A damaged non-homogeneous Timoshenko beam model for a dam subjected to aging effects

A hemi-variational formulation for a damaged non-homogeneous Timoshenko beam is proposed here for the purpose of fast simulation of the properties of a dam. The dam is therefore modeled as a damaged non-homogeneous Timoshenko beam embedded in 2D space. The damage evolution and the mechanics of the beam are governed both by the hemivariational principle and by the assumption on the form of the deformation energy functional, that comprehends the dissipative part owing to damage-aging effects. In the formulation, the Karush–Kuhn–Tucker (KKT) condition is derived and the damage, which locally decreases the stiffness (i.e., axial, bending, and shear stiffness) of the beam, becomes relevant once a measure of the elastic energy, called the normalized undamaged elastic energy, reaches a certain threshold that is constitutively prescribed ab initio. The aging effect is assumed by reducing such a threshold in the neighborhood of the bottom of the beam, where the concentration of the chemical and physical attacks is higher. As a result, we show a method for the calculation of the mechanical failure condition. In addition, we observe that, even though this threshold is assumed to decrease in a large region far from the bottom of the dam, the damage is confined at the bottom of it. Thus, we have proved that the trapezoidal shape that is used in the dam design is useful not only to decrease the state of stress in the elastic regime, but also to confine the effects of damage evolution owing to the aging effects.