A General Tensorial Formulation of Acoustoelasticity and Its Representation in Cylindrical Coordinates

Acoustoelasticity provides the physical sensing principle for ultrasonic stress measurement. However, most existing formulations are restricted to isotropic media, simple stress conditions, and Cartesian coordinate systems, which limits their applicability in practical sensing scenarios involving curved and anisotropic structures. In this work, a general tensorial formulation of acoustoelasticity is developed based on the theory of incremental deformation. The proposed governing equations describe the motion of incremental displacement with explicit dependence on initial stress or strain, and are applicable to materials with arbitrary symmetry and general initial stress states. Owing to its coordinate-independent tensorial nature, the formulation can be expressed in any curvilinear coordinate system. To facilitate practical ultrasonic sensing applications, the general equations are further expanded in a cylindrical coordinate system for orthotropic materials. This enables the analysis of elastic wave propagation in curved structures such as pipelines, pressure vessels, and boreholes. The formulation establishes a direct relationship between initial stress and effective elastic properties, which determine wave velocities measurable by ultrasonic sensors, such as time-of-flight and phase velocity. The proposed approach provides a rigorous theoretical foundation for ultrasonic stress sensing and nondestructive testing, particularly for curved and anisotropic structures, and supports improved accuracy in sensor-based stress evaluation.