A theory of caustics for mixed-mode fast running cracks

The method of caustics (shadow spot method) has proven to be a powerful optical method to measure stress intensity factors in static and dynamic fracture mechanics problems. In this paper, a theory of caustics was developed for elastodynamically propagating cracks under inplane mixed-mode conditions. Complex potentials for the general solutions of a near-tip field which have been previously derived by the authors were used in this theoretical development. Completely analytical expressions were derived for the caustic curves as well as for the initial curves for fast running cracks under inplane mixed-mode conditions. The effects of crack velocity and mixed-mode condition on the caustic pattern and the initial curve were investigated. New procedures were also proposed for the evaluation of the dynamic stress intensity factors K_I and K_II using the overall dimensions of the caustic pattern. The method of caustics developed here enables one to study quantitatively various mixed-mode dynamic fracture phenomena such as crack branching, crack curving, and crack kinking.     

Propagation of a slant crack under impact studied by caustics

In the present paper the propagation of a slant crack under an impact load was studied by the method of caustics. An air gun was employed for the production of the load pulse. It was concluded that the steady crack velocity increases linearly with the angle of crack inclination, reaching a maximum value when the initial crack is normal to the load direction. The crack starts to propagate in the presence of both K_I- and K_II-stress intensity factors. The K_II-factor has a maximum value of one-tenth of the K_I-factor and an oscillatory character. This becomes zero in the first third of the crack-path, as the crack tends, gradually, to follow its final direction, normal to the direction of the applied pulse. An apparently distinctive effect relating the crack velocities and stress intensity factors may be concluded from the experimental results. For the same values of K_I and K_II, the crack propagates in some cases with different velocities, depending, apparently, on the initial crack inclination. It is believed that it is related to the validity of energy fracture-criteria and requires further study.     

Optically exact equations of caustics for slant cracks biaxially loaded

The equations of caustics have been derived based on the exact theory of geometrical optics, and they have been adapted to the problem of a slanting internal crack in a disk under biaxial loading. A comparison of the exact caustics with those derived from the far-field theory used in the past in the applications showed negligible difference between the two theories. Thus, it is shown that the approximate theory of caustics used before for determining singular fields in mechanics is sufficiently accurate for engineering applications.     

Elastodynamic forms of caustics for running cracks under constant velocity

For the study of elastodynamic problems of propagating cracks it is necessary to evaluate the dynamic stress intensity factor K^d_I, which depends on the form of expressions for the stress components existing at the running crack tip at any instant of the propagation of the crack and the corresponding dynamic mechanical and optical properties of the material of the specimen under identical loading conditions. In this paper the distortion of the form of the corresponding reflected caustic from the lateral faces of a dynamically loaded transparent and optically inert specimen containing a transverse crack running under constant velocity was studied on the basis of complex potential elasticity theory and the influence of this form on the value of the dynamic stress intensity factor was given. The method was applied to the study of a propagating under Mode I crack in a PMMA specimen under various propagation velocities and the corresponding dynamic stress intensity factor K^d_I, evaluated.     

The stress intensity factors of internal radial cracks in rotating disks by the method of caustics

A technique based on the method of caustics is used to evaluate the stress intensity factors of internal radial cracks in rotating disks. The measurement strategy, experimental procedures and results are presented. The accuracy of the technique is demonstrated by comparing experimental results to theory.     

Dynamic caustics in unidirectional fiber-reinforced composites with mode I cracks: Experiment and numerical simulation

The stress singularities of a dynamic crack tip in orthotropic composites were studied through caustics. The parametric equations of the caustic and its initial curve surrounding the dynamic crack tip were derived through the elastic dynamic crack solutions of orthotropic composites and the basic principle of reflective caustics. Theoretical caustics and initial curves for three kinds of orthotropic composites were simulated, and the effects of crack velocity on the caustics and initial curves were analyzed. In comparison with numerical results, the dynamic caustic experiment was performed for dynamic cracks along the material axes in unidirectional, fiber-reinforced composites under drop-hammer, three-point-bend loading.     

