Fractional Modeling of Thermoelastic Fracture Behavior in a Cracked PZT-4 Strip under Transient Thermal Loading

Abstract

This paper investigates the thermoelastic fracture response of a transversely isotropic piezoelectric strip containing a vertical insulated crack under transient thermal shock loading and pre-existing stress fields. The analysis is conducted within the framework of generalized fractional heat conduction using the Ezzat model, which incorporates thermal relaxation and memory-dependent effects. The problem is formulated as a mixed boundary value problem governed by fractional thermoelastic equations. The Laplace transform technique is employed to obtain temperature and coupled fields in the transform domain. The resulting system of singular integral equations is solved using the Lobatto-Chebyshev collocation method to determine the displacement discontinuity and the associated thermal stress intensity factors at the crack tips. The transient response in the time domain is recovered through numerical inversion of the Laplace transform using the Stehfest algorithm. Numerical results for PZT-4 are presented to examine the influence of fractional order, thermal relaxation time, pre-existing stresses, and geometric parameters on temperature distribution, thermoelastic stress fields, and stress intensity factors. The results demonstrate significant deviations from classical Fourier predictions, revealing wave-like thermal behavior and inherent memory effects associated with fractional heat conduction. The present formulation establishes a unified framework for the analysis of thermoelastic fracture in piezoelectric ceramics and provides insights into the design and reliability of smart structures operating under severe thermal conditions.

Figures (9)

• Figure 1: Schematic diagram of a PZT-4 piezoelectric strip containing an internal vertical crack of length $2c$
• Figure 2: Variation of nondimensional temperature with Fourier number for different values of fractional order $\gamma$, thermal relaxation time $\tau_q$, strip thickness $H$, and normalized depth $x_3/H$.
• Figure 3: Variation of nondimensional temperature with normalized depth $x_3/H$ for different values of fractional order $\gamma$, thermal relaxation time $\tau_q$, Fourier number $F$, and strip thickness $H$.
• Figure 4: Variation of nondimensional thermal stress $\tau_0^T/(k_{11}T_0)$ with normalized depth $x_3/H$ for different values of fractional order $\gamma$, thermal relaxation time $\tau_q$, thermal moduli ratio $k_{33}/k_{11}$, and Fourier number $F$.
• Figure 5: Contour plot of the normalized temperature distribution $(T - T_I)/T_0$ as a function of the Fourier number $F$ and normalized thickness $x_3/H$, showing transient heat propagation and gradual transition to steady state.