Rate-Independent Gradient-Enhanced Plastic Deformation Model of Euler-Bernoulli Beams

This work presents a rigorously formulated rate-independent, gradient-enhanced continuum plasticity framework tailored for Euler-Bernoulli beams, accounting for both geometric and material nonlinearities. The model is grounded in thermodynamic consistency, exploiting the principles of maximum dissipation and the Clausius-Duhem inequality to derive constitutive relations via an additive decomposition of strain, a convex yield surface, and an associative flow rule. Gradient enhancements are incorporated through nonlocal terms in the free energy functional, yielding regularization effects that circumvent strain localization and ensure mathematical well-posedness. The governing equations are derived from variational principles, specifically the principle of virtual work and incremental energy minimization, resulting in a unified framework valid for both quasi-static and dynamic loading conditions. A novel contribution includes the derivation of a generalized consistent tangent operator that embeds inertial effects, critical for stability and convergence in implicit time integration schemes. Theoretical guarantees of existence and uniqueness are established via convexity and variational inequality formulations. The model naturally integrates shear effects and finite rotations through higher-order kinematic assumptions, and it rigorously addresses open problems in beam plasticity, including dynamic plastic collapse, interaction between bending and shear yield mechanisms, and plastic hinge migration under large displacements. The framework is extensible to layered composites, functionally graded materials, and beams with stochastic or temperature-dependent yield behavior, positioning it as a foundational advancement in gradient plasticity and computational structural mechanics.