Coupled Axisymmetric Thermoelectroelasticity Problem for a Round Rigidly Fixed Plate

Introduction. Various mathematical models are used to improve the functionality of piezoceramic sensors [1–3] based on the interdependence of thermoelectroelastic fields. To more accurately account for the effect of coupling of these fields, it is needed to construct closed solutions. Some simplifications are used to solve systems of initial nonself-adjoint differential equations. Thus, the problems can be considered in an uncoupled formulation, or the problems consider and analyze elements that have a degenerate geometry. An uncoupled stationary problem for a long electroelastic cylinder is considered in [4][5], and article [6] is devoted to the analysis of thermal stresses in a hollow sphere. Papers [7][8] are related to the determination of the temperature field in a piezoceramic shell and a round plate in solving uncoupled problems. Coupled dynamic problems for a homogeneous piezoceramic layer, as well as dynamic problems in a coupled formulation for a gradient–inhomogeneous piezoceramic layer, were considered in [9][10]. In [11][12], fields in an unbounded medium were analyzed. In [13][14], a long hollow cylinder was considered, and thermoelectroelastic fields were analyzed.

Currently, the literature does not describe the results of constructing closed solutions to the mentioned non-stationary problems in a three-dimensional coupled formulation. Therefore, in this paper, we consider a round plate made of piezoceramic composition and having a rigid fixation, for which a new closed solution to the problem of thermoelectroelasticity is obtained. The use of a limit on the rate of temperature change on its front surface [10] makes it possible not to include the inertial characteristics of the system under study and apply the equilibrium equations in the calculated ratios.

Materials and Methods. In the process of solving, a generalized finite biorthogonal transformation was used, which provided the reduction of the dimension of a nonself-adjoint system of equations and the construction of a closed solution through significant simplifying research in the image space.

Mathematical model. Consider certain area  which is occupied by a piezoceramic solid circular plate in the cylindrical coordinate system  Arbitrary temperature boundary conditions can be used for the problem under study. However, for the certainty of the solution, on the upper  front surface, the temperature change  at a given ambient temperature  on the lower plane ( — time) is considered. The cylindrical thermally insulated surface is rigidly fixed: there is no radial component of the displacement vector and the angle of rotation, and its lower part is fixed in the vertical plane. The lower plane of the round plate in question is grounded. The front electrodated planes of the plate are connected to the measuring device. The design scheme of the plate is shown in Figure 1.

The mathematical formulation of the problem under consideration in a dimensionless form for an axially polarized piezoceramic material with a hexagonal crystal lattice of 6 mm composition has the form:

where

 — temperature increment in dimensional form;  — components of the displacement vector, electric field potential; σ_zz (r, z, t), σ_rz (r, z, t)  — components of the mechanical stress tensor; D_r (r, z, t) — radial component of the electric field induction vector; Λ, k, α_t — coefficients of thermal conductivity, volumetric heat capacity, and linear thermal expansion;  — electric potential induced on the upper front surface; γ_ii, g_ii — components of the tensor of temperature stresses and pyroelectric coefficients (i = 1,3, γ_ii = С_iiα_t); e₁₅, e₃₁, e₃₃, ε₁₁, ε₃₃ — piezoelectric modules and permittivity coefficients; Θ^* = T – T; T, T₀  — current temperature and temperature of the original state of the body; β_rel  — relaxation time; α  —  heat transfer coefficient,  — known rate of temperature change ;

To determine the potential of the electric field induced under deformation on the upper front surface   in the case of connecting electrodes to a measuring device with a large input resistance, an additional boundary condition is used:

 (5)

where D_z (r, z, t) — axial component of the induction vector; S — surface area.

Construction of a general solution. To fulfill the condition of fixing the cylindrical surface of the plate in the vertical plane, new functions w (r, z, t), W₁(t) are introduced:

 (6)

this makes it possible to form a boundary value problem with respect to functions U, w, ϕ, Θ, which is investigated by the method of finite Fourier-Bessel transformations:

where j_n — positive zeros of the function  — Bessel functions.

