Equations Of Caustics Research Articles

Pearcey beam tuning and caustic evolution.

Based on the principle of catastrophe theory, by adding an additional phase factor, we adjust Pearcey beams, which therefore have a more flexible and controllable light-field structure. The basic optical structure and evolution characteristics of caustics are also investigated. In particular, we derive analytical equations of caustics for Pearcey beams by exactly considering the specially engineered phase factor. Experimentally, binary masks are used to encode light-field information with the superpixel method so that the theoretically designed Pearcey beam can be generated. Theoretical analysis and numerical simulations indicate that the caustics remain unchanged but exhibit lateral shift for a series of phase parameters during propagation in free space. This phenomenon has potential applications in the field of optical manipulation.

Expérimentation et analyse thermique d’un concentrateur hémisphérique stationnaire sous les conditions climatiques à Ouagadougou, Burkina Faso

This technology is used in drying, domestic water heating or in the production of electricity. However, it uses a solar tracking system that is too complex and expensive for countries with less equipment and high solar potential. In this study, we are interested in the experimentation and thermal analysis of a stationary hemispheric concentrator. The numerical resolution of the caustic equations of a spherical concentrator allowed to determine the dimensions and the position of the receiver, necessary for the design of the physical model and the assembly of the experimental device. The results of the 3D numerical simulation with the Comsol5.3a software allowed to highlight the ray tracing and the profile of the flow concentrated on the receiver. The results obtained experimentally show that the receiver and the air inside reached a maximum temperature of 224°C and 97.6°C respectively. The solar concentration device studied is therefore technically favorable for thermal applications requiring intermediate temperatures.

Contact length in discs and rings by the reflected caustics method

Based on the existing analytic solution of two cylindrical bodies in contact, the method of reflected caustics is here employed to obtain the contact length, realized between disc/rings, made of a transparent optically isotropic elastic material, and the jaws of the International Society for Rock Mechanics (ISRM)’ standardized apparatus for the implementation of the Brazilian-disc test. In this context, assuming a negligible refraction index, because of a small thickness of the specimen, the equations of caustics are here rederived, giving emphasis to those due to light reflection from the rear face of the specimen. Revealing some further features relating reflected caustics from the front and rear face of the disc, the contact length is obtained in closed form by the aid of an existing simple linear formula, usually ignored because of unspecified data, defined here in the light of the new evidence.

Investigation of dynamic mode I matrix crack-fiber bundle interaction in composites using caustics

In this work, the dynamic mode I matrix crack-fiber bundle interaction was studied using caustics. First, the strain fields at the dynamic mode I matrix crack tip ahead of the fiber bundles were deduced using transformation toughening theory. Subsequently, the caustic equation at mode I dynamic matrix crack tip ahead of the fiber bundles was established to investigate the influences of the fiber bundle and the crack propagating velocity on the initial curves and caustic curves at the dynamic matrix crack tip. Finally, a series of dynamic caustic spots surrounding the propagating crack tip were recorded using optical caustic experiment, and dynamic stress intensity factor were extracted from the shadow spots.

Experimental study on interaction between matrix crack and fiber bundles using optical caustic method

The stress singularity of the matrix crack perpendicular to the fiber bundles was studied by the method of optical caustics. First, the strain fields at the matrix crack tip in the neighbor of the fiber bundles were derived based on transformation toughening theory and Eshelby inclusion method. Then the caustic equation at the matrix crack tip in the neighbor of the fiber bundles was established. Finally, an optical caustic experiment for the interaction between the matrix crack and the fiber bundles was conducted to visualize the stress singularity at the matrix crack tip in the form of caustic spots.

Analytic aspheric coefficients to reduce the spherical aberration of lens elements used in collimated light.

We provide closed-form formulas for aspheric terms for either plano-convex or convex-plano aspheric lenses as functions of the paraxial parameters involved in the process of refraction. These formulas are obtained through an expansion in Taylor's series from the exact caustic equation produced by aspheric lenses considering a plane wavefront propagating parallel to the optical axis and impinging on the refracting surface. A comparison of the aspheric coefficients obtained through our analytic formulas and commercial optical design software is presented, showing good agreement. This is useful in reducing spherical aberration.

Generalized Dual- Reflector Axisymmetric Antennas [Antenna Designer's Notebook

Practically suitable generalized parametric equations of non-parabolic reflectors are obtained. These reflectors are chosen as the main reflectors during radio-physical designing of new dual-reflector axisymmetric antennas. An example of caustic equations is given. Equations of the subreflector are explicitly given.

