A Probabilistic Framework for Finite Strain Damage Response of Thick Curved Beams Including the Shear Effect.

Curved beams are widely used in engineering applications due to their unique geometry and mechanical advantages. In this study, the mechanical response of hyperelastic curved beams under cyclic loading is investigated, considering the Mullins stress-softening effect and the probabilistic variability of material parameters. A Neo-Hookean hyperelastic model integrated with the Ogden-Roxburgh damage formulation is employed to capture the cyclic stress-softening behavior. The governing partial differential equations are derived in cylindrical coordinates under plane-stress conditions and solved numerically. Unlike classical pure bending assumptions, the present formulation captures the coupled radial, circumferential, and shear stress components arising from the finite thickness and curvature of the beam. The model is validated through comparison with two-dimensional finite element solutions. The influence of material uncertainty is further examined via a probabilistic parametric analysis, and statistical measures including mean, standard deviation, skewness, and kurtosis are evaluated. The results demonstrate that Mullins-induced stress softening is concentrated near the inner curvature and beam root, and that inherent material variability significantly affects the damage distribution. The proposed framework provides a fast and accurate method for predicting the mechanical behavior of hyperelastic curved beams under cyclic loading.