  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.
