Universal exact solutions for multiphysical inhomogeneities and inclusions in Fourier space.

Physical inhomogeneities-spanning atomic-scale crystal defects and quantum dots to macroscopic features like bone cavities and geological faults-are fundamental to both engineered and natural systems. The study of how these inclusions influence material fields has driven critical advances across multiple disciplines, from composite materials and solid-state physics to biomechanics and geophysics. Eshelby revolutionized this field with the equivalent inclusion method (EIM), transforming inhomogeneous problems into homogeneous ones by introducing an equivalent eigen-strain. However, determining Eshelby's equivalent eigen-strain for arbitrary inclusions has remained a long-standing challenge. Here, we resolve this challenge and applied the EIM to multiphysics problems, deriving three-dimensional universal exact solutions in Fourier space for the generalized equivalent eigenfield, encompassing eigen-strain, eigen-electric, and eigen-magnetic fields. These solutions apply to both single and multiple inclusions of arbitrary shape, inhomogeneity, and anisotropy. Our results yield exact expressions for the effective properties of magnetoelectroelastic composites. We demonstrate the framework's versatility through complex multiphase applications, including a general expression for stress intensity factors of arbitrarily shaped cracks, a criterion for designing auxetic materials via engineered elastic inhomogeneities and an optimized pathway to enhance magnetoelectric coupling. This work establishes a universal approach to multiphysics inhomogeneous inclusion problems, rooted in multiple disciplines and spanning multiple length scales. It offers both profound fundamental insights and practical utility, paving the way for advancements in diverse fields.