Response of composite materials to hypervelocity impact

Investigation of composite materials response to hypervelocity impact by space debris has been carried out. In order to simulate hypervelocity impact, a unique laser driven flyer plate (LDFP) system was used, generating hypervelocity debris with velocities of up to 3 km/s. The materials studied in this research were Kevlar 29/epoxy and Spectra1000/epoxy thin film micro-composites (thickness of about 100 μm). Both Spectra and Kevlar fibers are used in long-duration spacecraft outer wall shielding to reduce the perforation threat. The micro-mechanical response of different composites was studied and correlated to the fiber, the matrix and the fiber/matrix interface properties. Visual and microscopic examinations of the damaged area identified fiber debonding as the prevailing failure mechanism. On the basis of a simple energy balance model it can be stated that for Spectra/epoxy composite the dominant mechanism is new surface creation, whereas for Spectra surface-treated fibers/epoxy the fiber pull out is the dominant mechanism. For Kevlar/epoxy fiber, pull out mechanism plays an important role.     

Response of reinforced concrete beams to hydrostatic pressure acting within primary cracks

This paper investigates the response of reinforced concrete beams to hydrostatic pressure acting within primary cracks. Notched beams were initially pre-cracked before pressurised water was introduced into the primary crack. The deflection and strain increase at tensile reinforcement level (due to the pressurised water load) was measured. Tests were carried out using applied hydrostatic pressures of 0.2 and 0.325 MPa. Results show that both deflection and strain at tensile reinforcement level increases immediately after the introduction of hydrostatic pressure into open primary cracks. If the crack is held open and hydrostatic pressure is allowed to build up within the depth of the crack, additional deformation occurs. A finite element (FE) model was constructed to investigate the effects of greater hydrostatic pressures acting within the primary crack. The FE model was first validated against the test data, before being used to assess the structural response of the reinforced concrete section to applied hydrostatic pressures of up to 1 MPa. It was found that section deformations increased as hydrostatic pressure was increased. At applied hydrostatic pressures of 0.8 MPa and above, the increases in strain at tensile reinforcement level were shown to be significant. DNV-0S-C205, which is the Det Norske Veritas (DNV) standard for offshore concrete structures states that the ‘effects of water pressure within cracks may be neglected for structural elements exposed to less than 100 m (0.981 MPa) of water head.’ The current research suggests that the effects of water pressure within cracks for structural elements exposed to slightly less than 100 m water head may also be significant (based on a 10 % threshold criteria). However, it is accepted that a more comprehensive parametric study would be required to determine whether or not the DNV should be redressed.     

Response of boron carbide subjected to high-velocity impact

This article presents an evaluation of the response of boron carbide (B₄C) subjected to impact loading under three different conditions. Condition A is produced by plate-impact experiments where the loading condition is uniaxial strain and the stresses and pressures are high. Under plate-impact loading the material fails at the Hugoniot Elastic Limit (HEL) and the failed material undergoes high confining pressures and relatively small inelastic strains. Condition B is produced by projectile impact onto thick targets where the stresses and pressures are dependent on impact velocity, but they are generally lower than those from plate impact. Under thick-target impact/penetration most of the material fails under compression, the inelastic strains are large and the material appears to exhibit more ductility than under condition A. Lastly, condition C is produced by projectile impact and perforation of thin targets where the stresses and pressures are a combination of compression and tension. Under thin-target perforation the material fails in both tension and compression. The Johnson–Holmquist–Beissel (JHB) constitutive model is used to evaluate the material behavior for each of the three conditions, but it is not possible to accurately reproduce the experimental results of the three conditions with a single set of constants. Instead, three different sets of constants are required to accurately model the three impact conditions. These three models/constants are used to provide insight into the complex response of B₄C, and to identify possible mechanisms that are not included in the JHB model.     

Stress intensities in elastic-plastic plane-stress fields by caustics

Summary The experimental method of reflected causties was extended in this paper for the study of the strength of the plastic singularity and the respective plastic stress intensity factor at the tip of a mode-I crack in an elastic-strain hardening material. The elastic-plastic field around the crack tip was defined by means of the singular plastic solutions of Hutchinson, Rice and Rosengren (HRR-asymptotic field) which gives satisfactory results for small scale yielding types of deformation. The initial curve, generating the caustic by reflection from the front face of the deformed specimen lying well within the plastic zone, was accurately determined by considering the contribution of the elastic and plastic components of stresses and strains within the plastic zone.Results with experiments executed on steel specimens single-edge notched with various values of the hardening exponentn coincided with the form and shape of the theoretically derived caustics thus confirming the validity of the developed theory. The values of the plastic stress intensity factor thus derived corroborated the already existing results from previous experiments.     

