Initiation of a periodic array of cracks in the thermal shock problem: A gradient damage modeling

This paper studies the initiation of cracks in the thermal shock problem through the variational analysis of the quasi-static evolution of a gradient damage model. We consider a two-dimensional semi-infinite slab with an imposed temperature drop on its free surface. The damage model is formulated in the framework of the variational theory of rate-independent processes based on the principles of irreversibility, stability and energy balance. In the case of a sufficiently severe shock, we show that damage immediately occurs and that its evolution follows first a fundamental branch without localization. Then it bifurcates into another branch in which damage localization will take place finally to generate cracks. The determination of the time and mode of that bifurcation allows us to explain the periodic distribution of the so-initiated cracks and to calculate the crack spacing in terms of the material and loading parameters. Numerical investigations complete and quantify the analytical results.