Interaction of damage processes in continuum damage mechanics

Abstract This work presents novel mathematical formulations for the interaction of damage processes within the framework of continuum damage mechanics, focusing on advanced numerical applications. The approach involves an innovative analogy between the decomposition of the damage variable/tensor and the rule of mixtures, leading to both scalar and tensorial representations. Four distinct cases are explored for each formulation: (1) basic interaction of two damage processes, (2) exponential interaction of two damage processes, (3) basic interaction of three damage processes, and (4) unsymmetrical interaction of two damage processes. The study further investigates a detailed plane stress example, where a system of nine coupled algebraic interaction equations is derived for each scenario. In particular, it is shown that these equations reduce to three core interaction equations in a special case, one of which is identified as the coupling interaction equation. The paper emphasizes mathematical rigor, with the goal of extending this research to tackle real-world problems and enhance numerical modeling techniques in future work.