Undular bores generated by fracture

Undular bores, or dispersive shock waves, are nonstationary waves propagating as oscillatory transitions between two basic states, in which the oscillatory structure gradually expands and grows in amplitude with distance traveled. In this work we report an important mechanism of generation of nonlinear dispersive shock waves in solids. We demonstrate, using high-speed pointwise photoelasticity, the generation of undular bores in solid (polymethylmethacrylate) prestrained bars by natural and induced tensile fracture. For the distances relevant to our experiments, the viscoelastic extended Korteweg-de Vries equation is shown to provide very good agreement with the key observed experimental features for suitable choice of material parameters, while some local features at the front of the bore are also captured reasonably well by the linearization near the nonzero prestrain level. The experimental and theoretical approaches presented open avenues and analytical tools for the study and applications of dispersive shock waves in solids.

Obtaining the Transfer Function of optical instruments using large calibrated reference objects

It has been suggested recently that the Transfer Function of instruments such as Coherence Scanning Interferometers could be measured via a single measurement of a large spherical artefact [Appl. Opt.53(8), 1554-1563 (2014)]. In the current paper we present analytical solutions for the Fourier transform of the 'foil' model used in this technique, which thus avoids the artefacts resulting from the numerical approach used earlier. The Fourier transform of a partial spherical shell is found to contain points of zero amplitude for spatial frequencies that lie within the Transfer Function. This implies that the Transfer Function is unmeasurable at these points when a single spherical artefact is used, in situations where the foil model is a valid representation of the physical system. We propose extensions to the method to address this issue.

Nuclear magnetic resonance measurements of velocity distributions in an ultrasonically vibrated granular bed

We report the results of nuclear magnetic resonance imaging experiments on granular beds of mustard grains fluidized by vertical vibration at ultrasonic frequencies. The variation of both granular temperature and packing fraction with height was measured within the three-dimensional cell for a range of vibration frequencies, amplitudes and numbers of grains. Small increases in vibration frequency were found--contrary to the predictions of classical 'hard-sphere' expressions for the energy flux through a vibrating boundary--to result in dramatic reductions in granular temperature. Numerical simulations of the grain-wall interactions, using experimentally determined Hertzian contact stiffness coefficients, showed that energy flux drops significantly as the vibration period approaches the grain-wall contact time. The experiments thus demonstrate the need for new models for 'soft-sphere' boundary conditions at ultrasonic frequencies.

Simultaneous wavenumber measurement and coherence detection using temporal phase unwrapping

Wavelength scanning interferometry and swept-source optical coherence tomography require accurate measurement of time-varying laser wavenumber changes. We describe here a method based on recording interferograms of multiple wedges to provide simultaneously high wavenumber resolution and immunity to the ambiguities caused by large wavenumber jumps. All the data required to compute a wavenumber shift are provided in a single image, thereby allowing dynamic wavenumber monitoring. In addition, loss of coherence of the laser light is detected automatically. The paper gives details of the analysis algorithms that are based on phase detection by a two-dimensional Fourier transform method followed by temporal phase unwrapping and correction for optical dispersion in the wedges. A simple but robust method to determine the wedge thicknesses, which allows the use of low-cost optical components, is also described. The method is illustrated with experimental data from a Ti:sapphire tunable laser, including independent wavenumber measurements with a commercial wavemeter. A root mean square (rms) difference in measured wavenumber shift between the two of ~4 m⁻¹ has been achieved, equivalent to an rms wavelength shift error of ~0.4 pm.

Stress measurement in East Asian lacquer thin films owing to changes in relative humidity using phase-shifting interferometry

In this paper, we address the behaviour of lacquer coatings similar to that found on the Mazarin Chest, an important Japanese lacquerware artefact currently held by the Victoria and Albert Museum (V&A) in London. The response of Japanese lacquer ( urushi ) to changes in environmental conditions was investigated by examining the deflection of a glass substrate coated with a thin film of urushi subjected to changes of humidity. This deflection, measured using phase-shifting interferometry, was then used to determine the two in-plane hygral stress components. Results were compared for two sample conditioning regimes—subjected to intense UV ageing and no ageing—each at a range of relative humidity (RH) steps. The changes in humidity were found to cause rapid stress changes in the lacquers, which then relax over much longer time scales. A simple one-dimensional model of the moisture transport and the stress development is shown to be effective in describing the response of the material to changes in environmental RH.

Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms

Phase contrast magnetic resonance velocity imaging is a powerful technique for quantitative in vivo blood flow measurement. Current practice normally involves restricting the sensitivity of the technique so as to avoid the problem of the measured phase being 'wrapped' onto the range -pi to +pi. However, as a result, dynamic range and signal-to-noise ratio are sacrificed. Alternatively, the true phase values can be estimated by a phase unwrapping process which consists of adding integral multiples of 2pi to the measured wrapped phase values. In the presence of noise and data undersampling, the phase unwrapping problem becomes non-trivial. In this paper, we investigate the performance of three different phase unwrapping algorithms when applied to three-dimensional (two spatial axes and one time axis) phase contrast datasets. A simple one-dimensional temporal unwrapping algorithm, a more complex and robust three-dimensional unwrapping algorithm and a novel velocity encoding unwrapping algorithm which involves unwrapping along a fourth dimension (the 'velocity encoding' direction) are discussed, and results from the three are presented and compared. It is shown that compared to the traditional approach, both dynamic range and signal-to-noise ratio can be increased by a factor of up to five times, which demonstrates considerable promise for a possible eventual clinical implementation. The results are also of direct relevance to users of any other technique delivering time-varying two-dimensional phase images, such as dynamic speckle interferometry and synthetic aperture radar.

Convection in vibrated annular granular beds

The response to vibration of a granular bed, consisting of a standard cylindrical geometry but with the addition of a dissipative cylindrical inner wall, has been investigated both experimentally (using positron emission particle tracking) and numerically (using hard sphere molecular dynamics simulation). The packing fraction profiles and granular temperature distributions (in both vertical and horizontal directions) were determined as a function of height and distance from the axis. The two sets of results were in reasonable agreement. The molecular dynamics simulations were used to explore the behavior of the granular bed in the inner wall-outer wall coefficient of restitution phase space. It was observed that one could control the direction of the toroidal convection rolls by manipulating the relative dissipation at the inner and outer walls via the coefficients of restitution, and with several layers of grains it was seen that double convection rolls could also be formed, a result that was subsequently confirmed experimentally.

Numerical solution of the Smoluchowski equation for a vibrofluidized granular bed

A stochastic approach, similar to that used to describe Brownian motion, was used to model the displacement probability of grains in a three-dimensional vibrofluidized granular bed. As neither an analytical description nor measurements of the diffusion coefficients were available, the governing partial differential equation, namely, the Smoluchowski equation, was solved numerically using an iterative procedure, modifying the granular temperature profile at each step. The results of this stochastic model were compared to experimental measurements of the displacement probability density made using positron emission particle tracking. The results indicate that methods based on hard elastic systems such as the Smoluchowski equation are appropriate to granular systems, particularly over timescales greater than the mean collision time.

Three-dimensional noise-immune phase unwrapping algorithm

The classical problem of phase unwrapping in two dimensions, that of how to create a path-independent unwrapped map, is extended to the case of a three-dimensional phase distribution. Whereas in two dimensions the path dependence problem arises from isolated phase singularity points, in three dimensions the phase singularities are shown to form closed loops in space. A closed path that links one such loop will cross a nonzero number of phase discontinuities. In two dimensions, path independence is achieved when branch-cut lines are placed between singular points of opposite sign; an equivalent path-independent algorithm for three dimensions is developed that places branch-cut surfaces so as to prevent unwrapping through the phase singularity loops. The placing of the cuts is determined uniquely by the phase data, which contrasts with the two-dimensional case for which there are many possible ways in which to pair up the singular points. The performance of the new algorithm is demonstrated on three-dimensional phase data from a high-speed phase-shifting speckle pattern interferometer.

Granular temperature profiles in three-dimensional vibrofluidized granular beds

The motion of grains in a three-dimensional vibrofluidized granular bed has been measured using the technique of positron emission particle tracking, to provide three-dimensional packing fraction and granular temperature distributions. The mean square fluctuation velocity about the mean was calculated through analysis of the short time mean squared displacement behavior, allowing measurement of the granular temperature at packing fractions of up to eta approximately 0.15. The scaling relationship between the granular temperature, the number of layers of grains, and the base velocity was determined. Deviations between the observed scaling exponents and those predicted by recent theories are attributed to the influence of dissipative grain-sidewall collisions.

Vibration-induced phase errors in high-speed phase-shifting speckle-pattern interferometry

We present results from numerical simulations of a dynamic phase-shifting speckle interferometer used in the presence of mechanical vibrations. The simulation is based on a detailed mathematical model of the system, which is used to predict the expected frequency response of the rms measurement error, in the time-varying phase difference maps, as a result of vibration. The performance of different phase-shifting algorithms is studied over a range of vibrational frequencies. Phase-difference evaluation is performed by means of temporal phase shifting and temporal phase unwrapping. It is demonstrated that longer sampling windows and higher framing rates are preferred in order to reduce the phase-change error that is due to vibration. A numerical criterion for an upper limit on the length of time window for the phase-shifting algorithm is also proposed. The numerical results are finally compared with experimental data, acquired with a phase-shifting speckle interferometer of 1000 frames/s.

