Application of the Abel inversion in case of a tensor field

Abel's integral equations arise in many areas of natural science and engineering, particularly in plasma diagnostics. The axisymmetric physical field is determined using the Abel inversion. Until now, the Abel inversion has been applied almost exceptionally for the determination of scalar fields, i.e. fields, which at a point are characterized by a scalar. In several areas of engineering, the need to determine axisymmetric tensor fields arises, for example by measuring residual stress in axisymmetric glass articles with photoelasticity. In this article, we show how the Abel inversion can be used for the determination of an axisymmetric stress tensor field. The peculiarity in determining the tensor field is that on every ray two integrals of the field are measured and for complete determination of all the components of the stress tensor, equations of the theory of elasticity are used. The method is illustrated by an example.