Whole-field determination of isoclinic parameter by five-step color phase shifting and its error analysis

Combining color imaging with phase shifting, a technique named five-step color phase shifting is presented to determine the whole-field isoclinic parameter. Relevant theory is derived and explicit conditions for directly determining the isoclinic parameter in the range of [0,π/2] are given. The unloaded light intensity of the model is systematically studied. A color camera recorded five isoclinic images coupled with isochromatics from a plane polariscope with five different settings, respectively. Experiments have been carried out with a circular disk under diametral compression and errors have been analyzed and estimated. This technique utilizes white light, which avoids undefined isoclinics near the locations where the isochromatics exist and will have active effect on experimental stress analysis and structural strength design.     

An investigation of cracking in brittle solids under dynamic compression using photoelasticity

Crack initiation in brittle materials was experimentally studied using photoelasticity under dynamic loading conditions with particular attention to the frictional characteristics of the microcracks. Two pieces of Homalite-100 were bonded except central region to prepare plate specimens with an inclined center crack. An edge of the specimen was impacted with and without lateral confinement. In situ photoelastic (isochromatic) fringes were obtained using a high-speed camera. Initial direction and profile of wing crack was the same as in static loading tests. Effect of crack surface roughness and lateral confinement on fringe pattern is discussed. Average speed of wing crack propagation was about 100 m/s and wing cracks from a crack with higher friction coefficient propagated faster than from a smooth crack.     

Whole-field digital measurement of principal stress directions in photoelasticity

A method that uses three light-field isoclinic images and the associated light-field unloaded model (or white) images for whole-field digital determination of the principal stress directions in photoelasticity is presented. Relevant theory is derived and explicit conditions for directly determining the directions to the range 0–π/2 are given. Tests of this method on a directly loaded two-dimensional disc and a stress frozen photoelastic slice are demonstrated. The results agree well with the values obtained from the manual method and/or the theory.     

Integrated photoelasticity for nondestructive residual stress measurement in glass

The paper gives a review of integrated photoelasticity and of its application for residual stress measurement in glass. By considering the basic theory of the method, two particular cases, the case of weak birefringence and that of constant principal stress axes, are picked up. It is shown that integrated photoelasticity is actually optical tensor field tomography. Its peculiarities in comparison with scalar field tomography are considered. Since directly integrated photoelasticity allows for the measurement of only some of the stress components, analytical or numerical methods are to be used for complete determination of the stress field. Nonlinear optical phenomena, interference blots and fringe bifurcation, are briefly considered. Several examples illustrate the application of the method.     

The influence of the quarter wave plates in automated photoelasticity

During the last decades, several methods have been proposed to automate photoelastic analyses. Some procedures are based on the circularly polarised light by using quarter wave plates. However, quarter wave plates are typically matched for a specific wavelength, and an error is introduced at different wavelengths. The error of quarter wave plates affects the measurement of isochromatic and isoclinic data. In this paper, the influence of the errors of quarter wave plates in some of the most common automated photoelastic methods is reviewed. The errors in the photoelastic data are given and the procedures to reduce, or eliminate, them are also suggested.     

Extraction of isotropic points using simulated isoclinics obtained by photoelasticity-assisted finite element analysis

A method for extracting isotropic points in structures using simulated isoclinics obtained from a combination of photoelastic experiments and finite element analysis (FEA) is presented. This method is divided into two parts. The first part involves the confirmation of the boundary condition necessary for FEA using isochromatics obtained by photoelastic experiments. The second part involves the determination of isotropic points using simulated isoclinics obtained by FEA under the boundary condition confirmed by photoelastic experiments. This method is applied to a ring and T-shaped plates subjected to a compressive load. The results show that the isotropic points in the ring and T-shaped plates could be accurately and easily extracted by this method.     

Numerical analysis of phase-stepping interferometric photoelasticity for plane stress separation

A phase-stepping method for interferometric photoelasticity is proposed to determinate whole-field plane stress components. Isopachic phase can be obtained by rotating polarizer, second quarter-wave plate and analyzer at definite optical arrangements. On the other hand, isochromatic and isoclinic phases can be determined in a circular polariscope arrangement. Furthermore, a load-stepping method is adopted to overcome the wrong mathematic sign of the isochromatics in the ambiguity regions and the influence of initial interferometric fringes on the isopachics. Light intensities and phase-stepping formula for the proposed method are derived using Jones calculus. Simulation of a circular disk under diametral compression demonstrates the feasibility of the proposed method.     

