Failure thresholds in hierarchical and euclidian space by real space renormalization group

The scaling of the failure strength as a function of the size L and the influence of disorder in hierarchical and Euclidian systems are discussed within a real space renormalization group (RG). In hierarchical models for which the RG is exact, two universality classes are found : for « small disorder », the system is strong and its strength increases as a power of its size whereas for « large disorder », the system has a finite strength which does not increase as L increases. However, the exponents are not universal and depend upon the detailed hierarchical construction and on disorder. An approximate RSRG is also proposed to treat the case of two and three dimensional Euclidian systems. We find that the system is always strong in the sense that its strength increases as a powerlaw of its size, with an exponent which is weakly dependent on disorder and is thus not strictly universal.