Symbolic derivation of the equations of caustics about a crack tip

Abstract

For the determination of the shape of the initial curve of a caustic about a crack tip in plane elasticity problems (whence the shape of the caustic itself is automatically obtained) we need to solve a nonlinear algebraic equation with unknown the distance of each point of the initial curve from the crack tip and parameter the polar angle. Here this nonlinear equation is solved (for a particular crack problem) by the classical method of successive substitutions (a one-point iterative method) in numerical analysis, but symbolically and not numerically, with respect to the polar angle. This yields a semianalytical equation for the initial curve (and, further, for the caustic itself) of particular importance for the study of its properties from the analytical point of view. On the other hand, the present results show the usefulness of symbolic computations in crack problems in fracture mechanics.