Nonlocal state - space strain gradient wave propagation of magneto thermo piezoelectric functionally graded nanobeam

: In this work, the state - space nonlocal strain gradient theory is used for the vibration analysis of mag - neto thermo piezoelectric functionally graded material ( FGM ) nanobeam. An analysis of FGM constituent proper - ties is stated by using the power law relations. The re ﬁ ned higher order beam theory and Hamilton ’ s principle have been used to obtain the motion equations. Besides, the governing equations of the magneto thermo piezoelectric nanobeam are extracted by developed nonlocal state - space theory. And to solve the wave propagation pro - blems, the analytical wave dispersion method is used. The e ﬀ ect of magnetic potential, temperature gradient, and electric voltage in variant parameters are presented in graph.


Introduction
Functionally graded materials (FGMs) are a type of composites initiated by a group of Japanese scientists to control the volume fraction of the mixture of two or more materials.The nonlinear vibration of the piezoelectric nanobeams based on the nonlocal and Timoshenko theory, the influence of the nonlocal parameter, temperature change, and external electric voltage on the size dependent nonlinear vibration characteristics of the piezo electric nanobeam are exposed [1].Researchers [2] studied the natural frequencies along with mechanical and thermo electric vibration of piezoelectric nanobeams based on the nonlocal theory.Ebrahimi [3] reported the scattering of waves of FG nanobeam of viscoelastic nature.In the framework of third-order shear deformation theory [4], the vibration characteristics of functionally graded (METE-FG) nanobeams were analyzed.And the free vibrations of FG nano plates resting on elastic foundation via Hamilton principle was dealt in detail [5].Alibeigi et al. [6] introduced the buckling retaliation of nanobeams on the basis of the Euler-Bernoulli beam model with the von Kármán geometrical nonlinearity.Bending of flexo electric magneto-electro-elastic (MEE) nanobeams lying over Winkler-Pasternak according to nonlocal elasticity theory has been studied [7].Several studies were conducted on [8][9][10] hygro-thermal loading, the bending analysis of magneto-electro piezoelectric nanobeams system, dynamic analysis of smart nanostructures, and frequency analysis of thermally post buckled FGM thin beams.Stress-driven vs strain-driven elastic nanobeams have been discussed via integral elasticity [11,12].Using the kinematic model, Kiani and Eslami [13] reported the buckling of beams made of FG under different types of thermal loading.The propagation of wave of infinite functionally graded plate in thermal environment was reported by Sun and Luo [14].A consistent refined HSDT is designed to probe the free vibration of GF plates on elastic foundation and the influence of boundary condition on the natural frequency [15].The different working conditions of the nano sized elements were studied by Thai et al. [16].By considering the nonlocal elasticity [17], a prediction has been made that the essential behaviors of the nanostructures cannot be same as the macro scale structures.The Euler-Bernoulli beam theory was used to study the bending analysis of microtubules (MT) by Eringen [18].Based on Euler-Bernoulli beam theory, the bending analysis of MT was studied using the method of Differential Quadrature (DQ) by Civalek and Demir [19].Using finite element method (FEM) [20], the nonlinear bending in nanobeams were discussed.Reddy and El-Borgi [21] investigated the dispersion of waves with the effects of surface stresses in smart piezoelectric nanoplates.Bi-directional FGM nanobeams with the characteristics of bending, buckling, and vibrational nonlocal elements were concentrated in some of the previous studies [22][23][24].The natural frequency variation located on a viscoelastic sheet was surveyed by using the nonlocal theory [25].The size-dependent elements of beam were analyzed by Ebrahimi and Barati [26].Nonlocal elasticity and its running conditions are discussed in details in the literature [27].Based on the nonlocal strain gradient theory (NSGT) [28], the thermo-mechanical buckling problem of graphene sheets was proposed.Stiffness, softening, and hardening effect of FG beam were studied by Li et al. [29].Solving the wave dispersion problem of nanoplates was accomplished by Ebrahimi and Dabbagh [30] with the application of infusing NSGT and surface-related elasticity for responsive piezoelectric materials.With the small-scale effect, the free vibration of 3D FGM Euler-Bernoulli nanobeam was studied by Hadi et al. [31].Alibeigi et al. [32] exposed the buckling response of a nanobeam on the basis of the Euler-Bernoulli beam model using a couple stress theory under various types of thermal loading and an electrical and magnetic field.Timoshenko beam theory was investigated by Ke and Wang [33] with the rise in uniform temperature, magnetic potential, and external electric potential via nonlocal form to MEE vibrations.Bending of MEE nanobeam was studied in detail by Ebrahimi et al. [34].Along with that, Ebrahimi et al. [35] investigated the bending of MEE nanobeams relating the nonlocal elasticity theory under hygro-thermal loading embedded in Winkler-Pasternak foundation.The size dependent problems using nonlocal elasticity theory, nonlocal couple stress theory, and shear deformation theory were reported [36,37].Ebrahimi et al. [38] discussed the effects of various parameters on the wave dispersion characteristics of size-dependent nanoplates.The thermal effects on the buckling and free vibration of the FG nanobeams is documented well in the literature [39].Ebrahimi and Barati [40] discussed the damping vibration characteristics of the hygro-thermally affected FG viscoelastic nanobeams.The thermal effect on buckling and free vibration characteristics of size-dependent Timoshenko nanobeams, and the free vibration of curved FG nano size beam in thermal environment been discussed in the literature [41,42].The buckling and vibration properties of sandwich FG beams were studied by Vo et al. [43].Jalaei et al. [44] studied the thermal and magnetic effects on the FG Timoshenko nanobeam.Studies over the hygro-thermal wave characteristic of nanobeam of an inhomogeneous material with porosity under magnetic field is notable [45].
Hence, this work shows the wave propagation analysis of FG nanobeam with the help of a nonlocal statespace strain gradient viscoelasticity.The magneto thermo material properties of the nanobeam also graded and implemented via power law relations and the motion equations are deduced through the Hamilton's principle.Furthermore, the dispersion for computed external electric voltage, magnetic effect, and the gradient of temperature are presented with graphical solution.

