Séminaire de Cédric Pozzolini

Viscoelastic Beam contact problems : From Euler-Bernoulli to Enriched Timoshenko Formulations

This work addresses the modelling and numerical simulation of unilateral contact induced by dynamic loading in viscoelastic beam-like structures, with direct applications to the nuclear industry, including steam generator tubes, fuel assemblies and control rods. In such systems, vibrations under seismic or accidental conditions may lead to repeated impacts against rigid stops, generating highly non-smooth dynamical responses. Although industrial models often rely on simplified beam descriptions combined with penalty-based contact formulations, such approaches may suffer from limited accuracy and strong parameter sensitivity. The governing equations involve linear differential operators coupled with contact forces defined through Signorini complementarity conditions, leading to variational inequalities and differential inclusions in time. To overcome the limitations of penalty methods, this work investigates the application of Nitsche’s method to beam contact problems. While Nitsche’s formulation is well established in three-dimensional contact mechanics and offers both variational consistency and the absence of additional unknowns, its direct application to classical Euler–Bernoulli and Timoshenko beam models is shown to degenerate into a simple penalty formulation due to insufficient kinematic richness. To restore the theoretical advantages of Nitsche’s method, an enriched Timoshenko beam model incorporating through-the-depth pinching kinematics is introduced. This enhancement enables the definition of physically meaningful contact stresses consistent with the underlying three-dimensional framework.
From a mathematical viewpoint, the viscoelastic impact problem is analysed both in its continuous and semi-discrete finite element settings, leading to systems of differential inclusions. Existence, uniqueness and regularity results are obtained by extending singular mass techniques initially developed for purely elastic cases. Ongoing numerical investigations focus on the dynamical behaviour of the system through bifurcation diagrams, probabilistic characterisation of chaotic regimes and energy evolution, as well as on the sensitivity to initial conditions and parameters for the construction of efficient shooting and continuation algorithms. Comparisons with classical penalty-based Timoshenko beam models demonstrate the improved robustness and physical consistency of the proposed Nitsche-based enriched formulation.