2D Analysis of Piezoelectric Layer Over a Rotating Micro-elongated Thermoelastic Medium with DPL Model

The Lord-Shulman theory with one relaxation time and the dual-phase-lag model with two relaxation times of thermoelasticity are used in this article to study the influence rotation micro-elongated thermoelastic layer, when a piezo electric layer is above it. To convert a partial differential equation to an ordinary differential equation, the normal mode method is utilized. Numerical computations are implemented for aluminum epoxy, and the results are charted. A comparison is made among the two theories in the complete absence and the presence of a rotation. The presence of a rotation has a major effect on all physical quantities.

Keywords: Piezoelectric, Normal Mode Analysis, Rotation, Dual-Phase-Lag Model, Micro-Elongated Thermo elasticity

Nomenclature: T Absolute temperature
Ω Angular velocity
α_t1, α_t2 Coefficient of linear thermal expansion where 10(32),tβλμα=+ 21(32)tβλμα=+
σ_ij Component of stress tensor for micro-elongated medium
ρ Density in micro-elongated medium
ρ^P Density in piezoelectric layer
u Displacement vector in micro-elongated medium
u^P Displacement vector in piezoelectric layer
Eⁱ Electric field in piezoelectric layer
τ^q Heat flux parameter
λ,μ Lame's constants in micro-elongated medium
a₀,λ₀,λ₀ Micro-elongational constants
ϕ Micro-elongational scalar
j₀ Microinertia
T₀ Reference temperature
c_e Specific heat at constant strain in micro-elongated medium
c^P_e Specific heat at constant strain in piezoelectric layer
τ_θ Temperature gradient parameter
ε_ij The dielectric moduli in piezoelectric layer
C_ijkl The elastic parameters tensor in piezoelectric
D_i The electric displacement in piezoelectric layer
ϕ^P The electric potential in piezoelectric layer
e_kij The piezo electric moduli in piezoelectric layer
k Thermal conductivity in micro-elongated medium
k^p Thermal conductivity in piezoelectric layer

Piezoelectric is the conversion of mechanical stress to electrical charge, and vice versa. Piezoelectric substances are produced industrially in single-crystal and ceramic shapes, and they are the second most common application of dielectric materials, after semiconductor materials (Othman et al. [1]). Based on observation distribution generated electric potential difference throughout the disk thickness, Ashida and Tauchert [2] proposed a finite-difference composition for assessing the time-varying, axisymmetric ambient temperature on the face of a piezoelectric circular disk. In the 2006 Haussuhl et al. [3] proposed Bismuth triborate (BiB3O6) is a nonlinear optical material with piezo-electric and elastic properties. Li and He [4] recently introduced the study of a general linear piezoelectric-thermoelastic problem with a nonlocal influence and temperature-dependent characteristics.

A micro-elongated elastic solid has four degrees of freedom, three of which are for translation and one for micro-elongation. According to micro-elongation theory, material particles can really only carry out volumetric micro elongation in addition to classical medium deformation. Such medium's material points can lengthen and contract independently of their translations. And also, micro-elongated media include solid liquid crystals, composite materials reinforced with chopped elastic fibers, and porous media with pores filled with non-viscous fluid or gas. In the context of generalized thermo-elastic theory, Shaw and Mukhopadhyay [5] examined the response due to periodically differing heat sources in vicinity of the origin of a functionally graded isotropic boundless micro-elongated medium in 2012. One year later, Shaw and Mukhopadhyay [6] again presented another study to investigate the implications of a moving heat source in an infinitely long micro-elongated, isotropic, homogeneous, thermoelastic medium. In 2015, Sachdeva and Ailawalia [7] presented research on two-dimensional deformation in a thermoelastic micro-elongated solid, when mechanical force applied along the user interface of fluid half-space and thermoelastic micro-elongated half-space. Also, in the same year, Ailawalia et al. [8] focused on to study the deformation caused by an internal heat source in a thermoelastic micro-elongated solid with an h-thick elastic layer on top. later, Ailawalia et al. [9] reintroduced the previous study in a different way, where he changes the elastic layer from a bounded layer to an unbounded layer. Utilizing the normal mode analysis technique, Ailawalia et al. [10] developed a model for a thermoelastic micro elongated solid keep in mind the micro elongation effect and laser pulse heating. The eigenvalue approach was used to solve an axisymmetric problem of a micro-elongated thermoelastic medium with an infinite circular plate under the effect of thermo-mechanical sources has been discussed by Kumar et al. [11]. Ailawalia and Singla [12] studied two-dimensional deformation caused by laser pulse heating in a thermoelastic micro elongated layer with a thickness of 2d that is engrossed an infinite inviscid liquid.

