Fracture in a higher-order elastic continuum

We have obtained analytically the mode I and mode II full-field solutions for a semi-infinite crack in an infinite solid characterized by the higher-order elastic continuum theory. The theory is the limit of the strain gradient plasticity theory with the plastic work hardening exponent n = 1. It also represents the macroscopic behavior of cellular materials. The analytical solution shows the transition from remotely imposed classical K field to the asymptotic field near the crack tip over the internal material lengths associated with the stretch gradient and rotation gradient of deformation. It is established that the asymptotic crack tip fields have no domain of physical validity because the stress tractions have the incorrect sign within a zone on the order of internal material length l₁ associated with the stretch gradient of deformation. This analytical full-field solution can be used as an important benchmark for the various finite elements developed for strain gradient plasticity.