Response of space structures to orbital debris particle impact

All long-duration space and aerospace and transportation systems, such as the Space Station Freedom and the Space Shuttle, are susceptible to impacts by pieces of orbital debris. These impacts occur at high speeds and can damage the flight-critical systems of such spacecraft. Therefore, the design of a structure that will be exposed to a hazardous orbital debris environment must address the possibility of such hypervelocity impacts and their effect on the integrity of the entire structural system. A technique is developed for analyzing the response of dual-wall structures to oblique Hypervelocity projectile impact. Ballistic limit curves that predict the potential of an impacting projectiles to perform the main wall of a dual-wall strucutral system are obtained using the techniques and are compated against experimentally derived curves. Comparisons are performed for a variety of impact velocities, trajectory obliquities and projectile masses. It is shown that the results obtained using the technique developmed herein compare very well with experimetanl results.     

Gradient piezoelectricity for cracks under an impact load

The flexoelectric effect on elastic waves is investigated in nano-sized cracked structures. The strain gradients are considered in the constitutive equations of a piezoelectric solid for electric displacements and the higher-order stress tensor. The governing equations with the corresponding boundary conditions are derived from the variational principle. The finite element method (FEM) is developed from the principle of virtual work. It is equivalent to the weak-form of derived governing equations in gradient elasticity. The computational method can be applied to analyze general 2D boundary value problems in size-dependent piezoelectric elastic solids with cracks under a dynamic load. The FEM formulation is implemented for strain-gradient piezoelectricity under a dynamic load.     

Light caustics of hydrodynamic solitons illuminated by a laser beam

Properties of caustics of plane curves are used for interpretation of optical experiments with hydrodynamic solitons in a one-dimensional water channel. The water-free surface acts as a curvilinear plane mirror for incoming light beams. Experiments with single solitons and with couples of colliding solitons are presented and discussed within the framework of geometrical optics.     

On the equivalent order of crack-tip singularity defined by caustics

The problem of a transverse Griffith crack in an infinite plate submitted to simple tension at infinity was studied by using its closed form solution described by the elastic potential function (z). The exact form of the caustic and its generatrix curve formed around the crack tips was exactly described by using the (z)-function. These exact forms were compared with the respective forms given either by the singular one-term solution of the problem and accepting that the order of singularity at the crack tip is (1/2), or by a solution defining the order of singularity and the respective stress intensity factor by taking into consideration the influence of the distance from the crack tip where these quantities are evaluated. It was shown by comparing the first stress invariant I ₁, whose gradient defines the respective caustic, that the differences between the exact values and the values of I ₁ derived by the proposed method with variable order of singularity is much smaller than the differences between the exact solution and the singular solution. The singular solution is based on the assumption of a constant value of the order of singularity.

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Résumé On a étudié le problème d'une fissure de Griffith transversale dans une plaque infinite soumise à traction simple à l'infini en utilisant une solution de forme fermée décrite par la fonction des potentiels élastiques (z). La forme exacte de la caustique et de sa génératrice au voisinage de l'extrémité de la fissure a été décrite avec exactitude en recourant à la fonction (z). On a comparé ces formes exactes avec les formes données respectivement par une solution singulière à un terme du problème, en acceptant que la singularité à l'extrémité de la fissure soit d'ordre 1/2, ou par une solution dans laquelle l'ordre de la singularité et les facteurs d'intensité de contraintes sont définis en tenant compte de l'influence de la distance qui sépare l'extrémité de la fissure dulieu où ces quantités sont évaluées. On montre que, pour le premier invariant de contrainte I ₁, les différences entre valeur exacte et valeur déduite de la méthode proposée, avec un ordre de singularité variable, sont plus faibles que les différences entre la solution exacte et la solution singulière, laquelle est basée sur l'hypothèse de valeurs constantes de l'ordre de singularité.

     

Properties of caustics produced by a positive lens: meridional rays

We study the formation of the caustic surfaces formed in both convex-plane and plano-convex spherical lenses by considering a plane wave incident on the lens along the optical axis. Using the caustic formulas and a paraxial approximation we derive analytical expressions to evaluate the spherical aberration to third order. Furthermore, we apply the formulas to evaluate the circle of least confusion for a positive lens.