Mathematical formulations
Based on the state-space nonlocal strain gradient theory, the length L, width b, and thickness h of a viscoelastic FG nanobeam has been investigated (Figure 1).The two parts of constituent FGM are composed of ceramic part and metallic part.The components of FGM are considered to be temperature dependent to evolve a realistic viscoelastic study.
In this section, power-law relations have been used to compute the properties.To calculate the temperature variable towards the thickness direction, the volume fraction of each phase must be calculated by using the power-law model.Hence, the volume fraction of ceramic partis given as follows: where the thickness h and the exponential power law p probe each phase in the material with its distribution of properties.By considering any desired material property at the local temperature (2.2) According to Eringen's nonlocal theory, the stress state at a point inside a body is a function of the strains at all points in the neighboring regions.The basic equations with zero body force can be defined as follows: where σ ε D E , , , ij ij i i , respectively, represent the stress, strain, electric displacement, and electric field and B Ω , i i are the magnetic induction and magnetic field.α kl and T Δ stand for thermal expansion and temperature difference.C e , ijkl mij , and χ ij are the elastic, piezoelectric, and magnetic constants, respectively, and = / τ e a l 0 defines the scale coefficient, e 0 is the material constant, and a and l are the characteristic length in the internal and external sides.
The governing equations of the nanobeams are obtained by an accurate kinematic theory.Higher order shear deformation theory also reveals stress-strain changes in solid bodies.From the previous study [8], we can take the refined shear deformable beam's displacement as follows: where Π , w b , and w s are the longitudinal displacement, bending, and shear components of the transverse displacement.Furthermore, in order to distribute the shear strain, ( ) * f z is the shape function which is designed as follows [4]: (2.5) To capture the shear strain and stress, the deformed structural cross section is uncertain with this function.
At free surfaces, it is required to satisfy the assumption of shear strain nonexistence.By continuum infinitesimal strain tensor, the nonzero strains can be measured as follows: where ( )

