An Insight into Computational Challenges in Damage Mechanics: Analysis of a Softening Hooke’s Spring

While many efforts are being currently spent to forge reliable damage laws based on the physics of the materials to be studied, damage modeling is still addressed numerically too naively in many situations. This article highlights some topical conceptual aspects that have been up to now dealt with too superficially by comparing the performances of different numerical algorithms in solving Karush–Kuhn–Tucker conditions for a simple linearly softening Hooke’s spring. It is concluded that even such a primitive model, because of the multiplicity of solutions satisfying simultaneously equilibrium, damage law and irreversibility conditions, actually requires well-established numerical algorithms to face unexpected challenges. A comparison between different numerical strategies, beyond highlighting critical behaviors of traditional algorithms, permitted to observe an appealing robustness shown by an iterative strategy based on the fixed-point theorem. As a closure remark, evidences collected within this contribution naturally lead to the following question, which is left open for future studies: Is it possible to envisage the formulation of a criterion– possibly an energetic one, like that distinguishing stable and unstable solutions in elasticity—to establish which solution should be considered as valid in a given situation?