Interframe intensity correlation matrix for self-calibration in phase-shifting interferometry

A new method of estimating reference phase shifts in phase-shifting interferometry is proposed. The reference phase shifts are determined from a matrix that represents the interframe intensity correlation (IIC) of phase-shifted interferograms. The root-mean-square error of intensity measurement is automatically obtained from the smallest eigenvalue of the IIC matrix. The proposed method requires only four interferograms, unlike others, and can extract phase shifts reliably even from interferograms without well-defined fringes, such as speckle patterns. In typical conditions, reference phase shifts and wave-front phases can be determined with an accuracy of lambda/6310 and lambda/150, respectively. The validity of the method is tested by comparing it with other methods in experiments and simulations.     

Exact solution to the pairing equations for bosons in a one-dimensional model

Pairing equations for bosons interacting with a delta-function potential in one dimension are reduced to two coupled equations involving complete elliptic integrals. These equations are solved numerically and the results are compared to the exact Lieb and Liniger solution for the problem of one-dimensional bosons with delta-function interactions. One finds that the pairing solution gives too large a potential energy and too small a kinetic energy. We also compare the exact and paired excitation spectra. We discuss the relevance of these results to realistic problems.Work supported in part by the National Science Foundation.     

Calibration of a neutron time-of-flight multidetector system for an intensity interferometry experiment

We present the details of an experiment on light particle interferometry. In particular, we focus on a time-of-flight technique which uses a cyclotron RF signal as a start and a liquid scintillator time signal as a stop, to measure neutron energy in the range of E_n≈1.8–$\text{150}\text{MeV}$. This dynamic range (up to $\text{300}\text{ns}$) is much larger than the beam bunch separation $\text{(54}\text{ns}\text{)}$ of the AGOR cyclotron (KVI). However, the problem of a short burst period is overcome by using the time information obtained from a fast projectile fragment phoswich detector. The complete analysis procedure to extract the final neutron kinetic energy spectra, is discussed.     

Wavelet–Galerkin solutions for one-dimensional partial differential equations

In this paper we describe how wavelets may be used to solve partial differential equations. These problems are currently solved by techniques such as finite differences, finite elements and multi-grid. The wavelet method, however, offers several advantages over traditional methods. Wavelets have the ability to represent functions at different levels of resolution, thereby providing a logical means of developing a hierarchy of solutions. Furthermore, compactly supported wavelets (such as those due to Daubechies¹) are localized in space, which means that the solution can be refined in regions of high gradient, e.g. stress concentrations, without having to regenerate the mesh for the entire problem. In order to demonstrate the wavelet technique, we consider the one-dimensional counterpart of Helmholtz's equation. By comparison with a simple finite difference solution to this problem with periodic boundary conditions, we show how a wavelet technique may be efficiently developed. Dirichlet boundary conditions are then imposed, using the capacitance matrix method described by Proskurowski and Widlund² and others. The convergence rates of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions. Preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method, at least for problems with simple geometries.     

Stress intensity factor equations for branched crack growth

Overload-induced fatigue crack branching is a well-known crack growth retardation or arrest mechanism, which can quantitatively explain such effects even when arguments based on plasticity induced crack closure cannot be applied, e.g. in high R-ratio or in plane strain controlled fatigue crack growth. However, the few results available for branched cracks cannot be used to predict the subsequent crack growth nor account for the delays observed in practice. In this work, specialized finite element (FE) and fatigue life assessment software are used to solve this problem. The crack path and associated stress intensity factors (SIF) of kinked and bifurcated cracks are numerically obtained by the FE program for several angles and branch lengths, and the companion life assessment program is used to estimate the number of delay cycles associated with them. From these results, crack retardation equations are proposed to model the number of delay cycles and the retardation factor along the crack path, allowing for a better understanding of the influence of crack deflection in the propagation life of structural components.     

Two-wavelength phase-shifting interferometry insensitive to the intensity modulation of dual laser diodes

We have constructed two-wavelength phase-shifting interferometry that is insensitive to the intensity changes in interferograms associated with the current variations in two laser-diode (LD) sources by using a newly developed phase-extraction algorithm. The tested phase at a synthetic wavelength can be measured from six interferograms with different phase shifts. The algorithm becomes a simple form for seven interferograms and reduces to a minimum of five phase-shifted data in the proper conditions. We shifted the phases equally in opposite directions to one another by separately varying the stepwise currents in dual LD's on an unbalanced interferometer. The measurement accuracy has been improved compared with that of the two-wavelength four-step method. The phase error caused by the power changes in the dual LD's has been investigated theoretically and experimentally. The experimental results are shown to measure a step object with a 4.6-μm synthetic wavelength.     

Compensation of wavelength dependent image shifts in imaging optical interferometry

In modern interferometers for absolute length measurements, when applying phase stepping interferometry, several different wavelengths are used as light sources, and the interferogram is projected to a CCD-camera array. Such interferometers are equipped with wedged optical components as windows and beam splitters, to prevent additional interferences. The wedged optics causes the position of a test piece within the interferogram related to the camera pixel coordinates to be dependent on the wavelength used. This effect depends on the wedge angles and the thicknesses of optical components as well as on their distances within the interferometer's optical pathway. We give a quantitative analysis and suggest a compensation of this dispersion effect by an additional wedge plate outside the interferometer.     