It should be noted here that to satisfy the last boundary condition (2), it is necessary to assume that the original temperature of plate T₀ is equal to the ambient temperature , and the temperature increment function on the upper front surface ω₁(1, t) = 0. These assumptions, without much error, allow assuming that on the cylindrical surface of the plate, Θ(1, z, t) = 0.

As a result of using the transformation algorithm in the image area, the following initial boundary value problem is obtained:

where 

At the next stage of the solution, the introduction of functions U_H(n, z, t), W_H(n, z, t), φ_H(n, z, t), Q_H(n, z, t) using the following relations:

 (12)

allows the reduction of conditions (9) to homogeneous.

Here,  — twice differentiable functions.

Substitution (12) in (9) – (11) when the conditions are satisfied:

 (13)

provides the formulation of the following task:

(14)

where

Using the biorthogonal finite transformation (CMD) [15], we obtain a solution to problem (14) – (16). CMD with unknown components of vector functions of transformations is introduced on the segment [0, h] K₁(λ_in, z)…K₄(λ_in, z), N₁(μ_in, z)…N₁(μ_in, z):

 (17)

where λ_in, μ_in — the eigenvalues of the corresponding problems with respect to the components of the vector functions of the CMD (k = 1…4).

In the course of transformations, we obtain a task for determining transformants G(n, λ_in, t):

whose solution has the following form:

(20)

in addition, two homogeneous problems with respect to components K₁(λ_in, z)…K₄(λ_in, z),

(21)

(22)

and N₁(μ_in, z)…N₄(μ_in, z):

where

m_1in, m_2in — roots of the characteristic equation: 

Constructed homogeneous problem (23), (24) with respect to functions N₁(μ_in, z)…N₄(μ_in, z) is invariant to the initial calculated relations (14), (15).

Systems (21), (23) are reduced to the following equations with respect to K₂(λ_in, z), N₂(μ_in, z):

(25)

In the paper, coefficients  are not given due to the limitation of its volume.

In equation (25), the left part is decomposed into commutative factors, presented below:

(26)

where  — real positive roots of the following characteristic equations:

When examining a round rigidly fixed piezoceramic plate, the general integral of equations (26) has the following form:

(27)

where 

It should be noted here that the condition of the actual positive values of coefficients B_1in, S_1in, A_1in…A_4in is fulfilled for most structures made of piezoceramic material. Otherwise, the formula structure (26), (27) simply changes.

Considering that the connections were previously obtained as a result of reducing (21), (23) to (25), we get expressions for functions K₁(λ_in, z), K₃(λ_in, z), K₄(λ_in, z), N₁(λ_in, z), N₃(λ_in, z), N₄(λ_in, z).

Substituting K₁(λ_in, z)…K₄(λ_in, z), N₁(μ_in, z)…N₁(μ_in, z) in conditions (22), (24) provides determining constants D_1in…D_8in, E_1in …E_8in and eigenvalues λ_in, μ_in.

The final expressions of functions U (n, z, t), W (n, z, t), ϕ (n, z, t), Θ (n, z, t) are obtained by applying the inversion formulas (17), (8). Then, taking into account (6), (12), we have:

 (28)

Functions f₁(z)… f₁₂(z) are calculated from the simplification condition F₁… F₄ when conditions (13) are satisfied:

Function W₁(t) is determined from condition W(1, h, t) = 0:

For a qualitative assessment of the induced electric pulse on its upper front surface, it is required to form two electrodes with a radius of separation R and connect them to a measuring device. In this case, potential ϕ₀(r, t), induced on two equipotential surfaces is represented as:

(29)

where H(…) — the Heaviside step function.

