Closed form solutions of the material failure process as hinges of beams by the embedded discontinuity model

Abstract Variational formulations of beam members with strong discontinuities adapted to represent strain localization are developed. The formulation for thin beams with embedded discontinuities takes into account only the internal strain energy due to bending induced strains and the formulation for thick beams with embedded discontinuities additionally takes into account the internal energy of shear strain. The developed energy functionals of the beam with discontinuities have as stationary conditions the corresponding boundary value problem. Closed form solutions of benchmark problems in beam members are developed by solving the corresponding boundary value problem. To validate the developed closed form solutions, representative examples are presented. The relationship of the total area under the moment-rotation jump and shear-displacement jump curves with the moment fracture energy and the shear fracture energy of the material, respectively, guarantees the correctness of the developed closed form solutions.