This chapter discusses the Schwarzschild black hole. It demonstrates how, by a judicious change of coordinates, it is possible to eliminate the singularity of the Schwarzschild metric and reveal a spacetime that is much larger, like that of a black hole. At the end of its thermonuclear evolution, a star collapses and, if it is sufficiently massive, does not become stabilized in a new equilibrium configuration. The Schwarzschild geometry must therefore represent the gravitational field of such an object up to r = 0. This being said, the Schwarzschild metric in its original form is singular, not only at r = 0 where the curvature diverges, but also at r = 2m, a surface which is crossed by geodesics.
S-N curve characterisation and prediction of remaining fatigue life are studied using polyethylene terephthalate glycol-modified (PETG). A new simple method for finding a data point at the lowest number of cycles for the Kim and Zhang S-N curve model is proposed to avoid the arbitrary choice of loading rate for tensile testing. It was demonstrated that the arbitrary choice of loading rate may likely lead to an erroneous characterisation for the prediction of the remaining fatigue life. The previously proposed theoretical method for predicting the remaining fatigue life of composite materials involving the damage function was verified at a stress ratio of 0.4 for the first time. Both high to low and low to high loadings were conducted for predicting the remaining fatigue lives and a good agreement between predictions and experimental results was found. Fatigue damage consisting of cracks and whitening is described.
Motivated by the hindrance of defining metric tensors compatible with the underlying spinor structure, other than the ones obtained via a conformal transformation, we study how some geometric objects are affected by the action of a disformal transformation in the closest scenario possible: the disformal transformation in the direction of a null-like vector field. Subsequently, we analyze symmetry properties such as mutual geodesics and mutual Killing vectors, generalized Weyl transformations that leave the disformal relation invariant, and introduce the concept of disformal Killing vector fields. In most cases, we use the Schwarzschild metric, in the Kerr–Schild formulation, to verify our calculations and results. We also revisit the disformal operator using a Newman–Penrose basis to show that, in the null-like case, this operator is not diagonalizable.
Previous special relativistic calculations of gravitational redshift, light deflection and Shapiro delay are extended to include perigee advance. The three classical, order G, post-Newtonian predictions of general relativity as well as general relativistic light-speed-variation are therefore shown to be also consequences of special relativistic Newtonian mechanics in Euclidean space. The calculations are compared to general relativistic ones based on the Schwarzschild metric equation, and related literature is critically reviewed.
The achievement of smoothness is one of the main purposes of graduation. Smoothness, however, has never been defined except in terms of concepts which themselves defy definition, and there is no accepted way of measuring it. This paper presents an attempt to supply a definition by constructing a quantitative measure of smoothness, and suggests that such a measure may enable us in the future to graduate without prejudicing the process by an arbitrary choice of the form of the relationship between the variables.
S-N Curve Characterisation for Composite Materials and Prediction of Remaining Fatigue Life Using Damage Function
