Longitudinal Vibration of Uncertain Beam Through the Second Strain Gradient Theory

This article explores the use of periodic structures to control the propagation of mechanical waves, a field of interest for critical applications that require high robustness in the face of uncertainties in numerical models. This field is crucial for applications requiring high robustness against numerical model uncertainties. Using a one-dimensional spectral formulation of the wave finite element method (WFEM) based on the non-classical Second Strain Gradient elasticity theory (SSG), we present a probabilistic approach applied to Bloch analysis for these structures. The Wave Finite Element (WFE) approach, which typically uses deterministic inputs, may have drawbacks when used on structures that exhibit variability because of the substantial influence of uncertainties at high frequencies. These uncertainties are influenced by geometric or mechanical properties, excitation forces, and condition variability. In order to tackle this complexity, probabilistic approaches must be investigated. Uncertainties in material properties and geometric dimensions predominantly emerge during manufacturing and assembly processes. Incorporating uncertainties in frequency band structure analysis is essential for accurate and reliable results. Neglecting these uncertainties can lead to inaccurate results and potentially unsafe conclusions. This stochastic SWFE approach, founded on a probabilistic parametric technique, incorporates inherent uncertainties in waveguides.

This study focuses in particular on beam structures, which are analyzed using the Euler-Bernoulli model considering loading modes such as longitudinal modes. Changes are made to the density and Young’s modulus, which impact the stiffness and mass matrices, in order to simulate uncertainties in the structure’s mechanical characteristics. Numerical tests highlight the effectiveness of this stochastic WFEM in estimating the standard deviations of directional wave propagation. Comparative analysis with theoretical and classical CT solutions confirms the robustness and stability of the proposed approach.