Towards effective phase unwrapping in digital photoelasticity

Phase-shifting techniques in photoelasticity have a peculiar problem of isoclinic–isochromatic interaction, which lead to ambiguous zones in phase map. A comprehensive discussion on the reason for the formation of ambiguous zones is presented in this paper. A new method is developed to remove this ambiguity in the phase map. The new methodology is validated with theoretically and experimentally generated phase maps. A flexible algorithm for phase unwrapping is developed using tiling. The applicability of this is demonstrated using various benchmark problems consisting of simply and multiply connected bodies and slices cut from stress frozen models.     

Computation of three-dimensional Brinkman flows using regularized methods

The Brinkman equations of fluid motion are a model of flows in a porous medium. We develop the exact solution of the Brinkman equations for three-dimensional incompressible flow driven by regularized forces. Two different approaches to the regularization are discussed and compared on test problems. The regularized Brinkman model is also applied to the unsteady Stokes equation for oscillatory flows since the latter leads to the Brinkman equations with complex permeability parameter. We provide validation studies of the method based on the flow and drag of a solid sphere translating in a Brinkman medium and the flow inside a cylindrical channel of circular cross-section. We present a numerical example of a swimming organism in a Brinkman flow which shows that the maximum swimming speed is obtained with a small but non-zero value of the porosity. We also demonstrate that unsteady Stokes flows with oscillatory forcing fall within the same framework and are computed with the same method by applying it to the motion of the oscillating feeding appendage of a copepod.     

Digitally whole-field analysis of isoclinic parameter in photoelasticity by four-step color phase-shifting technique

This paper presents a whole-field method for automatic determination of the isoclinic parameter φ in photoelasticity. The method determines φ in the true phase interval (-π/2<φ+π/2) based on the four-step color phase-shifting approach and a new unwrapping method based on simple logic operations, a binary image and a gray-scale mask. A plane polariscope with a white light source is used for capturing raw photoelastic fringe images. The theoretical validation of the method is done with the problem of a circular disk under compression. Comparisons between theoretical and experimental results are also demonstrated. The experimental confirmation for the proposed method is tested with the photoelastic model having an isotropic point(s) and/or a singular point(s). The results show the method permits the reliable isoclinic parameter to be determined in the true phase interval with almost no defects of the isochromatic parameter.     

Visualization of three-dimensional temperature distributions in a large-scale furnace via regularized reconstruction from radiative energy images: numerical studies

In this paper, the possibility of visualization of three-dimensional (3-D) temperature distributions in large-scale boiler furnaces from radiative energy images captured by multiple charge-coupled device (CCD) cameras mounted around the furnace is studied numerically. For the calculation of the radiative energy image formation, a fast algorithm proposed by the authors for pinhole imaging is used in this paper, which is based on the Monte Carlo method and combined with a concept of angular factor effective for image formation. This algorithm is applicable for the emitting, absorbing, and isotropic scattering medium. For the inversion of the 3-D temperature distributions which is an ill-posed problem, a modified Tikhonov regularization method is improved, where the finite difference regularizer is defined and can be used in 3-D cases, and the optimal regularization parameter is suggested to be selected by using a post-treatment method. For a 3-D unimodal temperature distribution, the numerical simulation results show that the reconstruction errors for the 3-D temperature distribution can be maintained at levels similar to the measurement error and the visualization quality of the temperature distribution is satisfactory. For a kind of bimodal temperature distribution, the reconstruction errors are higher than those for the unimodal distribution, but the bimodal feature of the temperature distribution can also be reproduced clearly.     

Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique

In this paper a new whole field technique for determination of characteristic parameters in integrated photoelasticity using phase-shifting methodology is proposed. Appropriate optical arrangements necessary for this are presented and the intensity equations are obtained by Jones calculus. The new methodology is verified experimentally for the problem of a circular disk under diametral compression viewed in oblique incidence.     

Visualization of isoclinics and isochromatics in a birefringent slice optically singled out in a three-dimensional model

The speckle patterns scattered from the boundary surfaces of a birefringent slice are imaged on a photographic plate. The method consists in taking account of the variations of the statistical properties of the two superimposed speckle patterns. The results are similar to those obtained by observing this thin slice in a classical parallel or crossed polariscope. This method does not need the slicing of the model.     