Motion equations
In accordance with Hamilton's principle, the extended Lagrangian can be given as follows: So, the Hamilton's principle can be given as follows: In Eq. (2.8), variables ∏ V , , and T are the strain energy, work done, and kinetic energy, respectively.Hence, the virtual strain energy can be given as follows: (2.9) and the stress resultants can be obtained as follows: (2.12) The kinetic energy of the system can be determined as follows:

14)
Nonlocal state-space strain gradient wave propagation of nanobeam  3 Infusion of Eq. (2.4) in Eq. (2.14) results in the following: (2.15) In accordance with the magnetic and temperature effect, where f B , η, A, and Ω x stand for the magnetic force, mag- netic field permeability, cross sectional area of the nanobeam, and the magnetic potential of the longitudinal magnetic field.For an FG nanobeam, it is assumed that the temperature can be distributed uniformly across its thickness and the temperature gradient at stress free state is T Δ .
In the above definition, the inertia of mass moments can be defined as follows: and the work done N x with a temperature gradient of thermal effect can be defined as follows: By inserting the Eqs.(2.11) and (2.15) in Eq. (2.8), the equation of the beam in Euler-Lagrange can be derived and the outcome can be coupled as follows: (2.17) (2.20)

Nonlocal state-space model
This section demonstrates the nonlocality stress and strain effects on the time-space domains, when the wave length or excitation frequency interferes with time and intrinsic characteristic length.Nonlocal time-space viscoelasticity problems are based on the combination of the Boltzmann superposition integral and the Eringen concept of nonlocal elasticity.Accordingly, integral stress in nonlinear state and strain equations are stated as follows: where σ ij and ε kl are the stress and strain tensor arrays and the nonlocal kernel functions are ( Eq. (3.1) can be read in the following form via Fourier, inverse Fourier, and Taylor series, To balance the absence of stiffness-hardening behavior, the nonlocal strain gradient elasticity must be incorporated in the equation.The following relation can be used to derive the nonlocal strain gradient viscoelasticity with fraction. (3. 3) The Kelvin-Voigt relation of viscoelastic material with three parameters in a solid state is given as follows: where = λ l ε and = μ l σ are length scale and nonlocal parameters, respectively.The relation between rheological character and spatial nonlocality of the system is designed as . From Eqs. (2.12) and (2.13), the displacement field having resultant stress over the area of cross section of the beam can be computed as follows: (3.7) (3.8) 15 11 (3.9) where cross sectional rigidities are  (4.1)

Analytical solution
The governing equation obtained in Section 4 will be solved in this section.The analytical wave dispersion method is used in this case to solve the problems of wave propagation for various types of structures, including beams, plates, and shells.A higher order beam's solution function can be in the form as follows: where Π w , b , and w s are the anonymous amplitudes of propagating waves.Here ω & β are frequency and wave number, respectively.By substituting the above expression in Eqs.(4.1)-(4.3), the achieved form will be as follows: Nonlocal state-space strain gradient wave propagation of nanobeam  5 where damping and mass matrices are given by [ ] K , [ ] C , and [ ] M respectively.The components of these symmetric matrices are as follows: (5.3) (5.4) (5.5)