Evaluation of Concentrated Oblique Load at the Apex of a Wedge by the Method of Caustics

The experimental method of caustics was used for the evaluation of the stress-singularities created by concentrated oblique load applied at the apex of a wedge or on the horizontal straight boundary of a half-plane. The wedge was considered to be elastic, isotropic and under generalized plane stress conditions. According to the method of caustics, the light rays impinging normally at the thin plate are partly reflected from either the front or the rear faces of the plate. The reflected rays are deviated because of the important constraint of the plate at the vicinity of the applied concentrated load and the significant variation of the refractive index there. The deviated light rays, when projected on a reference screen, are concentrated along a singular curve which is strongly illuminated and forms a caustic. It was proved that the shape and size of the caustic depend on the stress singularity at the point of application of the load. The parametric equations of caustics created by such a singularity were studied in relation with the loading mode of the wedge. Thus, by measuring the dimensions of the caustic, one can evaluate the state of stress at the singularity. The experimental evidence, based on the above method on Plexiglas (PMMA) wedges, corroborated existing elastic theories concerning the order of singularities for concentrated loads. The relations for the calculation of the load, the components of stress and the concentration factor based on the geometrical characteristic of caustics, which were reflected from the front face of the wedge, were given.

Recent application of caustics on experimental dynamic fracture studies

Combining the caustic method with high-speed photography is an efficient optical measurement technique to study the dynamic fracture behaviours of homogenous and isotropic material. In the last decade, the main emphasis is extended to study dynamic fracture of anisotropic material and dynamic propagation of multi-cracks and interface cracks in practical engineering materials. In this paper, the recent advances and applications about the dynamic caustic method in China are reviewed, such as impact response and dynamic fracture of composite materials (fibre composites, functionally gradient materials and nanometre composites), and dynamic interaction and propagation of multi-cracks and interface cracks. Particularly, some new numerical methods were developed to solve the complicated caustic equations by introducing both the maximum characteristic size and the relevant angles in caustic patterns. Also, some important experimental results in fracture mechanics are described, and the potential research prospects about dynamic caustics are included as well.

A numerical replacement of computer algebra methods for the derivation of polynomial equations in mechanics

The classical computer algebra methods of Gröbner bases and characteristic sets have been already used for the derivation of polynomial formulae, mainly in classical geometry. Here we derive analogous formulae in applied mechanics by using a completely different approach: the numerical determination of the coefficients in the required polynomial formulae without the use of computer algebra techniques. Two compatibility equations for the stress components in plane elasticity and the equation of caustics in fracture mechanics are used as the vehicle for illustration of the present elementary numerical approach. Beyond the simplicity of the present method, its powerfulness consists mainly in the fact that the well-known disadvantages of computer algebra methods (in the required computer memory and, mainly, computer time) are now avoided. Generalizations of the present approach follow easily (e.g. from polynomial to differential equations). © 1997 Elsevier Science Limited.

The equation of caustics in cartesian coordinates for mixed-mode stress intensity factors

The equation of the classical caustic about a crack tip in cartesian coordinates ( x, y) is derived in the case of mixed-mode stress intensity factors. This is achieved by using the original parametric equations of the caustic and eliminating the parameter (polar angle). Both Groebner basis and resultant algorithms in the computer algebra system Maple V were used for this elimination. The resulting bivariate equation (in cartesian coordinates) is seen to be of the sixth degree exactly, as is also the related equation for mode I stress intensity factors.

Mode I caustics for the infinite cracked plate using the exact solution: One parameter dependent curves

A dimensional analysis is carried out for the caustic equations expressed in terms of the Muskhelishvili complex potential. It is shown that, for the infinite cracked plate under mode I loading conditions, the caustic dimensions, appropriately normalized, depend on a single overall parameter that includes the crack length. The formulation presents itself for a straightforward investigation of the problem with respect to the derived parameter and a complete numerical analysis is carried out. The predicted values for the commonly used characteristic dimensions of caustics, obtained from the exact theory and its singular term approximation, are compared. The expected errors for a singular term evaluation of measurements are also presented in convenient single curve plots and may be used as “add on” corrections. The errors predicted were found to be from non-negligent to rather appreciable in the region near the crack tip where experiments are “often” conducted, as had originally been alleged but was later disputed.

Derivation of the equation of caustics in cartesian coordinates with maple

The well-known equation of caustics about crack tips in experimental fracture mechanics is usually expressed in the form of two equations for the Cartesian coordinates ( x,y) with a parameter (an appropriate polar angle playing the rǒle of this parameter). Here we derive the equivalent unique equation (but without a parameter) also in Cartesian coordinates with the help of the computer algebra system Maple V by using the available Gröbner-bases algorithm. The obtained Cartesian equation is of the sixth degree and it can be solved in closed form with respect to y yielding an explicit result y = y( x). The same equation is also checked in two special cases, where it gives the same results as the equivalent pair of parametric equations. Analogous, more general results can also be derived.

Analysis of dynamic mixed mode stress fields in bimaterials by the method of caustics

The equations of caustics for dynamically extending curvilinear interface cracks are derived where optical isotropy and anisotropy of the material have been considered, respectively. The equations have been obtained by applying the method of generalized complex potentials. Moreover, an algorithm for the determination of stress intensity factors from experimentally obtained caustics is presented for the case of dynamic crack extension.