Dynamic photoelasticity using a laser scanner

Over the years, several techniques have been developed to record dynamic fringe patterns in photoelasticity. Due to the whole field recording capability, the photoelastic technique continues to be of interest to the experimental mechanics community. A low-cost dynamic recording system employing an out-dated fax scanner is presented in this article. Details of the optical and electronic functions are discussed. The feasibility of the system is demonstrated by recording dynamic fringe patterns off a simply supported beam subjected to impact loading. Further modifications to the set-up to improve the quality of the fringe patterns are discussed briefly.     

Atmospheric trace gases profile retrievals using the nonlinear regularized total least squares method

We discuss the regularized total least squares (RTLS) method for the nonlinear overdetermined ill-posed problem of atmospheric trace gas retrieval. The RTLS method is used for the automatic determination of the regularization strength of the ill-conditioned matrix in an iterative process, and a mixed quadratic and cubic line search method is used for the nonlinear retrievals. Additional retrieval metrics, such as the model resolution matrix and the degrees of freedom in the retrieval, which characterize the vertical resolution of the retrievals, are also derived. Simulated retrievals as well as the retrieval from the data of a balloon-based spectroscopic measurement will be discussed. Retrieval results obtained using O₃ and CH₄ as test cases will be presented.     

Fringes of equal tangential inclination by curvature-induced birefringence

A new kind of interference fringes, fringes of equal tangential inclination by curvature-induced birefringence, is presented. These are two-beam interference fringes produced by bending a thin sheet of birefringent material into a part of an exact cylinder such that the curvature is constant. Due to this curvature there is a uniform birefringence being induced. The change in birefringence induced by applying different radii of curvatures to a Fortepan sheet is measured. The stored (fixed) or natural birefringence of this sheet is deduced.     

Stable solutions for qq^- bound states with a regularized Breit potential

The Breit interaction contains singular terms which may lead to an instability in quark-antiquark bound state calculations. We regularize the Breit interaction by multiplying the singular terms in momentum space by the form factor μ^2/（q^2＋μ^2） such that the interaction is not singular at the origin and the intermediate-and long-range parts of the interaction remain unchanged. The singular terms in the Breit potential find their stable contributions in the calculations after being multiplied by the form factor with different powers. Such a regularized Breit potential with a linear and a relativistically corrected confining potential are applied to the study of qq^- bound states. The spectra for most familiar mesons are consistently obtained and agree well with the experimental data.     

Limitation of carrier fringe methods in digital photoelasticity

The carrier fringes method has been proposed in digital photoelasticity in combination with techniques such as Fourier transform and phase shifting method, without considering the influence of the isoclinics on the isochromatic patterns analysis. Unlike other optical methods as moiré and holographic interferometry, in photoelasticity the light intensity emerging from a circular polariscope is related to both the isochromatic retardation and the isoclinic parameter. As it is shown by the theoretical analysis, owing to the misalignment between the principal stresses in the model and in the carrier, the computed retardation is affected by an error which is the same for all photoelastic methods based on the use of carrier fringes. Consequently, the photoelastic analysis carried out by methods that use carrier fringes cannot be applied as a full-field technique. In detail, numerical simulations show that the retardation error is comparable (less than 0.05 fringe orders) with that of other photoelastic methods provided that the misalignment between the principal stresses in the model and in the carrier is less than 30°. On the contrary, in the model zones where the misalignment is higher than 30°, the retardation measurement can be affected by non negligible errors (up to 0.25 fringe orders).     

A review of the scattered-light method in photoelasticity

The literature on the scattered-light method is reviewed and the theory, basic methods of analysis and applications of the method summarised.

The theory of fringe formation and methods of analysis are found to be well-established. The accuracy of the method is confirmed for many cases of stress analysis, but little experience has been gained in the areas where the method has the most potential: three-dimensional thermal, dynamic and elastoplastic stress analysis.

Difficulties associated with the necessity of immersing the photoelastic model and the effort required to obtain and analyse fringe data restrict the practical use of the method to these special areas of stress analysis. Wider use of the method within these areas is envisaged, however, together with developments in scattered-light equipment.