A Least-Squares Approach to the Practical Use of the Hole Method in Photoelasticity

The use of a small circular hole in elastostatic photoelasticity to determine the stress tensor for any two-dimensional general loading situation is well known. The original application required fringe-order information at four points on the boundary, on opposite sides, along the axes of symmetry or principal stress directions. Later, to obtain greater precision, it was adapted so that fringe information inside the field could be used. This led to the also limited use of fringe-order information from four points at 1.4 and two times the radius of the hole, along the principal axes of symmetry. More recent work has even allowed the use of fringe-order information, at a fixed radius, anywhere along the two principal axes of symmetry. The greatest limitation of all of these approaches is that the majority of the fringe-order information that is available, away. from the axes of symmetry, is not utilized at all. The current work presents a least-squares approach to the hole method that allows the simultaneous use of information anywhere and at any radial distance from the center of the hole inside the stress field. The objectives of this paper are: to apply the use of the least-squares approach to the hole method in photoelasticity; and, to show the consistent and practical application of this least-squares approach to the hole method. The achievement of this last objective permits the use of the values of specimen birefringence at a large number of points, taken from anywhere in the field around the hole.