Asymptotical analysis and padé approximation in problems on diffusion controlled cracks propagation

in this work, we consider the diffusion-controlled axisymmetric fracture in an infinite space, and half-space. an important example of diffusion-controlled fracture growth is given by hydrogen induced cracking. in metals, hydrogen is typically dissolved in the proton form. when protons reach the crack surface, they recombine with electrons and form molecular hydrogen in the crack cavity. then, the fracture can propagate even in the absence of any external loading, that is, only under the excessive pressure of gas hydrogen accumulated inside the crack. our results show that in the long-time asymptotic approximation (based on the quasi-static solution), the diffusion-controlled delamination propagates with constant velocity. we determine a maximum critical concentration that limits the use of the quasi-static solution. a transient solution, representing a short-time asymptotic approximation, is used when the concentration of gas exceeds the critical concentration. we then match these two end-member cases by using the method of padé approximations and present closed-form solutions for both internal and near-surface diffusion-controlled crack propagation at different time scales.