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

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.