Various theoretical applications of the maximum modulus principle to the experimental method of caustics (Q1342166)

The basic complex variable equation of the initial curve of the caustic in plane elasticity is \(| C \phi '' (z) |= 1\), where \(C\) is an overall optical-mechanical constant of the experimental set up, and \(\phi '' (z)\)is the second derivative of the first complex potential \(\phi (z)\) of Kolosov-Muskhelishvili. The values of \(z\) satisfying the above equation constitutive the initial curve \(L\) of the caustic of the plane specimen. Unfortunately, \(\phi (z)\) and \(\phi '' (z)\) are not analytic in the domain \(S\) bounded by \(L\). When we apply the maximum modulus principle to obtain theoretical bounds on some quantities of probable interest in fracture mechanics, we need to modify the form \(\phi '' (z)\) so that the analyticity of the modified form can be assured inside and on \(L\). In this paper, examples illustrate how the potential \(\phi (z)\) must be modified in some special cases including concentrated forces, edge dislocation, pair of concentrated forces, straight crack, and circular inclusion.