New applications of the method of caustics

The method of caustics is a very powerful technique for measuring the value of the stress-intensity factor. This method, however, cannot be applied to the three-dimensional stress analysis technique of stress-freezing, and it is rather difficult to apply to materials which exhibit small deformations such as ceramics.

In this paper, several new application techniques for caustics are shown. These new applications are: (i) the analysis of the individual values of the optical constants using a simple method based on the theory of caustics, (ii) the analysis of the reason why the caustic pattern cannot be obtained in the frozen-stress model in the immersion liquid by the conventional caustics technique using those optical constants, and the proposition of a new technique to apply the caustic method on such a frozen-stress model, (iii) the caustic coating technique, (iv) the multiplication technique of the caustic, and (v) some examples of the application of caustics to ceramics.     

Optical method of caustics applied in viscoelastic fracture analysis

The optical method of caustics is developed here to study the fracture of viscoelastic materials. By adopting a distribution of viscoelastic stress fields near the crack tip, the method of caustics is used to determine the viscoelastic fracture parameters from the caustic patterns near the crack tip. Two viscoelastic materials are studied. These are PMMA and ternary composites of HDPE/POE-g-MA/CaCO₃. The transmitted and reflective methods of caustics are performed separately to investigate viscoelastic fracture behaviors. The stress intensity factors (SIFs) versus time is determined by a series of shadow spot patterns combined with viscoelastic parameters evaluated by creep tests. In order to understand the viscoelastic fracture mechanisms of HDPE/POE-g-MA/CaCO₃ composites, their fracture surfaces are observed by a Scanning Electron Microscope (SEM). The results indicate that the method of caustics can be used to characterize the fracture behaviors of viscoelastic materials and further to optimize the design of polymer composites.     

The influence of singular field dominance on the caustic methods

The caustic method as applied to fracture mechanics for measuring the stress intensity factor, contains several approximations which limit its applicability and accuracy. The accuracy of the caustic method for the determination of the stress intensity factor is based on the restriction that the K-dominant singular field is fully established at which the experimental data are taken. In this paper, we attempt to identify the regions in which experimental measurements by caustic methods can be performed with confidence. The reliability of the method of caustics applied to the static problem and the case of a stationary crack subjected to dynamic loading are investigated in detail. The results demonstrate that significant deviation can occur in the determination of the stress intensity factor from shadow spot measurements. Conclusions regarding the method of caustics in the determination of the stress intensity factor are extracted from these results.     

Shear wave caustics and the shadows of surface breaking cracks

A refinement to the detection of defects in rods and pipes by means of their ultrasonic shadows is presented. It has been previously shown that in standard non-destructive test configurations ultrasonic waves inside cylindrical test objects form caustics, surfaces of high sound intensity. Here it is demonstrated that when a crack edge crosses a caustic the diffraction of sound around the defect is enhanced, which allows the point at which this intersection occurs to be detected. Because the caustic is a well-defined geometric entity the position of the crack edge is now known. This effect presents a novel method of sizing defects that extend in from the surface. Experiments were performed in water immersion using 5 MHz sound. The test specimens were 76 mm diameter aluminium cylinders into which slots were cut to act as artificial defects.     

Boundary-limited thermal conduction of crystalline rods oriented near phonon-focusing caustics

We apply the Casimir model for boundary-limited heat conduction to single-crystal rods oriented near phonon-focusing caustics. We show that rods with axes close to the direction of an external conical refraction caustic, a highly degenerate caustic that exists for certain hexagonal crystals, exhibit a thermal conductivity that diverges logarithmically on approaching the caustic. For rods with axes close to the directions of the more generic fold and cusp caustics, the conductivity remains finite, but displays singular behavior with a 1/2- or 1/3-power law falloff with angular deviation from the caustic. Moreover, in the direction of a fold caustic, the Casimir conduction is not necessarily a maximum. Numerical results are presented for zinc, with the quasi-transverse branch providing examples of the external conical refraction and fold caustics, and in a certain sense, also the cusp caustic.     

The optical method of caustics

An overview of the basic principles of the optical method of caustics for the determination of stress intensity factors in crack problems is presented. The method is based on the assumption that the state of stress in the neighborhood of the crack tip is plane stress. However, the state of stress changes from plane strain very close to the tip to plane stress at a critical distance from the tip through an intermediate region where the stress field is three-dimensional. The caustic is the image of the so-called initial curve on the specimen and, therefore, depends on the state of stress along the initial curve. For the determination of stress intensity factors the values of the stress-optical constants are needed. These values depend strongly on the state of stress being plane stress, plane strain or three-dimensional. This complicated the experimental determination of stress intensity factors. For the characterization of the state of affairs near the crack tip a phenomenological triaxiality factor is introduced. A methodology based on the use of optically birefringent materials is developed for the determination of stress intensity factors without paying attention to the location of the initial curve in the plane stress, plane strain or three-dimensional region. Finally, a comparison of the methods of photoelasticity and caustics takes place, and the potentialities and limitations of both methods for the solution of crack problems are explored.     

