  Hertzian contact of a rigid sphere and a highly deformable soft solid is
investigated using integrated photoelasticity. The experiments are performed by
pressing a styrene sphere of 15 mm diameter against a 44 x 44 x 47 mm$^3$
cuboid made of 5% wt. gelatin, inside a circular polariscope, and with a range
of forces. The emerging light rays are processed by considering that the
retardation of each ray carries the cumulative effect of traversing the
contact-induced axisymmetric stress field. Then, assuming Hertzian theory is
valid, the retardation is analytically calculated for each ray and compared to
the experimental one. Furthermore, a finite element model of the process
introduces the effect of finite displacements and strains. Beyond the
qualitative comparison of the retardation fields, the experimental,
theoretical, and numerical results are quantitatively compared in terms of the
maximum equivalent stress, surface displacement, and contact radius dimensions.
A favorable agreement is found at lower force levels, where the assumptions of
Hertz theory hold, whereas deviations are observed at higher force levels. A
major discovery of this work is that at the maximum equivalent stress location,
all three components of principal stress can be determined experimentally, and
show satisfactory agreement with theoretical and numerical ones in our
measurement range. This provides valuable insight into Hertzian contact
problems since the maximum equivalent stress controls the initiation of plastic
deformation or failure. The measured displacement and contact radii also
reasonably agree with the theoretical and numerical ones. Finally, the
limitations that arise due to the linearization of this problem are explored.
