Exact Dispersion and PDE Mapping in Fractional Generalizations of Eringen’s Nonlocal Elasticity

Abstract Classical formulations of Eringen’s integral nonlocal elasticity, typically based on exponential or Bessel-type kernels, offer limited flexibility in modeling long-range interactions and do not adequately capture memory effects observed in microstructured and scale-dependent materials. Moreover, their differential reformulations are often associated with boundary-condition paradoxes and inconsistencies with the original integral theory. To overcome these limitations, this study introduces a fractional-kernel generalization of Eringen’s integral framework, in which conventional kernels are replaced by fractional-order power-law kernels governed by a tunable index α ∈ (0, 2). This formulation naturally captures long-range spatial interactions and intrinsic memory effects within a unified and physically interpretable framework. Exact analytical solutions for time-harmonic antiplane shear waves are derived using exponential trial functions and Laplace transform techniques, leading to generalized dispersion relations that recover local elasticity as a limiting case and extend classical nonlocal models. An equivalent fractional partial differential equation representation is established through spectrally matched pseudo-differential operators, ensuring consistency on unbounded or periodic domains. The fractional order α emerges as a continuous control parameter governing nonlocality strength, dispersion intensity, and wave speed. Application to functionally graded nanoplates demonstrates the framework’s effectiveness in modeling scale-dependent wave dynamics relevant to MEMS/NEMS. The proposed approach provides a robust foundation for advanced nonlocal modeling and numerical implementation in complex materials.