The classical theory of linear elastic fracture mechanics for a quasi-static crack in an infinite linear elastic body has two significant mathematical inconsistencies: it predicts unbounded crack-tip stresses and an elliptical crack opening profile. A new theory of fracture developed by Sendova and Walton, based on extending continuum mechanics to the nanoscale, corrects these erroneous effects. The fundamental attribute of this theory is the use of a dividing surface to describe the material interface. The dividing surface is endowed with an excess property, namely surface tension, which accounts for atomistic effects in the interfacial region. When the surface tension is taken to be a constant, Sendova and Walton show that the theory reduces the crack-tip stress from a square root to a logarithmic singularity and yields a finite angle opening profile. In addition, they show that if the surface tension depends on curvature, the theory completely removes the stress singularity at the crack-tip, for all but countably many values of the two surface tension parameters, and yields a cusp-like opening profile. In this work, we develop a numerical model using the finite element method for the Sendova-Walton fracture theory applied to the classical Griffith crack problem in the case of constant surface tension. We show that the numerical model behaves as predicted by the theory, yielding a reduced crack-tip singularity and a finite opening angle for all nonzero values of the constant surface tension. We also lay the groundwork for the numerical implementation of the curvature-dependent model by constructing an algorithm to determine the appropriate threshold values for the surface tension parameters that guarantee bounded crack-tip stresses. These values can then be directly applied to the forthcoming numerical model.

Many chemical reactions can be described as the crossing of an energetic barrier. This process is mediated by an invariant object in phase space. One can construct a normally hyperbolic invariant manifold (NHIM) of the reactive dynamical system which is an invariant sphere that can be considered as the geometric representation of the transition state itself. The NHIM has invariant cylinders (reaction channels) attached to it. This invariant geometric structure survives as long as the invariant sphere is normally hyperbolic. We applied this theory to the hydrogen exchange reaction in three degrees of freedom in order to figure out the reason of the transition state theory (TST) failure. Energies high above the reaction threshold, the dynamics within the transition state becomes partially chaotic. We have found that the invariant sphere first ceases to be normally hyperbolic at fairly low energies. Surprisingly normal hyperbolicity is then restored and the invariant sphere remains normally hyperbolic even at very high energies. This observation shows two different energy values for the breakdown of the TST and the breakdown of the NHIM. This leads to seek another phase space object that is related to the breakdown of the TST. Using theory of the dividing surface including reactive islands (RIs), we can investigate such an object. We found out that the first nonreactive trajectory has been found at the same energy values for both collinear and full systems, and coincides with the first bifurcation of periodic orbit dividing surface (PODS) at the collinear configuration. The bifurcation creates the unstable periodic orbit (UPO). Indeed, the new PODS (UPO) is the reason for the TST failure. The manifolds (stable and centre-stable) of the UPO clarify these expectations by intersecting the dividing surface at the boundary of the reactive island (on the collinear and the three (full) systems, respectively).