Efficient Formulas for the Maslov Canonical Operator near a Simple Caustic

We revive well-known old formulas for the asymptotics of rapidly oscillating integrals with two nearly coincident saddle points and present an efficient approach to the computation of functions represented by Maslov’s canonical operator near a simple caustic in an algorithmic fashion convenient for implementation on software like Wolfram Mathematica. The exposition is elementary and assumes that no preliminary information about the canonical operator is be known to the reader.     

Dynamic optical visualization on the interaction between propagating crack and stationary crack

Dynamic interactions between the propagating crack and the static crack in PMMA material are studied by combining high-speed Schardin camera with optical caustic method. A series of dynamic optical bifocal patterns (the specimen-focused image and the off-focused image) around the propagating crack tip and the static crack tip are recorded for PMMA thin strip which contains two collinear-edge-cracks subjected to tensile loading, the variations of the caustic diameter and the distortion of the caustic shape are revealed due to the influence of local stress singularity at the crack tip. Interactions between the moving crack and the static crack are analyzed by means of the evolution of dynamic fracture parameters. The influence of crack interaction on fracture parameters is discussed based on both a K-dominance assumption and a higher order transient crack-tip expansion. These results will be useful to the evaluation of dynamic properties and the design of structures in the cracked polymer material.     

Efficient Generation of Customized Caustics: Nondiffracting Deltoid and Pentagram Beams

This study puts forward a powerful new strategy that allows highly efficient generation of the shaping transverse nondiffracting caustics that concentrate around the expected curve. First, under the condition of stationary phase approximation, as the typical example of caustic beams, the analytical formula of nondiffracting deltoid and pentagram caustics is theoretically derived. Based on the obtained analytical formula, the desired caustics of deltoid and pentagram are numerically simulated. Hence, their optical shape and the propagation characteristics of nondiffracting caustics can be investigated. Then, based on the given generating mechanism, the corresponding experimental system to experimentally explore the nondiffracting deltoid and pentagram caustics with good self-healing properties is constructed. Compared with the previous generating methods, the proposed method has a high light energy utilization.     

Generalized ray tracing and the caustic surface

The caustic surface can be a useful tool for diagnosing and correcting image errors in the process of designing an optical system. However, because of their complicated structure, caustics are hard to visualize. Therefore, a computer programme has been formulated to plot the two sheets of a caustic separately, each as a stereo pair. Results obtained for a simple plano-convex lens are presented.     

Lattice dynamics of InSb from phonon imaging

The ballistic heat flux radiating from a point source of heat in a cold crystal displays a complex pattern of caustics due to phonon focusing. The caustics correspond to folds in the elastic-wave surface of the crystal. The pattern of caustics is independent of phonon frequency unless the phonon wavelength is comparable to the lattice spacing; i.e., for values of wavevectork approaching /a. We have measured both the shift in the caustic pattern (angular dispersion) and the increasing time-of-flight (velocity dispersion) for ballistic phonons in InSb with wavevectors up to 40% of the Brillouin-zone boundary. Comparison with existing lattice-dynamics models favors the Bond Charge Model (BCM). The phonon-imaging method gives information about theshapes of the wave surfaces which is complementary to the dispersion curves measured only along symmetry directions by neutron scattering.     

Theoretical investigations on optical caustics of spherical microresonators: Analytical expressions of caustics and their asymptotic behaviors,computational simulations

This paper introduces analytical expressions of optical caustics r₁ and r₂ in micro-resonators (respectively called inner and outer caustics in micro-optics) in terms of radial and azimutal modal numbers n and l mutually. Such a theoretical study focuses on spherical micro-resonators and is carried out according to two complementary ways: Electromagnetic (E.M) approaches based respectively on an eigenvector problem resolution called modal field in integrated optics and an analogy with Quantum Mechanics (Q.M) using the notions of potential and geometric eikonal principle. Thus, using quantization and phase considerations, fundamental new formulation of caustics, that depend on the two previous quantum modal numbers simultaneously, have been shaped and confronted with numerical and graphical simulations. Both ways lead to results in good agreement (correlation coefficient superior to 0.99) and allow us to specify the evolution of such caustics according to each modal number of the quantification; such an analytical caustics formulation involves a specific function f whose study is expanded for various values of n ranging from 1 to 500. Moreover a whole study of asymptotic behaviors of f-function and its inverse f^? 1 on specific n and l values, directly related to the previous caustics expressions, is conducted, allowing the expansion of new analytical expressions of caustics in various limit cases and asymptotic behaviors. Then, the evolution of the positions of caustics in terms of modal numbers is interpreted, and the physical meanings of optical properties such as quasi-resonance, position and confinement of field with resonance, and associated electromagnetic fields are discussed.     

