Hillar Aben

Modern Photoelasticity for Residual Stress Measurement in Glass

Residual stress is one of the important indicators of the quality of any glass product. The paper gives a review of the methods, which are nowadays used in glass industry for measuring residual stress both in architectural and automotive glass panels and in hollow glassware.

Isochromatic fringes in photoelasticity

The structure of the isochromatic fringe pattern in a two-dimensional photoelastic model is investigated with the phase diagram method of the theory of dynamic systems. Isochromatics are interpreted as phase paths (or level curves) of a Hamiltonian system. Possible singularities of the fringe pattern are analyzed.

Principal formulas of integrated photoelasticity in terms of characteristic parameters

Recent advances in integrated photoelasticity have opened the possibility of determining tomographically arbitrary three-dimensional stress fields. Since photoelastic tomography is based on experimental measurement of the characteristic parameters, the dependence of these parameters on the stress distribution on a light ray is considered in detail. The possibility of determining certain integrals of the stress components is analyzed, and the linear approximation of integrated photoelasticity has been rigorously treated.

On the optical theory of photoelastic tomography

In recent years many authors have considered the possibility of using tomography for nondestructive determination of three-dimensional stress fields. A natural starting point for this is integrated photoelasticity. The problem is complicated since the stress field is a tensor field, and in the general case in integrated photoelasticity the relationships between the measurement data and the parameters of the stress field are non-linear. To elucidate these relationships, we have systematically studied the propagation of polarized light in an inhomogeneous birefringent medium. The inverse problem of integrated photoelasticity is formulated in the general form, and particular cases in which the polarization transformation matrix is exactly determined by integrals of the stress tensor components are considered. The possibility of using the Radon inversion for approximate determination of the normal stress field in an arbitrary section of the test object is outlined.

Transformation equations in polarization optics of inhomogeneous birefringent media

The quaternion formalism has been used to derive new systems of equations that describe transformation of the polarization of light in inhomogeneous birefringent media. In quaternion algebra the problem of parametric representation of the unitary transformation matrix reduces to the problem of formulation of the quaternion in trigonometric form. It is shown that this can be done in 30 different ways and that to each trigonometric form corresponds its own system of transformation equations. The six simplest systems of transformation equations have been derived.

HYBRID MECHANICS FOR AXISYMMETRIC THERMOELASTICITY PROBLEMS

The present article extends the integrated photoelasticity technique for the case where stresses are due to an unknown axisymmetric temperature field. Relationships that express radial and circumferential stresses through the axial and shear stresses are derived. Since residual stresses in glass can be described by the equations of thermoelasticity, the results of this article offer a means for complete determination of the residual stresses in axisymmetric glass articles using integrated photoelasticity. The results are verified by numerical and physical experiments.

Alternative equations of magnetophotoelasticity and approximate solution of the inverse problem

In magnetophotoelasticity, photoelastic models are investigated in a magnetic field in order to initiate rotation of the plane of polarization that is due to the Faraday effect. The method has been used for the measurement of stress distributions that are in equilibrium on the wave normal and therefore cannot be measured with the traditional photoelastic technique. In this category belong bending stresses in plates and shells and residual stresses in glass plates. Two new systems of equations of magnetophotoelasticity are derived. One of them describes evolution of the polarization of light in a magnetophotoelastic medium in terms of eigenvectors, the other in terms of distinctive parameters. For the latter system, an approximate closed-form solution has been found. The integral Wertheim law has been generalized for the case of stress states in equilibrium when rotation of the plane of polarization is present.

On the generalized Wertheim law in integrated photoelasticity

Two-dimensional photoelasticity is based on the classical Wertheim law. The integral Wertheim law can be used in integrated photoelasticity only in the case when the directions of the secondary principal stresses are constant on the light beam. We generalize the integral Wertheim law for the case when a slight rotation of the secondary principal directions takes place.

Duality in optical theory of twisted birefringent media

Two classical treatments of the transformation of the polarization of light in twisted birefringent media belong to Neumann and Kuske [see H. Aben , Integrated Photoelasticity (McGraw-Hill, New York, 1979)]. Since these authors have chosen parameters that characterize the state of polarization differently, the equations derived by them have been considered to be independent of one another. We show that duality exists between these equations. By the appropriate exchange of parameters, the first system of equations is transformed into the second one and vice versa. This duality follows from the duality between the two different parametric representations of the unitary unimodular matrix that describes the transformation of polarization in twisted birefringent media.

Application of the Abel inversion in case of a tensor field

Abels 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.