One-dimensional Localization Solutions for Time-dependent Damage

This article presents analytical solutions for a class of one-dimensional time-dependent elasto-damage problems. The considered damage evolution law may be seen as a one-dimensional version of the Kachanov–Rabotnov creep damage model with classical loading–unloading conditions. We construct analytical solutions for the quasistatic one-dimensional problem. The evolution consists of a first regime, in which damage and strain grow uniformly, followed by a regime in which localization occurs. In the second regime, the uniqueness of the solution is lost and the deformation of the body is represented by a sequence of arbitrary alternate loading/unloading regions. Complex evolutions with progressive enlargement of the unloading regions in a finite number of steps are also constructed. We study analytically and numerically the features of the obtained bifurcated solutions. It is shown that, at every instant of time, a lower limit exists for the size of the localization zone. This lower limit is actually realized by the solution with successive unloadings constructed in this article. These features help us to understand the behavior of numerical solutions for time-dependent damage in the quasistatic approximation.