Average number and probability distribution of caustics in randomly inhomogeneous medium and behind a random phase screen

Formulas are derived for the average cross-sectional caustic density and the probability density of the distances to caustics behind a random phase screen. The constancy of the average number of caustics at great distances behind the screen as the probability density approaches zero is explained.Nizhny Novgorod Architecture and Construction Academy. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 4, pp. 471–478, April, 1994.     

Duality between phase vortices and Argand-plane caustics

Singularities in the mappings of complex two-dimensional vortical wavefields to Argand space are used here to investigate the duality between vortices in the physical field and caustic-like structures in the Argand plane. Such map singularities are exemplified via a method of optical vortex generation based on low-pass filtering of two-dimensional noise. The presence of Argand-plane fold caustics crossing the Argand-plane origin is tied to changes in vortex numbers in the physical wavefield. Increasing the low-pass filter cutoff leads to higher-order Argand-plane caustic singularities, such as the hyperbolic umbilic catastrophe. This work presents a duality between vortices and caustics, explores the Argand plane as a tool in analysing optical wavefields, and provides a new environment for studying catastrophes.     

A general method for automated analysis of caustics

In most published work on the optical method of caustics, experimental results were obtained by manually measuring some characteristic dimensions of the shadow pattern and, therefore, used only a limited amount of data from the caustic. However, some previous work has been done using digital image processing and light intensity plots combined with least-squares techniques to analyse the shape of a caustic obtained from a particular problem. All of these methods have been developed to solve only one specific problem. This paper highlights the essential guidelines of an automated method, which has the novelty of generalising the whole analysis, such that only the parametric equations of a particular problem are required to solve it. In the proposed method the image is digitised, the central line of the caustic pattern is identified, and the parameters in the mapping equation of the caustic are determined. The method has been validated, and the aim of this paper is to describe the key features of the method, i.e. the optical arrangement, digital image processing, and numerical analysis.     

Caustics method in dynamic fracture problem of orthotropic materials

Caustics method is a powerful optical technique in fracture mechanics because of its high sensitivity to stress gradients. In this paper, it is applied to resolve dynamic fracture problems in orthotropic composites. Considering most orthotropic materials are opaque, reflective caustics method is derived here by combining the fundamental principle of caustics method with the mechanical properties of orthotropic materials. Meanwhile, corresponding experiments are carried out for typical glass fiber-reinforced composites, where mode I and mixed-mode fracture states are taken into account. By recording and analyzing shadow spot patterns during the crack propagation process carefully, crack onset time, dynamic fracture toughness and crack growth velocity of orthotropic composite are determined. These results will be useful to evaluate the dynamic fracture properties of composites and further to optimize their designs.     

Experimental investigation of the travel-time variance of an acoustic wave propagating through the grid-generated turbulence

An experimental technique for the investigation of the behaviour of acoustic wave propagation through a turbulent medium is discussed. The present study utilizes the ultrasonic travel-time technique to diagnose a grid-generated turbulence. Travel-time variance is studied versus mean flow velocity, travel distance and outer turbulence scale. The effect of thermal fluctuations, which result in fluctuations of sound speed, is studied using a heated-grid experiment. Experimental data obtained using ultrasonic technique confirm numerical and theoretical predictions of nonlinear increase of the travel-time variance with propagation distance, which could be connected to the occurrence of caustics. The effect of turbulent intensity on the travel-time variance and appearance of caustics is studied. It is demonstrated experimentally that the higher turbulence intensity leads to the shorter distance, at which the first caustic occurs. The probability density for caustics appearance is analysed against the measured wave amplitude fluctuations. The analysis reveals that the region of high-amplitude fluctuations corresponds to the region where the probability of formation of random caustics differs from zero. Experimental results are in very good agreement with theoretical and numerical predictions.     

Focusing of an electromagnetic plane wave into a uniaxial crystal by a Wood lens

High frequency fields, refracted by a geometry containing a Wood lens placed at a certain distance from a planar uniaxial interface, are derived by using Maslov’s method. The geometrical optics approximation generally valid for high frequency fields fails in the vicinity of a caustic. Maslov’s method is a systematic procedure for predicting the field in the caustic region, combining the simplicity of the ray and the generality of the transform method. Numerical computations are made for the field pattern around the caustic by using Maslov’s method. The results are found to be in good agreement with those obtained using Kirchhoff’s approximation.      

Crossing sum rules applied to inequalities on ππ S and P waves

We use the crossing conditions for the physical absorptive parts to improve rigorous constraints of the Martin type for ππ S and P waves in the unphysical region 0 ? s ? 4. In particular, (i) inequaltities for the scattering lengths in terms of the amplitudes at s = 0, (ii) inequalities relating only I = 0 and I = 2 S waves, (iii) the inequalities of Auberson, Brander, Mahoux and Martin are considered. Though some of these new inequalities seem to be very tight, they are satisfied in nearly all tested models.