On the stochastic dynamics of nonlocal viscoelastic plates

The bending vibrations of nonlocal viscoelastic plates subjected to stochastic excitations are investigated within the framework of the axisymmetric Kirchhoff model. This study is particularly relevant to the design of mesoscale heterogeneous structures, biotissues, miniaturized two-dimensional structures and metamaterials, such as those employed in energy harvesters, sensors, actuators, wave energy converters, transistors, bioinspired devices, and microrobots, often fabricated from unconventional materials. For such systems, classical local continuum theories fail to accurately capture the underlying mechanics. To address this, the mechanical response is analyzed by accounting for two key features: viscoelasticity and nonlocality. The constitutive behavior is described through a stress-driven integral nonlocal model coupled with fractional-order viscoelastic stress–strain relation, allowing the formulation to incorporate both size-dependent and hereditary effects. Random excitation is introduced to account for the inherent variability of external dynamic environments, leading to a stochastic partial differential equation featuring fractional operators. Owing to the complexity of this equation, a semi-analytical solution procedure based on modal decomposition is developed in order to compute the time-dependent response and evaluate the power spectral densities. The results highlight the influence of nonlocal interactions and viscoelastic parameters on the dynamic response and natural frequencies of the system. These findings offer valuable insights for the design and optimization of advanced two-dimensional nano- and micro-scale devices and other devices where long-range interactions occur.