An asymptotic study of slow radial cracking

Radial cracking with closure

“…Significantly, the above expression differs fundamentally from that in [8]. It is important to note here that (93) has been derived without specifying any relationship between So and 34o; the closure conditions in (29) (29) …”

“…The cracks are assumed to be 'relatively' long in the sense that the three-dimensional contact problem can be described in a statically equivalent two-dimensional idealization. The viewpoint adopted in this paper forms one extreme in which one supposes that the large number of cracks formed permits a quasi-continuum axisymmetric approach rather than one involving the discussion of discrete sectors [8]. This supposition requires a formulation in which the interconnected action of both bending and stretching is treated as well as closure effects of the radial crack face contact.…”

“…Fortunately, if the solution is mainly independent of this small parameter, the hole radius can be taken to be zero without an essential change in the solution. For instance, consider the local conditions in the case of a plate loaded solely by a concentrated load P. With no concentrated radial actions applied the conditions at the plate center are as stated in (8). The 'hole solution' satisfies these requirements if its radius ro --+ O.…”

“…To date, the coupled plane-bending interaction of the radial crack faces has been ignored [4,5]. Historically, the difficulties introduced by such crack closure phenomena have long been recognized [6,7,8]; however, a framework for the application of fracture mechanics to cracked plates subjected to closure does not yet exist.…”

Topological Dynamics in Tandem with Permutation Groups

“…Reference [2] provides the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with most of the main areas of mathematics. Topological dynamics is defined as the study of asymptotic [11][12][13][14] or long term properties of families of maps in [15] of topological spaces. Abstract topological dynamics is usually developed in the context of flows as in [16].…”