Ozisik and Tzou [13] and Tzou [14, 15] developed a new model for the heat transport mechanism named the dual-phase-lag model, in which Fourier's law is replaced by an approximation to the modification of Fourier's law with two different time translations for the heat flux and the temperature gradient. In the dual-phase-lag model, Othman and Abd-Elaziz [16] focused on investigation the influence of thermal loading caused by laser pulses on thermoelastic medium with voids. Abbas [17] published a research paper to examine the influence of dual-phase-lag on thermoelastic interplay in an infinite fiber-reinforced anisotropic medium with a circular hole. Reflection of plane waves from electro-magneto-thermoelastic half-space using a dual-phase-lag model has been discussed by Abd-Alla et al. [18]. Othman and Abd-Elaziz [19] utilized a dual-phase-lag model to investigate the effect of gravitational and rotation fields on the plane waves of a linearly magneto-micropolar thermoplastic isotropic medium. Based on the dual-phase-lag modification of Fourier's law, Othman and Eraki [20] studied the induced thermo-elastic diffusion waves in a homogeneous isotropic medium due to an ultra-short-pulsed laser heating that decomposed significantly. Abdou et al. [21] explained the effect of rotation and gravity on generalized thermoelastic medium with double porosity under L-S theory.

In the background of Green and Lindsay's linearized theory, Othman [22] provided the normal mode analysis to two-dimensional problems of general linear thermoelasticity with two relaxation times under the influence of rotation. Othman [23] proposed a generalized thermo-viscoelastic plane wave model for a half-space whose surface is subjected to a thermal shock under the influence of rotation with one relaxation time. Othman and Singh [24] discussed the influence of rotation on general linear micropolar thermoelasticity in a half-space according to five theories. Othman and song [25] studied the rotational effects on general linear electro-magneto-thermo-viscoelasticity plane waves with two relaxation times. Later, Li et al. [26] proposed to investigate the rotational influences on plane waves of general linear electro-magneto-thermoelastic with diffusion in a half-space. Othman and Abbas [27] provided the multi-phase-lag theory in order to investigate the rotating on a 2-D analysis of the micropolar thermoelastic isotropic medium.

The main objective of this paper is to examine the influence rotation micro-elongated thermoelastic layer under dual-phase-lag model, when piezo electric layer is above it. The precise expression of the variables examined for the dual-phase-lag model of thermo-elasticity and the variation of the variables examined are depicted graphically using normal mode analysis.

The system of governing equations of a micro-elongated thermoelasticity with rotation, in a dual-phase-lag model (Fig. 1) can be written as [8, 12]

From equation (1) and equation (4) for displacement vector (x,z,t)=u(u₁,0,u₃), and the rotation Ω=(0,Ω,0) the equations of motion are given by.

For simplification we shall use the following non-dimensional variables

The displacement potentials Φ(x,z,t) and ψ(x,z,t) which relate to displacement components has been introduced, we obtain

Substituting from Eqs. (7) and (8) into Eqs. (2), (3), (5) and (6), we obtain

The solution of the considered physical variable can be decomposed in terms of normal modes as the following form:

where, ω is a complex constant, i=√-1,b is wave number in the x direction.

Using Eq. (13) into Eqs. (9)-(12), then we have

Eqs. (14-17) have a non-trivial solution if the determinant coefficients of the physical quantities equal to zero, then we get.

Eq. (18) can be factorized as:

Where, K²_n,(n=1,2,3,4)= are roots of the characteristic equation of Eq. (19)

The general solutions of Eq. (19) bound as =(z→∞) is given by.

Substituting from Eq. (20) into Eq. (8) we obtain the components of displacements.

Substituting from Eqs. (7) and (13) into (4) and with the help of Eqs. (20-22) we obtain the components of stresses.

where, the coefficient a_i, A,B,C,E and H_in are given in Appendix 1.

The system of governing equations of general piezoelectric are given by [28-30]

For simplification, we shall use the following non-dimensional variables [4]

Substituting from Eqs. (28) and (13) into Eqs. (24) and (25)

Eliminating u^p*₁ ,u^p*₃, ϕ^p* between Eqs. (29), (30) and (31), we obtain

Eq. (32) can be factorized as:

Where, r²_m,(m=1,2,3) are roots of the characteristic equation of Eq. (33) the solutions of Eq. (33) are of the form:

Substituting from Eqs. (28) and (13) into Eqs. (26) and (27) and with the help of Eq. (34),

we obtain the components of stresses and the electric displacement in a piezoelectric layer

The coefficient ,l_m′A₁,A₂,G,N,F,H_n'n and

L_m'm, L_m'(m+3), Where m'=1,2,......,6 are given in Appendix 2

The parameters M_n , (n = 1,2,3,4), R_m and R_m+3, (m = 1,2,3) have to be selected such that boundary conditions at the surface are

Where, P₁ is the magnitude of the mechanical force.