Results and discussion
This section illustrates the magneto-thermo vibration of FG nanobeam with the numerical examples.The material properties are composed of BaTiO 3 and CoFe O 2 4 , which are presented in Table 1.Table 2 shows the comparison of the buckling load for various power-law exponent values.Whereas, Table 3 gives the variation in the magnetic potential and electric voltages with the varying and increasing nonlocal parameter.Since an increasing nonlocal parameter shows a decrement with the magnetic potential (Ω) and the electric voltage (V ).Now, Figures 2 and 3 highlight the effect of external voltage (V) with the variation in the rising temperature (ΔT ) and the gradient index (p) for = μ 1.0 & 1.5.Whereas, with the increase in the external voltage (V = 0, 0.5, and 1), the temperature gradually decreases with respect to the gradient index (p) and nonlocal parameter.Figure 4 shows the decrement in the buckling load with an increase in the magnetic effect (Ω) and the nonlocal value (μ).Also, when there is no magnetic effect found, the buckling load at some point decreases with the increse in the value of the nonlocal (μ). Figure 5 presents the variation in the dimensionless buckling load with the nonlocal parameter (μ) in the effect of the external voltage (V).Since there is an increase in the voltage (V), the dimensionless buckling load declines and gets neutralized at some point of the nonlocal parameter.Figures 6 and 7 depict the effect of gradient index with respect to  stabilizes with an increase in the gradient index and magnetic potential.Thus, when the magnetic potential = Ω 0,          the temperature calms down and stabilizes at some point with respect to an increase in the gradient index (p) and nonlocal values.Figures 12 and 13 show the effect of an internal damping factor (τ) in the existence of wave number and wave frequency over nonlocal parameter (μ).
In these figures, the damping factor with ≥ τ 1.5 shows a raising linear value and when ≤ τ 0.5, there is an oscilla- tion in the wave number and wave frequency.

Conclusion
The above study shows the wave propagation analysis of piezoelectric FGM nanobeam.Magneto thermo properties of the FG nanobeam are considered to be the function of thickness according to the power-law model.The governing equations are extracted by substituting the structure displacement field equations in the beam's Euler-Lagrange equations and are framed as symmetric matrices components to arrive at required solutions.Hence, the upshots of the work are as follows: • The stability behaviors of FGM nanobeam are affected .bymagneto thermo piezo electricity and nonlocal values.

Wave number
Wave frequency 12: Damping factor in the presence of wave number with respect to wave frequency for μ = 1.0.• Physical variants could be controlled via applying a suitable value of damping factor.• Natural frequency reduces, while the nonlocal parameter and gradient index of the FG nanobeam amplify.• The increase in power-law index softens the volume fraction.• The bending rigidity and phase velocities are high in amplified wave numbers and get reversed in low wave numbers.

Figure 2 :
Figure 2: External voltage with the presence of rising temperature to gradient index for μ = 1.0.

Figure 3 :
Figure 3: External voltage in the presence of temperature with respect to gradient index for μ = 1.5.

Figure 4 :
Figure 4: Magnetic potential on the dimensionless buckling with respect to nonlocal parameter.

Figure 6 :
Figure 6: Gradient index with the presence of rising dimensionless buckling load to magnetic potential for μ = 1.0.

Figure 5 :
Figure 5: External voltage on the dimensionless buckling load with respect to nonlocal parameter.

Figure 7 :
Figure 7: Gradient index in the presence of dimensionless buckling load to magnetic potential μ = 1.5.

Figure 8 :
Figure 8: Gradient index in the presence of dimensionless buckling load to electric voltage when μ = 1.0.

Figure 9 :
Figure 9: Gradient index in the presence of dimensionless buckling load to electric voltage when μ = 1.5.

Figure 10 :
Figure 10: Magnetic potential in the presence of temperature with respect to gradient index for μ = 1.0.

Figure 11 :
Figure 11: Magnetic potential in the presence of temperature with respect to gradient index for μ = 1.5.

Figure 13 :
Figure 13: Damping factor in the presence of wave number with respect to wave frequency for μ = 1.5.

Table 1 :
MEE coefficients of material properties

Table 2 :
Comparison of FGM beam non-dimensional buckling for various power-law exponents

Table 3 :
Dimensionless frequency of an FG nanobeam varies with nonlocal parameters, electric voltages, and magnetic potentials