Some Closed-form Solutions to Simple Beam Problems Using Nonlocal (Gradient) Damage Theory

In this article, a family of damage models which leads to the analytical solvability of the nonlocal evolution problem of a homogeneous bar in tension is defined. Explicit gradient damage models and implicit gradient damage models are investigated in a simple structural framework. The natural boundary conditions are derived from a variational principle, and are obtained at the boundary of the damage zone. It is shown that these damage models are the only ones leading to a linear differential equation of the strain variable. Some closed-form solutions are then available, providing a useful framework for the verification of computational models. Furthermore, these continuum damage mechanics models are well suited for the tension behavior of quasi-brittle materials, such as rock or concrete materials. It is theoretically shown that the damage zone evolves with the load level. This dependence of the localization zone to the loading parameter, is a basic feature, which is generally well accepted, from an experimental point of view. The strain profiles are also theoretically obtained, and corroborate well with what is usually numerically found for such nonlocal models. An imperfection analysis shows that the softening evolution problem is well posed in presence of strength imperfections. However, explicit gradient damage models lead to physically questionable results, in presence of imperfections. It is then recommended to use implicit gradient damage models when modeling realistic structures. It is hoped that the present article could serve as a benchmark to numerical gradient-damage finite element codes.