How to use lumped damage mechanics: Two-dimensional solids.2026. V.C. Guimaraes, P.; Chang-Qi Pan.; A. Picón, R.; L.N.F. Amorim, D.; Yan-Gao Hu.; P.B. Proença, S.; Flórez López, J

This paper proposes an extension of Jean Lemaitre’s Continuum Damage Mechanics (CDM) into a unified framework denoted Lumped Damage Mechanics (LDM), capable of integrating diffuse micro-damage and localized fracture processes. The fundamental hypothesis assumes a sudden transition from distributed energy dissipation to localized mechanisms represented by zero-measure entities such as inelastic hinges, hinge lines, and cohesive process zones. LDM applications include two-dimensional solids, beams, frames, plates, and shell structures. Two complementary formulations are hereby addressed: (i) a differential formulation in terms of displacement discontinuities and lumped damage variables, and (ii) a variational formulation based on Nash non-cooperative equilibrium principles, ensuring mesh-objective responses under localization. Numerical implementation focusing on two-dimensional solids employs the concept of numerical extensometers (numexes), which enables a robust finite element discretization of both diffuse and localized damage. The approach is validated through experimental–numerical analyses of a fiber-reinforced concrete L-shaped plate and a tunnel segment. Digital image correlation and load–displacement curves are the experimental tools used to confirm the model’s predictive capacity. Results demonstrate that LDM can capture the evolution from diffuse damage to discrete crack patterns, reproducing complex crack networks beyond the capabilities of classical fracture mechanics (FM). Additional applications to reinforced concrete plate elements show that the formulation effectively accounts for cracking and reinforcement yielding, while preserving computational efficiency. Compared with conventional CDM and FM, the proposed LDM framework yields objective, scale-consistent predictions of failure processes in quasi-brittle and ductile materials, offering a generalized tool for structural analysis under progressive degradation.