Substituting (29) into (5) makes it possible to define expressions for determining potentials ϕ₀(t), ϕ₀₂(t):

(30)

As a result of solution (30), functions ϕ₀(t), ϕ₀₂(t) are defined as follows:

where

In this case, the potential difference V(t) is determined by the equality:

(31)

Research Results. Numerical results are presented for a plate made of piezoceramics of the composition
PZT–4 [4][11][16]:

{C₁₁, C₁₂, C₁₃, C₃₃, C₅₅} = {13.9, 7.78, 7.3, 11.5, 2.26}×10¹⁰ Pa, {ε₁₁, ε₃₃} = {6.46, 5.62} × 10^–9 F/m,

{e₁₅, e₃₁, e₃₃} = {12.7, –5.2, 15.1} C/m², Λ = 1,6 W/(m∙K), α_t = 0.4×10^–5 K^–1,

k = 3×10⁶ J/ (m³∙K), g₁₁ = g₃₃ = –0.6×10^–4 C/(m²∙K), β_rel = 10^–4 s, α = 5.6 W/(m²∙K).

The following case of temperature change   is investigated:

where  — maximum temperature value and the corresponding time 

Figure 2 shows graphs reflecting at various points in time  the change in temperature Θ^*(0, z, t) in the thickness of the plate (b = 14×10^–3 m, h^* = 1×10^–3 m).

According to the calculation result, it is observed that due to the high coefficient of thermal conductivity and the small thickness of the piezoceramic plate, the steady-state temperature regime is formed quite quickly  when it reaches Θ^*(0, z, t) on the lower front surface (z = h) 78°C (Fig. 2).

Figure 3 shows the change  in time  taking into account (represented by a solid line) and without account for (represented by dotted line, β = 0) the relaxation of the heat flux (b = 14×10^–5 m, h^* = 1×10^–5 m). It should be emphasized that the application of the hyperbolic Lord-Shulman heat conduction equation is needed only in the study of a piezoceramic micro-dimensional structure with a very rapid change 

The numerical results of determining function Θ^*(r, z, t) show that when conducting a study of a structure made of piezoceramic material, it is possible to neglect the impact of the rate of change of body volume and tension on the temperature field, i.e., to use only the equation of thermal conductivity in calculations.

Figure 4 shows a diagram of movements W^*(0, z, t) over time t, and Figure 5 shows the dependence of the change in the radial component of normal stresses σ_rr(r, z, t) along coordinate  at different points in time:  solid line — z = 0, dotted line — z = h.

It should be noted that under heating, the plate bends with increasing thickness; due to fixation, compressive normal stresses σ_rr(r, z, t) are formed at all points. In the case of complete heating of the structure (t = 10t_max), the value of normal stresses σ_rr(r, z, 10t_max) in the height of the section practically coincide (Fig. 5, Diagram 2, solid and dotted lines). At this, σ_rr(r, 0, t) remains constant over the entire time interval t ≥ t_max (Fig. 5, solid line), and on the lower plane at the initial moment of time σ_rr(r, h, t), it is significantly less (Fig. 5, Diagram 1, dotted line).

For a qualitative assessment of the induced electric pulse in the form of a potential difference V(t) (31), two electrodes with a radius of separation R = 0.7 and connected to a measuring device (Fig. 6, solid line) must be formed on the upper front surface of the element in question. At this, determination of V(t) by connecting the upper and lower (grounded) solid electrodated surfaces of the plate to the voltmeter (Fig. 6, dotted line) is ineffective.

Discussion and Conclusion. The developed closed solution of the coupled axisymmetric thermoelectroelasticity problem for a round plate made of piezoceramic material is more accurate than the solution that was developed when solving problems in an uncoupled formulation. This is due to the fact that the calculated ratios obtained make it possible to determine how the non-stationary temperature field affects the stress-strain state and the electric field of the element in question, which makes it possible to describe the behavior of a round piezoceramic plate under the influence of thermal and electrical loads with greater accuracy. In addition, it becomes possible to scientifically establish the dimensions of two uncoupled electrodes, which provides measuring the induced electric pulse most effectively.