Rate-Dependent Plastic Deformation Model of Euler-Bernoulli Beams

This work develops a thermodynamically consistent, geometrically exact, rate-dependent plasticity model for Euler–Bernoulli beams under finite deformation. Beginning with a rigorous formulation of continuum kinematics, the study systematically constructs the mechanical framework by employing the deformation gradient tensor and associated measures of strain, decomposed multiplicatively to separate elastic and plastic contributions. The mechanical balance laws are derived in their weak forms using the principle of virtual power and energy balance, followed by the derivation of the entropy inequality from the second law of thermodynamics, guaranteeing consistency with the Clausius–Duhem inequality. The constitutive framework is constructed via the Helmholtz free energy approach, incorporating both elastic and plastic energy storage, as well as temperature-dependent contributions. A Perzyna-type overstress flow rule is introduced to model the rate-dependence of plastic flow, complemented by a smooth yield function and isotropic hardening evolution. Dimensional reduction techniques specific to Euler–Bernoulli beam kinematics are applied, leading to beam-specific stress resultants and thermo-mechanical coupling relations. The strong and weak forms of the resulting governing equations are provided, and a locking-free mixed finite element discretization strategy is proposed. The model’s time integration is handled via an exponential map-based update for the plastic deformation gradient, ensuring preservation of plastic incompressibility. Further, the well-posedness of the problem is addressed through functional analytic formulation using Sobolev spaces, and existence, and uniqueness of solutions are discussed. In summary, this work provides a foundational, rate-dependent, large-deformation plasticity model for thermomechanically coupled Euler–Bernoulli beams, suitable for applications in computational mechanics involving thin structures undergoing large, irreversible deformations at finite rates.