A Discrete Algorithm of Integrated Photoelasticity for Axisymmetric Problems

Abstract

By measuring stresses in cylinders with no stress gradient in axial direction, the cross-section of the cylinder may be considered as consisting of concentric rings in each of which the stresses are constant. Axial stress distribution in the cylinder is determined stepwise starting with the outermost ring (the so-called onion-peeling method). This paper generalizes the onion-peeling method for the case of axisymmetric stress distributions when stress gradient in the axial direction is present. Measurement of the integrated isoclinic and optical retardation is carried out in two parallel sections which are perpendicular to the specimen axis, Δz apart from each other. The distributions of the axial stress and shear stress are determined directly from the measurement data, using linear approximation of integrated photoelasticity. Other stress components are determined using the equilibrium equation and the compatibility equation (if stresses are due to external loads) or using the generalized sum rule (if residual stresses in glass are measured). The method is less sensitive to measurement errors than the Abel inversion. In comparison with the polynomial approximation of the stress distributions, the onion-peeling method gives more adequate results if stress distribution is not smooth. The paper is illustrated with several applications.

Elastic and residual stresses around ball indentations on glasses using a micro-photoelastic technique

“…2, the arrows in stresses indicate the positive (+), or tensile, directions. These stress components in cylindrical symmetry can be determined using the onion-peeling method, if there is no stress gradient in the axial direction [21]. In the onion-peeling method, the circular crosssection of the cylinder is divided into a number of concentric rings, or onion skins, in each of which the axial stress is considered to be constant.…”

“…In the onion-peeling method, the circular crosssection of the cylinder is divided into a number of concentric rings, or onion skins, in each of which the axial stress is considered to be constant. As for the axisymmetric specimen with stress gradient in the axial direction, the generalized onion-peeling method [21] can be applied to obtain all components of the axisymmetric state of stress shown in Fig. 2.…”

“…In the generalized onion-peeling method, the z-axis dependence of the stresses is successfully obtained from the difference in BR parameters of the corresponding points on the two sections, which are main and auxiliary sections, apart from a given depth (one pixel in this study). The detailed algorithm used in the generalized onion-peeling method is explained elsewhere [21]. For example, the shear stress in a given pixel which exists in an outermost ring, or skin, can be obtained from the optical path length (one pixel) and from the retardation and azimuth values of the transmitted light in the given pixel.…”

Annealing of Compression Molded Aspherical Glass Lenses

“…This will allow the axial stress r z and shear stress s rz to be determined directly from isoclinic angle u and retardation D [14][15][16][17]. The relations between u, D, and the components of the stress tensor along a light ray L can be expressed as [14] Dk…”

“…To evaluate the impacts of the annealing experiments, residual stresses and refractive index variation of the molded glass lens of pre-and postannealing were compared. The residual stresses were studied by a circular polariscope based on the property of birefringence of a glass lens [14][15][16][17]. The refractive index variation was measured and reconstructed by an optical setup based on MachZehnder interferometer [7,18].…”