Fractional Order Thermo Piezoelectric Modelling of qP Wave Interaction and Energy Partition at Welded Interface

TL;DR

This work analyzes the reflection and transmission of quasi-longitudinal qP waves at a welded interface between a functionally graded piezoelectric material (FGPM) and a thermo-piezoelectric half-space under fractional-order Lord–Shulman thermoelasticity with rotation and initial stress. An analytical solution is developed, yielding closed-form reflection/transmission coefficients and energy-partition factors for two boundary-condition cases, while incorporating material gradation, memory effects via a fractional order $\\gamma$, and thermal relaxation time $\\tau_0$. Key findings show that initial stress, fractional-order memory, and relaxation time significantly influence amplitude and energy distribution, whereas rotation and Gradation (in the studied ranges) have minimal impact; energy conservation is verified across incidence angles. The results provide guidance for smart sensors, vibration control, aerospace structures, and energy harvesting in FGPM-based systems operating under coupled thermo-electro-mechanical effects with fractional-order dynamics.

Abstract

An analytical model is developed to investigate the interaction of quasi longitudinal (qP) waves with a perfectly bonded interface between a thermo piezoelectric half space and a functionally graded piezoelectric half space. The formulation is based on the fractional order Lord Shulman generalized thermoelasticity theory, which provides an enhanced description of coupled thermo electro mechanical wave behaviour. Rotational effects are incorporated into the constitutive relations and equations of motion for both media, while the lower half space is assumed to be subjected to initial stress. Closed form solutions for reflection and transmission coefficients are obtained, together with associated energy partition factors, allowing a comprehensive assessment of interface wave characteristics. Numerical simulations carried out using MATLAB demonstrate that the reflection and transmission responses are strongly influenced by initial stress, fractional order parameter, and thermal relaxation time. The calculated energy ratios of scattered waves satisfy the energy conservation condition, confirming the mathematical consistency of the formulation. The findings of this study are relevant to the design and analysis of smart sensors, rotating and aerospace structures, vibration control systems, and energy harvesting devices employing functionally graded thermo piezoelectric materials under fractional order effects.

Figures (12)

• Figure 1: Geometry of the Problem
• Figure 2: Amplitude ratio versus incidence angle $\theta_0$ for varying material gradient parameters in FGPM media
• Figure 3: Amplitude ratio versus incidence angle $\theta_0$ for varying initial stress in the thermo-piezoelectric media $(M_1)$.
• Figure 4: Amplitude ratio versus incidence angle $\theta_0$ for varying rotation parameters in the thermo-piezoelectric media
• Figure 5: Amplitude ratio versus incidence angle $\theta_0$ for varying order of fractional derivative in the thermo- piezoelectric media