Macro and micro cracking in one-dimensional elasticity

In classical fracture mechanics, the equilibrium configurations of an elastic body are obtained by minimizing an energy functional containing two contributions, bulk and surface. Usually, the bulk energy is convex and the surface energy is concave. While this type of minimization successfully describes macroscopic cracks, it fails to model micro-defects forming a so-called process zone. To describe this phenomenon, we consider, in this paper, a model with a non-concave, `bi-modal' surface energy, which allows the formation of both macro- and micro-cracks. Specifically, we consider the simplest one-dimensional problem for a bar in a hard device and show that if the surface energy is not subadditive, the solution exhibits a new mode of failure with a finite number of micro-cracks coexisting with one fully developed macro-crack. We present an explicit example of a `quantized' micro-cracking with a subsequent development into a single macro-crack.