Dimension reduction for gradient damage models in slender rods
This paper presents a method for reducing a three-dimensional gradient damage model to a one-dimensional model for slender rods (with a small radius-to-length ratio, $\delta = R/L \to 0$). The 3D model minimizes an energy functional that includes elastic strain energy, a damage-dependent degradation function $a_\eta(\alpha)$, a damage energy term $w(\alpha)$, and a gradient term penalizing abrupt damage variations. After non-dimensionalizing and rescaling, the problem is reformulated on a unit cylinder, and the behaviour of the energy functional is analyzed as $\delta$ approaches zero. Using $\Gamma$-convergence, we show that the sequence of 3D energy functionals converges to a 1D functional, defined over displacement and damage fields that are independent of transverse coordinates. Compactness results guarantee the weak convergence of strains and damage gradients, while lower and upper bound inequalities confirm the energy limit. Minimizers of the 3D energy are proven to converge to the minimizers of the 1D energy, with strains approaching a diagonal form indicative of uniaxial deformation.