Continuum damage mechanics is characterized by mesh-dependent results unless specific countermeasures are taken. The most popular remedies involve introducing either nonlocality via filtering or a gradient extension for the damage variable(s). Such approaches have their limitations, e.g., they are hard to integrate into conventional finite-element codes, involve parameters that are non-trivial to determine experimentally and are incompatible with a scale transition that is both physically and mathematically sensible. The work at hand considers an alternative route to obtain mesh-independent damage models, namely via convex relaxation. Such convex damage models were considered before, but they are usually not capable of representing softening behavior. Schwarz et al. (Continuum Mech. Thermodyn., 33, pp. 69–95, 2021) proposed such a strategy by considering the convex envelope of a rate-limited simple damage model, i.e., an isotropic damage model without tension-compression anisotropy at small strains. However, they were not able to compute the envelope explicitly and provided an approximation only. In the work at hand, we introduce a number of conditions on the damage-degradation function which permit us to compute the convex envelope analytically for a large class of damage-degradation functions used in small-strain isotropic damage models. Interestingly, the obtained models involve a one-dimensional damaged microstructure, i.e., damage distributions emerge naturally. The resulting model is structurally simple and purely local, i.e., gradient-free, thermodynamically consistent and readily integrated into standard finite-element codes via traditional user subroutines. We discuss the computational and solid mechanical aspects of the ensuing model and demonstrate its numerical robustness via dedicated computational experiments. We also show that the model permits to be homogenized by considering a representative volume element study for an industrial-scale fiber-reinforced composite.