Correlating elasticity and crack formation

For solving the longstanding materials science problem of correlating elastic properties of a solid material to the formation of cracks we present a new general concept. This concept is applied to the technologically most important cracks of loading mode I for which we establish exact correlations by introducing a localization length as a new material parameter. We study two limiting cases of crack formation making use of analytic models determining the material and direction dependent parameters by comparison to {\em ab initio} density functional results. This is done for a variety of real materials in order to test our approach for different types of bonding. For the most interesting ideal brittle cleavage we find that the localization length is -within a reasonable approximation- of constant value, which results in a simple relation for the critical stress, presumably being useful for materials engineering. Our results confirm that the proposed general concept results in meaningful physical models which -for the first time- allow a rigorous and simple correlation between elastic and mechanical properties.