field variables. Furthermore, the thermoelastic deflections, thermal stress, displacement, temperature, and moment strongly depend on the pulse of temperature.
The current investigation can be furthered to the analysis and design of micro-structures with geometrical shapes and different load conditions under different boundary conditions based on the Euler-Bernoulli beam theory, and the obtained results may be valuable to mechanical engineers in designing small scale resonators and MEMS applications.

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