Dynamics of dispersion of coaxial SH-waves in a pre-stressed piezo cylindrical column, with ideal and non-ideal coatings of epoxy graphite, zinc and isotropic layers

Technical Summary: Dynamics of Dispersion of Coaxial SH-Waves in a Pre-Stressed Piezo Cylindrical Column

Problem Statement
This study investigates the propagation dynamics of circumferential shear-horizontal (SH) waves within a composite cylindrical column. The structure consists of a pre-stressed piezoelectric core surrounded by a coating layer. The research addresses the wave dispersion characteristics under two distinct interface conditions: perfectly bonded and imperfectly bonded (non-ideal). To ensure practical applicability and material versatility, the coating layer is modeled using three different material classes: epoxy graphite (orthotropic), zinc (transversely isotropic), and isotropic materials. The model explicitly accounts for electromechanical coupling, material anisotropy, and the presence of initial stress within the piezoelectric medium.

Methodology
The mathematical formulation employs a cylindrical coordinate system $(r, \Theta, z)$ to describe the geometry. The displacement field is assumed to involve only axial shear deformation ($u_z$), with radial and circumferential displacements set to zero, consistent with SH-wave propagation.

1. Governing Equations: The equations of motion are derived based on Biot's theory for pre-stressed elastic solids and the constitutive relations for piezoelectric media. For the piezoelectric core, the electromechanical coupling is governed by Hooke's law extended with piezoelectric and dielectric constants. An auxiliary function is introduced to decouple the mechanical displacement and electric potential equations.
2. Analytical Solution: The method of separation of variables is applied to the governing differential equations. This yields solutions in terms of Bessel functions of the first and second kinds ($J_\nu$ and $Y_\nu$) for the mechanical displacement and Cauchy–Euler solutions for the electric potential auxiliary function.
3. Boundary and Interface Conditions:
   • Perfect Interface: Assumes continuity of displacement and stress across the interface.
   • Imperfect Interface: Modeled using a spring-like interface condition where the interfacial shear stress is proportional to the displacement discontinuity, governed by an interface stiffness factor ($\beta_{ij}$).
   • Free Surface: The outer surface of the coating is stress-free, and the inner core's electric potential is grounded.
4. Frequency Equation: By applying the boundary and interface conditions to the general solutions, a system of algebraic equations is formed. The dispersion equation is derived by setting the determinant of the coefficient matrix to zero, allowing for the calculation of phase velocities.

Key Contributions and Results
The study provides a numerical analysis of how various dimensionless parameters influence the phase velocity ($c/c_1$) of SH-waves. The key findings include:

• Dispersion Behavior: For all material models and interface conditions, the phase velocity decreases as the wave number increases, confirming the dispersive nature of SH-waves in this composite structure.
• Material Anisotropy: A comparative analysis reveals that the orthotropic epoxy-graphite coating yields the highest phase velocity, followed by the transversely isotropic zinc coating, with the isotropic coating exhibiting the lowest velocity. This highlights the significant role of material anisotropy in wave propagation speed.
• Interface Conditions: Perfectly bonded interfaces consistently result in higher phase velocities compared to imperfect interfaces. This is attributed to the better continuity of displacement and stress at the interface in the perfect bonding scenario.
• Parameter Sensitivity:
  • Piezoelectric Constant ($e_{15}^{II}$): An increase in the piezoelectric constant generally leads to an increase in phase velocity due to enhanced electromechanical coupling, which effectively increases the medium's stiffness.
  • Initial Stress ($P^{II}$): Higher initial stress increases the effective stiffness of the medium, resulting in increased phase velocity.
  • Dielectric Constant ($\varepsilon_{11}^{II}$): An increase in the dielectric constant tends to decrease the phase velocity. This is attributed to the increased energy storage capability of the electric field, which diminishes the strength of the electromechanical coupling.
  • Radius Ratio ($b/a$): An increase in the radius ratio (indicating a thicker coating layer relative to the core) leads to a decrease in phase velocity, likely due to reduced effective stiffness in the thicker coating.

Significance and Claims
The paper claims that the developed analytical model and the resulting dispersion equations offer a rigorous framework for understanding wave propagation in pre-stressed piezoelectric cylindrical structures with complex interface conditions. The study emphasizes that the inclusion of imperfect interfaces and initial stress provides a more realistic representation of physical systems compared to idealized models.

The authors state that these findings can serve as a useful guide for the design and optimization of various technologies, including piezoelectric waveguides, ultrasonic sensors and actuators, Surface Acoustic Wave (SAW) and Bulk Acoustic Wave (BAW) devices, Structural Health Monitoring (SHM) systems, Non-Destructive Evaluation (NDE) techniques, vibration control systems, and energy harvesting systems. Furthermore, the results are relevant for aerospace and automotive composites where accurate modeling of wave propagation is critical. The study validates its frequency equation by showing it matches previous work by Li and Wang [14] when initial stress is removed, ensuring the reliability of the derived mathematical model.