焦散线法及其与应力“解冻”相结合的应用

详细地阐述了焦散线法原理，对焦散线的形成作了说明，简述了焦散线法在平面问题中的应用，结合笔者研究工作介绍了将焦散线法拓广应用于三维问题的关键技术和实例。     

On the System of Hamilton–Jacobi and Transport Equations Arising in Geometrical Optics

Abstract

In the present article, we study the system of eikonal and transport equations arising in geometrical optics. The mathematical analysis is performed by using the suitable notion of solution, i.e., the viscosity solution for the Hamilton–Jacobi equation and the measure solution for the transport equation defined via the generalized Filippov characteristics. We study the stability as well as the geometry of the solution to the system.     

The effects of surface topography on the method of caustics

An investigation was performed to establish the effects of surface topography on the determination of stress intensity factors of opaque components using the method of caustics. Theoretical predictions of caustics produced by curved surfaces were generated and validated by experiment. The errors introduced for stress intensity factors determined at various radii of curvature were assessed. It was found that relatively small surface curvatures caused significant errors in the stress intensity factor. These errors were affected by the initial radius, load and material properties.     

Rayleigh wave propagation in curved waveguides

A JWKB asymptotic expansion describing inplane elastic waves is used to approximate a Rayleigh-like wave guided within a curved elastic waveguide whose curvature is small and changes slowly over a wavelength. The two lowest eigenmodes in a curved guide, taken together, constitute the Rayleigh-like wave. It is shown that this wave lies in the shadows of four, closely spaced, virtual caustics, two caustics per constituent eigenmode. If the curvature becomes too large one or more of the caustics ceases to be virtual and enters the guide after which a Rayleigh-like wave cannot propagate. The overall disturbance is shown to have an amplitude that is modulated because the wavenumbers of the constituent eigenmodes differ by a small amount. Moreover, the disturbance is shown to propagate with a wavenumber that, to leading order, has a linear dependence on the curvature causing the phase to be modulated, as well. Passing from a thin guide to a very thick one suppresses the amplitude modulation, making the phase modulation evident. Propagation into an environment of increasing curvature, for both thin and thick, shallowly curved guides is studied so that the modulations may be observed.     

Statistical models for spatial patterns of heavy particles in turbulence

The dynamics of heavy particles suspended in turbulent flows is of fundamental importance for a wide range of questions in astrophysics, atmospheric physics, oceanography, and technology. Laboratory experiments and numerical simulations have demonstrated that heavy particles respond in intricate ways to turbulent fluctuations of the carrying fluid: non-interacting particles may cluster together and form spatial patterns even though the fluid is incompressible, and the relative speeds of nearby particles can fluctuate strongly. Both phenomena depend sensitively on the parameters of the system. This parameter dependence is difficult to model from first principles since turbulence plays an essential role. Laboratory experiments are also very difficult, precisely since they must refer to a turbulent environment. But in recent years it has become clear that important aspects of the dynamics of heavy particles in turbulence can be understood in terms of statistical models where the turbulent fluctuations are approximated by Gaussian random functions with appropriate correlation functions. In this review, we summarise how such statistical-model calculations have led to a detailed understanding of the factors that determine heavy-particle dynamics in turbulence. We concentrate on spatial clustering of heavy particles in turbulence. This is an important question because spatial clustering affects the collision rate between the particles and thus the long-term fate of the system.     

Caustics,pseudocaustics and the related illuminated and dark regions with the computational method of quantifier elimination

