Transient Wave Propagation in Double–Porosity Spherical Cavities: A Spatiotemporal Nonlocal Elasticity Framework with Klein–Gordon Operator

This study introduces a novel Klein–Gordon-type spatiotemporal nonlocal elasticity framework to investigate transient wave propagation in spherical cavities characterized by double porosity. Unlike classical nonlocal models that rely solely on spatial parameters, this work innovatively integrates internal length- and characteristic time–scale parameters, simultaneously capturing size–dependent behaviors and temporal memory effects in porous microstructures. For the first time, this spatiotemporal approach is applied to the complex interaction between macro- and micro–voids in a spherical geometry. The governing equations for displacement, stresses, and coupled void volume fractions are derived, non–dimensionalized, and solved analytically using a time–harmonic approach with spherical Bessel functions. Numerical simulations demonstrate that increasing spatiotemporal nonlocality significantly smooths stress distributions and enhances damping near cavity surfaces, effectively mitigating stress concentrations that are prevalent in classical models. The analysis further uncovers distinctive wave modes arising from the interplay between dual porosity scales and applied radial stresses. As the first comprehensive treatment of spatiotemporal nonlocality in double–porosity spherical cavities, this work establishes a robust theoretical tool for predicting dynamic stability in advanced materials, with broad relevance for geomechanics, porous biomaterials, and subsurface engineering.