Convection in highly fluidized three-dimensional granular beds

Free, buoyancy-driven convection has been observed experimentally in three-dimensional highly fluidized granular flows for the first time. Positron emission particle tracking was used to determine the position of a tracer grain in a vibrofluidized bed, from which packing fraction distributions as well as the velocity fields could be determined. The convection rolls, although small compared to the magnitude of velocity fluctuations (<5%), were consistently observed for a range of grain numbers and shaker amplitudes. Density variations are a signature of free convection and, with negative temperature gradients also present, were interpreted as the mechanism by which the convection rolls were initiated.

Single-particle motion in three-dimensional vibrofluidized granular beds

A technique to probe the interior of three-dimensional dynamic granular systems is presented. Positron emission particle tracking (PEPT) allows a single tracer particle to be followed around a three dimensional vibrofluidized granular bed for periods up to six hours. At present the technique is able to resolve the position of the grains to +/-4 mm, with an average temporal resolution of about 7 ms. Packing fraction profiles are calculated by making use of the ergodicity of the system, and granular temperature profiles are obtained, in the dilute case, from the short time behavior of the mean squared displacement. At longer times, the mean squared displacement shows a range of behavior which can be explained by the presence of strong gradients in the packing fraction. Convection currents were observed, but were sufficiently small in magnitude to be ignored during the analysis of grain motion. The system was modeled using the Smoluchowski equation, which was solved numerically, and the results compared with the experimentally determined displacement probability density functions. Good agreement between experiment and numerical results was achieved using Brownian motion relationships modified to accommodate differences between granular systems and thermal systems.

Self-diffusion of grains in a two-dimensional vibrofluidized bed

The analogy of vibrofluidized granular beds with a thermal gas of hard discs has been tested. Analysis of the mean squared displacement behavior of the grains allowed comparison of the measured diffusion with the predicted value at a particular combination of granular temperature and packing fraction. High speed photography, image analysis, and particle tracking software enabled accurate location of the grains. Appropriate analysis of the three mean squared displacement regimes, ballistic, diffusive, and crossover between the two extremes, allowed both the diffusion coefficient and the granular temperature to be measured at the same packing fraction. Broad agreement between Chapman-Enskog theory relating temperature to self-diffusion and observation was observed up to packing fractions of eta approximately equal to 0.7. At higher packing fractions the grains showed evidence of caging and jump diffusion, with the observed diffusion rapidly diverging from that predicted by theory. Measurement of self-diffusion coefficients and subsequent use of kinetic theory was found to be an accurate method to determine the granular temperature for intermediate packing fractions (eta=0.4-0.6), and would be particularly suitable for those situations where the time resolution of the experimental facility is insufficient to resolve the speed of the grain between collisions.

Phase-shifted dynamic speckle pattern interferometry at 1 kHz

We describe a phase-shifting out-of-plane speckle interferometer operating at 1 kHz for studying dynamic events. The system is based on a Pockels cell that is synchronized to a high-speed video camera to ensure that the phase shifting occurs between frames. Phase extraction is performed by use of a standard four-frame algorithm, and temporal phase unwrapping allows sequences of several hundred absolute (rather than relative) displacement maps to be obtained fully automatically. The maximum theoretical surface velocity of 67 microm s(-1) is a factor of 40 greater than can be achieved with a speckle interferometer based on a conventional video camera. We test the system using a target that is displaced with constant speed in a direction normal to its surface by means of a piezoelectric transducer. The system's performance in a practical situation is illustrated with measurements on a thin plate undergoing out-of-plane deformation.

Simple model for image-plane polychromatic speckle contrast

A simple geometrical model was developed for calculation of the contrast of a polychromatic image-plane speckle pattern from a source of light with high spatial coherence. It is based on counting the number of independent speckle patterns that contribute to a given point in the image plane. This results in a simple equation for the contrast as a function of imaging geometry; relative orientation of the projection direction, observation direction, and specimen normal; bandwidth of the light source; and surface roughness. Its validity was established by comparison with an exact solution: rms errors in the calculated contrast were only 0.033 over a wide range of parameter values likely to be encountered in practice.

Temporal phase unwrapping: application to surface profiling of discontinuous objects

The recently proposed technique of temporal phase unwrapping has been used to analyze the phase maps from a projected-fringe phase-shifting surface profilometer. A sequence of maps is acquired while the fringe pitch is changed; the phase at each pixel is then unwrapped over time independently of the other pixels in the image to provide an absolute measure of surface height. The main advantage is that objects containing height discontinuities are profiled as easily as smooth ones. This contrasts with the conventional spatial phase-unwrapping approach for which the phase jump across a height discontinuity is indeterminate to an integral multiple of 2pi. The error in height is shown to decrease inversely with the number of phase maps used.

Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm

An algorithm for unwrapping noisy phase maps by means of branch cuts has been proposed recently. These cuts join discontinuity sources that mark the beginning or end of a 2π phase discontinuity. After the placement of branch cuts, the unwrapped phase map is unique and independent of the unwrapping route. We show how a minimum-cost-matching graph-theory method can be used to find the set of cuts that has the global minimum of total cut length, in time approximately proportional to the square of the number of sources. The method enables one to unwrap unfiltered speckle-interferometry phase maps at higher source densities (0.1 sources pixel(-1)) than any previous branch-cut placement algorithm.

Improved noise-immune phase-unwrapping algorithm

An algorithm for unwrapping noisy phase maps has recently been proposed, based on the identification of discontinuity sources that mark the start or end of a 2π phase discontinuity. Branch cuts between sources act as barriers to unwrapping, resulting in a unique phase map that is independent of the unwrapping route. We investigate four methods for optimizing the placement of the cuts. A modified nearest neighbor approach is found to be the most successful and can reliably unwrap unfiltered speckle-interferometry phase maps with discontinuity source densities of 0.05 sources pixel(-1).

Temporal phase-unwrapping algorithm for automated interferogram analysis

A new algorithm is proposed for unwrapping interferometric phase maps. Existing algorithms search the two-dimensional spatial domain for 2π discontinuities: only one phase map is required, but phase errors can propagate outward from regions of high noise, corrupting the rest of the image. An alternative approach based on one-dimensional unwrapping along the time axis is proposed. It is applicable to an important subclass of interferometry applications, in which a sequence of incremental phase maps can be obtained leading up to the final phase-difference map of interest. A particular example is quasi-static deformation analysis. The main advantages are (i) it is inherently simple, (ii) phase errors are constrained within the high-noise regions, and (iii) phase maps containing global discontinuities are unwrapped correctly, provided the positions of the discontinuities remain fixed with time. The possibility of real-time phase unwrapping is also discussed.

Parallel processing system for rapid analysis of speckle-photography and particle-image-velocimetry data

An automated system has been constructed to process double-exposure speckle-photography and particle-image-velocimetry images. A 3 × 3 array of laser beams probes the photograph, forming nine fringe patterns in parallel; these are then analyzed sequentially by digital computer and the use of a two-dimensional Fourier-transform method. Results are presented showing that the random errors in the measured displacements from such a system approach the expected speckle-noise-limited performance, with a total analysis time per displacement vector of 160 ms.

Maximum-likelihood analysis of speckle photography fringe patterns

A new method for analyzing the Young's fringe patterns from a double-exposure speckle photograph is proposed based on maximum-likelihood estimation. Unlike previous linear algorithms, which rely on Fourier spectral analysis, the method permits knowledge of the speckle-noise statistics (in particular, that the noise is multiplicative rather than additive) to be incorporated in a systematic way. As a result, random errors in the measured displacement components are reduced, in the case of good visibility fringe patterns by a factor of up to 6. The proposed method is also applicable to the general problem of measuring the spatial frequency components of a two-dimensional sinusoid in the presence of signaldependent noise.

Speckle interferometry: noise reduction by correlation fringe averaging

A method for noise reduction in double-exposure speckle interferometry is proposed, based on averaging independent spatially filtered correlation fringe patterns.

Speckle photography fringe analysis: effect of imaging geometry on displacement errors

Random errors in the measured displacement components from double-exposure speckle photographs recorded through a rectangular aperture are considered both by numerical simulations and by simple dimensional analysis.

Speckle photography fringe analysis: assessment of current algorithms

The accuracy and reliability of four fringe analysis algorithms have been assessed, using Young's fringe patterns calculated from computer generated speckle patterns. The 1-D integration, 1-D autocorrelation, 2-D Fourier transform, and 2-D Walsh transform methods all estimated the two in-plane displacement components with negligible systematic error; the random errors were found to vary in proportion to sigma(2)/VR (speckle diameter = sigma, fringe visibility = V, and radius of laser probe beam = R), with the lowest errors given by the 2-D Fourier transform method.

Noise-immune phase unwrapping algorithm

A new phase unwrapping algorithm is proposed which combines noise immunity with computational efficiency. It is based on the requirement that the unwrapped map should be independent of the route by which unwrapping takes place.

Speckle photography fringe analysis by the Walsh transform

Two-dimensional Walsh spectral analysis is presented as a new method of numerically processing the Young's fringes diffraction pattern from a double-exposure speckle photograph. The Walsh spectrum of the fringes is more complex than the Fourier spectrum but can be interpreted reliably by further cross correlation of the fringes with square waves. Compared with Fourier spectral analysis, the new technique gives results of almost identical accuracy but with substantially reduced computational effort. The ideas presented have relevance to the general problem of detecting and accurately determining the frequency components of a 2-D sinusoid in the presence of noise.