Transient dynamics of nonlocal beams with scale-dependent phase fractions

The transient bending response of Bernoulli–Euler nanobeams is investigated within the framework of two-phase strain-driven nonlocal integral elasticity. A central novelty of this study is the introduction of a physically motivated functional relationship between the nonlocal phase fraction and the nonlocal length-scale parameter, eliminating the conventional assumption that these quantities are independent. Three functional forms are proposed for this dependence: linear, exponential saturation, and power-law models, each satisfying the requirement that nonlocal contributions vanish in the local limit and grow monotonically with increasing internal length scale. The governing integro-differential equation of transverse motion is derived via Hamilton's principle and converted into an equivalent sixth-order partial differential equation supplemented by constitutive boundary conditions. The resulting initial-boundary value problem is solved analytically using the Laplace transform technique in the temporal domain. For the simply supported beam conFig.uration, closed-form time-domain solutions are obtained for impulsive, step, and harmonic loading cases through exact inverse transformation; for other boundary conditions, the full sixth-order system must be solved numerically in the Laplace domain. A systematic parametric study reveals that the scale-dependent phase fraction coupling amplifies the softening character of nonlocality relative to constant-fraction models, with significant consequences for natural frequencies, peak transient displacements, and resonance characteristics. The present analytical solutions serve as benchmarks for numerical implementations of two-phase nonlocal beam formulations.