Statics of nonlocal stress driven graded material beam - cauchy distribution approach

Nonlocal elasticity theories are essential for modeling size-dependent mechanical behavior in micro- and nano-structured components, yet existing stress-driven formulations rely almost exclusively on rapidly decaying kernels such as the Helmholtz or Gaussian forms. These kernels cannot represent algebraically decaying, scale-free interactions known to arise in materials with long-range microstructural coupling. This work addresses this limitation by developing the first stress-driven nonlocal Euler–Bernoulli beam model employing the Cauchy kernel. The governing equations are derived by coupling classical beam kinematics with a stress-driven integral constitutive law based on a normalized Cauchy attenuation kernel. Closed-form analytical solutions are obtained for simply supported, clamped, and cantilever beams where material gradations follow a power-law variation along the beam thickness. Parametric studies reveal that the Cauchy kernel induces stronger long-range nonlocal stiffening than exponential kernels, reduces maximum deflection more markedly, and maintains smooth convergence to the classical local model as the nonlocal parameter vanishes. These findings demonstrate that algebraic kernels fundamentally alter static response predictions and provide a physically motivated alternative for modeling nanoscale beams. The proposed formulation establishes a foundation for extending power-law-kernel SDM models to dynamics, instability, thermal fields, and experimental calibration.