Spectral Localization Analysis in Nonlocal Softening Materials

The paper shows that spectral wave propagation analysis reveals in a simple and clear manner the effectiveness of various regularization techniques for softening materials, such as plasticity and damage models. It is verified the effectiveness of the nonlocal integral-type approach with an unconventional nonlocal formulation introduced in 1994 by Vermeer and Brinkgreve, which was called ”over-nonlocal” by Di Luzio and Baˇzant in 2005 because it uses a linear combination of local and nonlocal variables in which a negative weight imposed on the local variable is compensated by assigning to the nonlocal variable weight greater than 1. The spectral approach readily confirms that the nonlocal integral-type generalization of softening plasticity with an additive format gives correct localization properties only if an over-nonlocal formulation is adopted. By contrast, the nonlocal integral-type generalization of softening plasticity with a multiplicative format provides realistic localization behavior, just like the nonlocal integral-type damage model, and thus does not necessitate an over-nonlocal formulation. The localization behavior of explicit and implicit gradient-type models is also analyzed demonstrating that they have very different localization properties.