Sprain energy and gap test consequences for damage localization and fracture mechanics

The smooth Crack Band Model (sCBM), conceived in 2021, incorporated a novel localization limiter that is imposed on the 'sprain' field, representing the second-gradient of displacement, to prevent spurious damage localization during fracture growth.A following study in 2023 presented an improved model, called the smooth Lagrangian Crack Band model (slCBM), in which the term "spress" was introduced as the force variable work-conjugate to the "sprain" tensor.More importantly, the numerical difficulty of the sCBM due to using the nodes of adjacent finite elements was overcome by treating displacement vectors and their gradients as independent fields with C 0 continuity in finite element implementation, constrained by second-order tensorial Lagrange multipliers.Combined with the microplane model M7 for triaxial softening damage, our numerical validation of the gap test results using the slCBM demonstrates accurate reproduction of size effects under varying crack-parallel stresses.The same, though with path-dependence limitations, is achieved by a simple formula for predicting the crack-parallel stress effects on the fracture energy.Traditional line crack models, including their phase-field reincarnation, give errors of up to 100%.Further it is demonstrated that the existing strain-gradient theories, lacking the resistance to material rotation gradients, predict incorrect fracture patterns with load errors up to 55% error in the case of Mode II and III fractures and for Mode I fractures mixed with shear loading.The crack-parallel stress effect appear to be universal for all materials, including atomistically sharp crystal cracks.There are fundamental implications for the theory of fracture mechanics.