The interpretation of optical caustics in the presence of dynamic non-uniform crack-tip motion histories: A study based on a higher order transient crack-tip expansion

The optical method of caustics is re-examined by considering the presence of dynamic non-uniform crack-tip motion histories. Based on the higher order transient expansion obtained by Freund and Rosakis (1990, Eleventh National Congress of Applied Mechanics; 1992, J. Mech. Phys. Solids40(3), 699–719) and Rosakis et al. (1991, Int. J. Fract. 50, R39–R45), in which dynamic transient effects were included in the near-tip deformation field, the exact mapping equations of caustics are derived for non-uniformly propagating cracks. The resulting equations indicate that the classical analysis of caustics based on the assumption of KId-dominance, is inadequate to interpret the experimental caustic patterns when dynamic transient effects become significant. In this paper, an explicit relation between the instantaneous value of the dynamic stress intensity factor KId(t) and the geometrical characteristics of the caustic is established. This relation shows that for the case of non-uniformly propagating cracks, the relation between the dynamic stress intensity factor and the geometrical characteristics of the caustic pattern depends on the crack-tip acceleration and on KId(t). It also reduces to the classical relation between KId(t) and the caustic diameter for the case of KId-dominance (when the crack-tip fields are well described by the r−½ singularity in stresses). The Broberg problem is used as an example problem to check the feasibility of analysing caustics in the presence of higher order transient terms. It is shown the value of the dynamic stress intensity factor obtained by the proposed method agrees remarkably well with the exact analytical value while large errors are introduced when the classical analysis (KId-dominant) of the method of caustics is used.

Analysis of quasistatic and dynamic interface crack extension by the method of caustics

The equations of caustics for quasistatically or dynamically extending curvilinear interface cracks are derived where optical isotropy and anisotropy of the material, respectively, have been considered. The equations have been obtained by applying the method of complex potentials in the cases of quasistatic and dynamic crack extension. The curvature of the crack path has been included in the equations by using the method of conformal mapping. In addition, a system of nonlinear equations has been obtained for the evaluation of caustics from shadow optical experiments.

Construction of the equation of caustics in dynamic plane elasticity problems with the help of reduce

The method of caustics has become a very efficient tool in crack, hole and many additional plane elasticity problems. Unfortunately, the fundamental equation of the caustics frequently requires complicated algebraic computations including that of a Jacobian determinant. Here we show that computer algebra software can prove very efficient in these computations, using as a vehicle for this illustration the already known fundamental equation of caustics in dynamic plane elasticity (both for crack problems in fracture mechanics as well as for hole and additional problems). We have used the programming capabilities of REDUCE, a very popular computer algebra system, for our algebraic computations. Moreover, we illustrate the ‘learning’ abilities of REDUCE especially for the derivation of the complex form of this equation. The case of static plane elasticity results simply as a special case of dynamic plane elasticity. Additional possibilities are suggested in brief.

Symbolic derivation of the equations of caustics about a crack tip

For the determination of the shape of the initial curve of a caustic about a crack tip in plane elasticity problems (whence the shape of the caustic itself is automatically obtained) we need to solve a nonlinear algebraic equation with unknown the distance of each point of the initial curve from the crack tip and parameter the polar angle. Here this nonlinear equation is solved (for a particular crack problem) by the classical method of successive substitutions (a one-point iterative method) in numerical analysis, but symbolically and not numerically, with respect to the polar angle. This yields a semianalytical equation for the initial curve (and, further, for the caustic itself) of particular importance for the study of its properties from the analytical point of view. On the other hand, the present results show the usefulness of symbolic computations in crack problems in fracture mechanics.

A preliminary report on 3d effects on caustics

A modification to the standard caustic equations was made to include the effect of a mixed plane stress-plane strain state across the specimen thickness near the crack tip. By assuming the thickness of the plane stress surface layer to be of the size of the plastic zone along the crack axis, a K ~ D 5 6 dependence for the stress intensity factor K vs the caustic diameter D was obtained. This dependence contrasts the classical K ~ D 5 2 relationship based on plane stress considerations, but it is in good agreement with our experimental data.

Determination of the dynamic stress intensity factorsK I d andK II d , for a mixed-mode propagating crack

In this paper, the dynamic propagation problem of a mixed-mode crack was studied by means of the experimental method of caustics. The initial curve and caustic equations were derived under the mixed-mode dynamic condition. A multi-point measurement method for determining the dynamic stress intensity factors,K I d , andK II d , and the position of the crack tip was developed. Several other methods were adopted to check this method, and showed that it has a good precision. Finally, the dynamic propagating process of a mixed-mode crack in the three-point bending beam specimen was investigated with our method.