The method of caustics is a powerful experimental method in elasticity and particularly in fracture mechanics for crack problems. The related method of pseudocaustics is also of interest. Here we apply the computational method of quantifier elimination implemented in the computer algebra system Mathematica in order to determine (i) the non-parametric equation and two properties of the caustic at a crack tip and especially (ii) the illuminated and the dark regions related to caustics and pseudocaustics in plane elasticity and plate problems. The present computations concern: (i) The derivation of the non-parametric equation of the classical caustic about a crack tip through the elimination of the parameter involved (here the polar angle) as well as two geometrical properties of this caustic. (ii) The derivation of the inequalities defining the illuminated region on the screen in the problem of an elastic half-plane loaded normally by a concentrated load with the boundary of this illuminated region related to some extent to the caustic formed. (iii) Similarly for the problem of a clamped circular plate under a uniform loading with respect to the caustic and the pseudocaustic formed. (iv) Analogously for the problem of an equilateral triangular plate loaded by uniformly distributed moments along its whole boundary, which defines the related pseudocaustic. (v) The determination of quantities of interest in mechanics from the obtained caustics or pseudocaustics. The kind of computations in the applications (ii) to (iv), i.e. the derivation of inequalities defining the illuminated region on the screen, seems to be completely new independently of the use here of the method of quantifier elimination. Additional applications are also possible, but some of them require the expansion of the present somewhat limited power of the quantifier elimination algorithms in Mathematica. This is expected to take place in the future.     

Experimental studies on dynamic fracture behavior of thin plates with parallel single edge cracks

Dynamic fracture behavior of polymer PMMA thin plates with three- and four-parallel edge cracks was studied by means of the method of caustics in combination with a high-speed Schardin camera. A series of dynamic caustic patterns surrounding the crack tip and fracture path of the specimen were recorded simultaneously by two types of focused images. Some dynamic fracture parameters such as the dynamic stress intensity factor, crack velocity and acceleration were determined. The evolution of dynamic stress intensity factors on the parallel edge cracks, due to the dynamic unloading effect, was analyzed from the viewpoint of the release of elastic strain energy.     

Influence of empty hole on crack running in PMMA plate under dynamic loading

The influence of original prefabricated empty hole on dynamic crack propagation was studied. The dynamic behavior of crack propagation due to the impacting of drop hammer on the defective PMMA medium of manufactured hole defect (different extent of left shift, respectively, for L = 110 mm, L = 111.25 mm and L = 112.5 mm) was conducted using the test system of digital laser dynamic caustics (DLDC). The stress intensity factor and velocity at the running crack tip were analyzed. Moreover, the recently developed distinct lattice spring model (DLSM) was validated using the experimental data. It was found that the numerical modeling can reproduce the experimentally observed phenomena. Combining the experimental and numerical results, it can be concluded that the influence of empty hole defects of brittle materials on dynamic fracturing is great.     

Comparison of caustics and the strain gage method for measuring mode I stress intensity factor of PMMA material

Both caustics and strain gage methods are shown to be effective in the determination of mode I stress intensity factor (SIF) at the crack tip. However, there is a lack of investigation with regard to the quantitative comparison of the two methods. In this paper, dynamic three-point bending tests were performed on polymethyl methacrylate (PMMA) specimens using the caustic and the strain gage methods simultaneously. The SIFs calculated by these two methods were analyzed and compared. The results show that the deviations of the SIF values between caustics and the strain gage method are slightly more significant when the orientation of the gage is obtuse rather than acute. Furthermore, this deviation increases with the gage mounted closer to the edge of the specimen when the orientation is acute. In addition, the whole cracking process could be clearly recorded by the caustic method, while the time-evolution of SIF can be roughly obtained with a limited number of strain gages during the crack propagation.     

Dynamic crack growth characteristics of half-circular disk PMMA specimens with prefabricated penetration defects under impact loading

In this paper, the dynamic fracture experiment of half-circular disk PMMA specimens with prefabricated penetration defects under impact loading was carried out on the digital laser dynamic caustics experimental system (DLDC), the law of crack initiation, growth and traversing under different tilt angles (30°, 45°, 60°) was examined, and the variations of crack growth trajectory, velocity and DSIF at crack tip were compared for analysis. In addition, the discrete lattice spring method (DLSM) as a new development was employed to simulate how the dynamic cracks grew in PMMA specimens with defects, it was found that the numerical model can reproduce the experiment phenomenon. On the basis, the effect of the elastic modulus of the penetration defect medium on crack growth was analyzed. Referring to the experiment and numerical results, it was thus concluded that the angle and the elastic modulus of penetration defect would dramatically influence the crack growth characteristics.