Dimension reduction for gradient damage models in slender rods

Abstract: This work addresses the reduction of damage in slender cylindrical structures, specifically the uniaxial deformation model of a cylindrical rod, through an energetic approximation. The research focuses on minimizing the damage energy in a three-dimensional environment, reducing the problem to one dimension (1D) through a homogenization process. A variational approach is proposed for a system of slender rods with specific boundary conditions, aiming to establish a minimization relationship between deformation and damage along the rod’s length. The damage energy is expressed in terms of deformation, material damage, and the energy associated with the spatial variation of damage. Through a sequence of approximations that transform the three-dimensional behavior into a unidimensional approximation, it is shown that the solutions converge to a uniaxial state, validating the model with a variational approach.

This work has key applications in materials and structural engineering under stress and damage conditions, and is fundamental for the efficient design of advanced materials requiring a deep understanding of mechanical properties and damage evolution in extreme conditions.