On the existence of a nucleation length for dynamic shear rupture

The paper derives the similarity-limit system a/b = W P + L̃(W), 0 = (1−P)(1−WP), shows that the elastic operator rescales distances by εLb (two half-spaces) or √ε Lbh (thin layer), and concludes that the nucleation half-length vanishes in the slip-law limit; in the two–half-space case it further gives an explicit interior semicircle W with L/(εLb) = 1/[π(1−a/b)^2] . The candidate solution reproduces exactly this scaling argument (via the same change-of-variable on the elastic operator) and the same semicircle construction and constant, hence arrives at L(ε)=εLb/[π(1−a/b)^2] for two half-spaces and L(ε)∝√ε for a thin layer, so both are correct and essentially the same proof .