The utilization of the expressions of the variables considered into above boundary conditions (37), (38) to obtain the equations that are satisfied by the parameters. And hence, ten equations will be obtained. If the inverse method matrix is applied on the ten equations, we get then the value constant M_n , (n = 1,2,3,4), R_m and R_m+3, (m = 1,2,3).

The analysis is conducted for aluminum epoxy-like material as [6]:

λ=7.59×10¹⁰N/m², a₀=0.61×10^-10N, a₀=0.61×10^-10N, ρ=2.19×10³ kg/m³

C_e= 966J/kg.k, β₀=β₁=0,05×10⁵N/m².k, k =252J/m.s.k, j₀=0.196×10^-4m²

λ₀=λ₁=0.37×10¹⁰N/m², T₀=293k, τ_θ= 0.002s, 0.009s,ω=ω₀+iζ

ω₀= -6.428×10^-4S^-1 ζ=0.06657S^-1, b=4m^-1, h=0.001m.

The material chosen for piezoelectric is taken as Cadmium Selenide (CdSe) having hexagonal symmetry (6 mm class) [31]

C₁₁= 7.41×10¹⁰N.m^-2, C₁₃= 3.93×10¹⁰N.m^-2, C₃₃= 8.36×10¹⁰N.m^-2, C₄₄= 1.32×10¹⁰N.m^-2, ρ= 5504kg.m^-3, e₃₁= -0.160C.m^-2, e₃₃= -0.347C.m^-2, e₁₅= -0.138C.m^-2, C^p_e= 260 J/kg.k, ε₁₁=8.26x10^-11C².N^-1.m^-2, ε₃₃=9.03x10^-11C².N^-1.m^-2

