Stress Singularities at Crack Tips; Some Developments in Greece

Abstract

The chapter discusses the recent developments in the field of fracture mechanics in Greece. In addition, it concerns about stress singularities at crack tips. The mathematical analysis in the mechanics of fracture is related to the development of several methods for the numerical solution of Cauchy-type one-, two-dimensional singular integral equations (S.I.E.) frequently encountered in the problems of elasticity containing singularities, because of the geometry, or the loading type of the stress field. An integral equation, or a system of equations, can be numerically solved by reducing it to an equivalent system of linear algebraic equations. To obtain this reduction, the S.I.E. is applied to a number of appropriately selected points of the integration interval and then a numerical integration rule is selected for the approximation of the integrals. The method is capable of solving efficiently crack and inclusion problems, as well as contact problems. A generalization of the method to the case of boundary-value problems of three-dimensional elasticity is at present under development. Another area of research is to develop methods that are liberated from the restriction of selecting special collocation points by using procedures of Lagrange interpolation to the functional equation derived from the numerical integration of the integral equation accurate to a (2n – l)-degree. Finally, combined methods of finite elements and singular integral equations, as well as hybrid methods using data from photoelasticity, moiré and other experimental methods, and combined with the solution of the appropriate singular integral equation, are in plain evolution.