Real-time heterodyne speckle pattern interferometry using the correlation image sensor

A real-time method for heterodyne speckle pattern interferometry using the correlation image sensor (CIS) is proposed. The CIS demodulates the interference phase of heterodyned speckle wavefronts pixelwise at an ordinary video frame rate. The proposed method neither suffers loss of spatial resolution nor requires a high frame rate. Interferometers for out-of-plane and in-plane deformation are developed with a 200 × 200 pixel CIS camera. Experimental results confirm that the proposed method realizes real-time imaging of a rough-surfaced object under deformation. The average standard deviations of demodulated phase-difference images for the out-of-plane and in-plane interferometers are 0.33 and 0.13 rad, respectively.     

Consistent diagonal mass matrices and finite element equations for one-dimensional problems

The conventional dynamic variational approach and finite element base functions lead to non-diagonal consistent mass matrics which are inappropriate for use with an explicit time integration scheme. In this work, it is shown that if orthogonal base function are used with a mixed variational formulation, then consistant diagonal mass matrices and corresponding sets of spatially discretized field equations are obtained. Although the approach is quite general, the theory is purposely illustrated by a detailed development for one set of base functions. Central difference time integration is incorporated for applications to one-dimensional wave propagation and to Euler-Bernoulli beams. Numerical examples are provided for elastic and elastic-plastic materials.     

Evaluation of the errors in equations for stress intensity factors of elliptical type cracks

A method of evaluation of the integral error in the approximate equations for stress intensity factors based on the energy balance equation is described. A review of the equations for calculation of stress intensity factors for cracks of elliptical, semielliptical, and quarter-elliptical form is given. The boundaries of applicability of the equations are given, the errors in approximation of the corresponding numerical solutions are noted, and the integral errors in the equations for stress intensity factors are also calculated.Institute of Problems of Strength, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Problemy Prochnosti, No. 10, pp. 53–58, October, 1989.     

Parametric equations for T-butt weld toe stress intensity factors

This paper describes the generation of parametric equations for weld toe stress intensity factors. The methodology employed used a two-dimensional finite element analysis to evaluate the ‘crack opening’ stress distribution in the uncracked plane of T-butt geometries. This was then used as input into a dedicated weight function solution for the determination of stress intensity factors. The final parametric equations describe the stress intensity factor distributions for tension and bending as a function of plate thickness, weld attachment width, weld angle, weld root radius, crack length and crack shape. The equations are compared and validated against a wide spectrum of published values and appear by comparison accurate and wide ranging. The validation exercise uncovered situations where present design guidance is unconservative.     

Modulating functions-based method for parameters and source estimation in one-dimensional partial differential equations

In this paper, modulating functions-based method is proposed for estimating space–time-dependent unknowns in one-dimensional partial differential equations. The proposed method simplifies the problem into a system of algebraic equations linear in unknown parameters. The well-posedness of the modulating functions-based solution is proved. The wave and the fifth-order KdV equations are used as examples to show the effectiveness of the proposed method in both noise-free and noisy cases.     

Simple bounds for the stress intensity factors by the method of singular integral equations

The method of singular integral equations has become a classical method for solving plane and antiplane, static and dynamic crack problems in isotropic and anisotropic elasticity, particularly in cases where no closed-form solutions are available. In this paper, very simple methods are suggested for obtaining upper bounds for the stress intensity factors at the crack tips from the corresponding singular integral equation without solving this equation, even numerically, and with very few computations. Naturally, such a simplicity should lead to very conservative bounds and this is really the case. But, clearly, in a lot of practical cases such bounds are sufficient. Numerical results in simple crack problems show the efficiency of the proposed methods.     

Investigation of crack closure in stainless steel by laser interferometry

The effect of strain-induced martensite transformed during fatigue on the fatigue crack propagation rate near ΔK _th, as well as low-cycle fatigue behaviour of three differently heat-treated stainless steels, was investigated. The heat treatments were chosen so that austenite stability during fatigue was different. The crack closure stress during fatigue crack propagation near the ΔK _th region was measured using laser interferometry. The sensitized specimen showed the highest value of closure load ratio (K _cl/K _max), which was considered to be due to the roughness-induced crack closure caused by intergranular facets. The specimen with the lowest austenite stability showing the largest amount of strain-induced martensite during fatigue, showed the highest crack growth rate. The effect of brittle fracture through the harder strain-induced martensite was larger than that of possible transformation-induced crack closure.     

Complex refractive-index measurement based on Fresnel's equations and the uses of heterodyne interferometry

The phase difference between s and p polarization of the light reflected from a material is used for measuring the material's complex refractive index. First, two phase differences that correspond to two different incidence angles are measured by heterodyne interferometry. Then these two phase differences are substituted into Fresnel's equations, and a set of simultaneous equations is obtained. Finally, the equations are solved by use of a personal computer by a numerical analysis technique, and the complex refractive index of the material can be estimated.     

Simple time and space adaptation in one-dimensional evolutionary partial differential equations

A simple technique for time and space adaptation in one-dimensional evolutionary partial differential equations, suggested by Sanz-Serna and Christie,¹ is tested. It is found that the equidistribution of hu″(x_i) greatly improves on the equidistribution of arc-length used by those authors. The time-step control is found to perform poorly in the integration of rough solutions and the reasons for this behaviour are analysed.     

In-plane dynamic speckle interferometry: comparison between a combined speckle interferometry/speckle correlation and an update of the reference image

A common problem during study of, for instance, tensile tests with interferometers is that the sample moves too much so that the speckles decorrelate and no phase information is obtained. Two ways to overcome this problem are compared: a combination of speckle interferometry and speckle correlation and a method in which the reference image is updated during the experiment. The comparison shows that both techniques can be used to measure the deformation of an object even if it is exposed to rigid body motions. Both techniques are applied to measurements of microscale deformation fields of an adhesive joint in a carbon-fiber epoxy composite.