The computations are implemented for the value dimensionless time t=0.021s in the range 0≤Z≤1.4 on the surface x=1.4m to all physical quantities except the temperature T and the micro-elongational scalar ϕ is in a range 0≤z≤3. The numerical technique presented here is used to distribute the horizontal displacement u₁ the vertical displacement u₃ the temperature ,T the micro-elongational scalar ,ϕ the stresses components ,σ_xx, σ_zz and σ_xz with distance z. To examine the effect of the presence and complete absence of rotation on the solution in the dual-phase-lag model and Lord-Shulman theory and the influence of the phase-lag temperature of gradient τ_θ, when the phase-lag of heat flux τ_q is constant on solution in just dual-phase-lag. This paper presents the numerical evaluation results in the form of charts. The results are depicted in Figs. 2-15 for the magnitude of mechanical force 1N10.P=N Figs. 2-8 exhibit the variation of the previous physical quantities with the distance zin the presence and absence of rotation (i.e., Ω=0.91S^-1,Ω=0) for both the dual-phase-lag model and Lord-Shulman theory. Fig. 2 illustrates the distribution of horizontal displacement u₁ versus a distance z. It obvious that the two curves based on the dual-phase-lag model start at the same point then decreasing up to disappear at z≥1.4, moreover, there are on other two curves based on Lord-Shulman theory initiate from the same point which is different from the previous point and also decreasing up to disappear at z≥1.4 Figs. 3 and 6 clarify the values of the vertical displacement and the force stress component always initiate from positive values and decrease in the range 0≤,z≤ 1.4, after that go to zero in the range z≥1.4 It is also clear that the values of u₃ and σ_xx based on the dual-phase-lag model are large compared with the values of u₃ and σ_xx based on the Lord-Shulman theory. Figs. 4 and 5 depict the change in temperature T and the micro-elongational scalar ϕ with distance z. In the presence of rotation (i.e., Ω=0.91S^-1,Ω=0) the values of T and ϕ always start with increasing to a maximum value, then decrease and finally converge to zero. It is also seen that the values of Tdependent on the dual-phase-lag model are small compared with the values of Tdependent on the L-S theory in the case of the presence of rotation in fig. 4, while occurs the opposite in fig. 5 on the same previous case. Figs. 7 and 8 exhibit the value of the forces stresses components σ_zz and σ_xz always initiate from negative values and increase in the range 0≤,z≤ 1.4 after that go to zero in the range z≥1.4 It is evident that the values of σ_zz and zdependent on the dual-phase-lag model initiate from the same point which is the same point in Figs. 7 and 8. Figs. 9-15 show the variation of the previous physical quantities with the distance z in the presence of rotation (i.e., Ω=0.91S^-1) for dual-phase-lag model at 5910,sτ_q=9×10^-5 S, τ_θ=10^-5, 2×10^-5, 3×10^-5 S^–1. Figs. 9, 10 and 13 depict the variation of the horizontal displacement u₁ and the vertical displacement u₃ and the force stress component σ_xx with distance z. The values of the previous physical decrease up to vanished at distance z≥ 1.8. It is also obvious that the influence of T is an increase in Fig. 9 whereas in figs. 10 and 13 is a decrease. Fig. 11 describes the change in the temperature T against distance z. It is clear that the values of Tincrease when the effect of the phase-lag of temperature gradient τ_θ increases. Fig. 12 presents the change in the micro-elongational scalar ϕ with distance z. The three values of ϕ initiate from zero then increasing to a maximum after that decrease even vanish. It is also evident that the values of ϕ decrease when the influence of the phase-lag of temperature gradient τ_θ increases. Figs. 14 and 15 exhibit the distribution of the forces stresses components σ_zz and σ_xz versus distance z. All curves begin from the negative values then increase even go to zero. In Fig. 14 the values of σ_zz decrease when the phase-lag of temperature gradient τ_θ decreasing, but in Fig. 15, the values of σ_xz increase when the phase-lag of temperature gradient τ_θ decreases. e phase-lag of heat flux τ_q is constant on solution in just dual-phase-lag. This paper presents the numerical evaluation results in the form of charts. The results are depicted in Figs. 2-15 for the magnitude of mechanical force 1N10.P=N. Figs. 2-8 exhibit the variation of the previous physical quantities with the distance zin the presence and absence of rotation (i.e., Ω=0.91S^-1,Ω=0) for both the dual-phase-lag model and Lord-Shulman theory. Fig. 2 illustrates the distribution of horizontal displacement 1u versus a distance z. It obvious that the two curves based on the dual-phase-lag model start at the same point then decreasing up to disappear at z≥1.4 moreover, there are on other two curves based on Lord-Shulman theory initiate from the same point which is different from the previous point and also decreasing up to disappear at z≥1.4 Figs. 3 and 6 clarify the values of the vertical displacement and the force stress component always initiate from positive values and decrease in the range 0≤Z≤1.4 after that go to zero in the range z≥1.4 It is also clear that the values of u₃ and σ_xx based on the dual-phase-lag model are large compared with the values of u₃ and σ_xx based on the Lord-Shulman theory. Figs. 4 and 5 depict the change in temperature T and the micro-elongational scalar ϕ with distance In the presence of rotation (i.e., Ω=0.91S^-1) the values of T and ϕ always start with increasing to a maximum value, then decrease and finally converge to zero. It is also seen that the values of Tdependent on the dual-phase-lag model are small compared with the values of Tdependent on the L-S theory in the case of the presence of rotation in fig. 4, while occurs the opposite in fig. 5 on the same previous case. Figs. 7 and 8 exhibit the value of the forces stresses components σ_zz and xzσalways initiate from negative values and increase in the range 0≤Z≤1.4 after that go to zero in the range z≥1.4 It is evident that the values of σ_zz and σ_xz dependent on the dual-phase-lag model initiate from the same point which is the same point in Figs. 7 and 8. Figs. 9-15 show the variation of the previous physical quantities with the distance z in the presence of rotation (i.e., Ω=0.91S^-1) for dual-phase-lag model at τ_q=9×10^-5 S,τ_θ=10^-5,2x10^-5, 10^- Figs. 9, 10 and 13 depict the variation of the horizontal displacement u₁ and the vertical displacement u₃ and the force stress component σ_xx with distance z. The values of the previous physical decrease up to vanished at distance z≥1.8. It is also obvious that the influence of τ_θ is an increase in Fig. 9 whereas in figs. 10 and 13 is a decrease. Fig. 11 describes the change in the temperature T against distance z. It is clear that the values of T increase when the effect of the phase-lag of temperature gradient τ_θ increases. Fig. 12 presents the change in the micro-elongational scalar ϕ with distance .z The three values of ϕ initiate from zero then increasing to a maximum after that decrease even vanish. It is also evident that the values of ϕ decrease when the influence of the phase-lag of temperature gradient τ_θ increases. Figs. 14 and 15 exhibit the distribution of the forces stresses components σ_zz and σ_xz versus distance z. All curves begin from the negative values then increase even go to zero. In Fig. 14 the values of σ_zz decrease when the phase-lag of temperature gradient τ_θ decreasing, but in Fig. 15, the values of σ_xz increase when the phase-lag of temperature gradient τ_θ decreases.

The method presented in this research applies to a wide range of thermodynamic problems. The current theoretical results may be of interest to experimental scientists' /researchers/ seismologists working in this subject. When the figures obtained under the two theories are compared, the following significant phenomena are observed:

1. The boundary conditions are satisfied by all physical quantities.
2. The physical quantities differ significantly between the DPL model and the L-S theory
3. The influence of rotation plays a big role in all physical quantities.
4. Recent interest in micro-elongated materials arises from their potential applications in smarter engineering structures.
5. Micro-elongated is presently experiencing a surge in basic research as well as technical applications.
6. The study of thermodynamic systems of bodies in equilibrium whose interactions with their surroundings are restricted to mechanical work, heat exchange, and external work is the focus of thermoelasticity.