The combination of digital image processing and optical measurement has been demonstrated to be very useful and powerful in recent research. Karmas and B.R.Pramod[1] used the digital image processing technique for crack tip analysis with photoelasticity. Chen and Taylor[2] utilized the computerized fringe analysis in photoelasticity, holographic interferometry and speckle interferometry. But in studying the crack tip vicinity photoelasticity, speckle interferometry and holographic interferometry usually do not result in a useful solution because the density of the fringes is too high to be distinguished. The results obtained by the photoelasticity method and the caustic method in dynamic crack growth behave quite differently[3-4] and there are many dvantages to the caustics analysis. The most important advantage is the simplicity of the caustic experiment. Therefore we have developed a new image processing system to analyze the crack tip with a caustics measurement tool.

We are interested in analyzing not only the caustic curve but also the interference fringe pattern from the front and rear face of the material. The caustic method has four main advantages:

1. This technique provides a full-field observation. We do not need to predict the direction of crack growth.

2. The specimen is observed continuously and mechanical events can be instantaneously analyzed.

3. This system can determine both the stress-intensity factor and the plastic zone size.

4. The method is a non-contact measurement and suitable for static or dynamic mechanical events.

The caustic's application to fracture studies was originally performed by Manogg in 1964[5]. He used only the transmitted caustic which is sometimes called the laser shadow spot. In 1969 Theocaris[6] made a big step in the development of the caustic method and mentioned use of the reflected caustic. A reflected caustic can provide an elementary understanding of how the fracture occurs. The reflected caustics include two pieces of information( front-face caustic and rear-face caustic) concerning the values of the stress-intensity factor K. Another type of information provided by the reflected caustics outside the caustic curve lies in the interference fringes and concerns the thickness variation of the specimen. With two CCD cameras recording the caustic image variations of the crack growth we use two sets of image processing techniques to obtain the

Interference fringes and caustic image. We then digitize the data and store it in two computers. In order to get the exact value of a every short time duration, a trigger circuit is used. The procedure of our system is presented in Fig.1.

A typical reflected caustic set-up is shown in Fig.2 (A). The optical set-up consists of a helium neon laser operating at 632.8 nm wavelength, producing a parallel light beam which is expanded by a spatial filter. A set of suitable apertures and lenses are used to achieve the required magnification. The laser beam can be regarded as a point light source. We here introduce two lenses, one lens converges the laser light leaving the spatial filter to a parallel beam and the other one converges the light again, to increase the flexibility of the optical method. The point light source is reflected by the surface of the crack tip, forming the caustic image. Many types of caustic experiments can be arranged by passing the laser light through some advanced optical filtering, although these are not easy to set up. The application of the caustic set-up for stress intensity factor K and plastic zone measurements has been widely discussed. For example, Gdoutos and Aifantis[7] study the caustics around cracks under mixed-mode loading conditions. Kamath and Kim[8] use the caustic set-up to study three dimensional deformation fields very near the crack tip. Theocaris[9] has extended the caustics method in resolving the evaluation of cod and the core region of plane strain. J. F. Kalthoff[10] has done beautiful dynamic K measurements by utilizing only transmitted caustic set-up.

Brown and Srawley[11] derived a simply stress intensity factor K_I solution for a single-edge crack's in tension as follows: be

where the K_I equation for a double-edge crack in tension is:

and the K_I equation for the crack in pure bending of a beam:

where S is the applied stress; ; P is the tensile stress; t is the thickness of the specimen; W is the width of the specimen; M is the momentum; and a is the crack length.

For comparison, a K_I [12] equation is established by reflected caustic theory and is expressed as follows:

whereDt is the transverse diameter of the reflected caustics and d;n ; C_1 and C_3 are constants depending on the characteristic distances(such as in Fig.2(A) f_1, f_2, z_0, z_i ... etc.) of

the experiment arrangement.

At the crack tip, the material will shrink in the thickness direction and assume a spheroid shape. In other words, the surface of the specimen is like a concave mirror for a reflected light or a lens for transmitted light. The caustic is obtained by illuminating the spheroidally shaped depression by a point-light source or a parallel beam. The phenomenon is illustrated in Fig.2(B), where reflected rays(incident on the screen) will form an envelope that is known as a caustic surface. We can move the screen toward the specimen to see the size of the circular image spot assume a different diameter. Because the caustics obtained by a sphere are well defined, so the deviation of the surface of the specimen(which defines the factor K) can be calculated, i.e., the characterization of the deformed surface near the crack tip is achieved by the description of the curve of the caustics.

However, the geometrical optical equations for mapping a light-ray path is very complicated. But here we adopt a new mixed optical system which consists of two sets of reflected caustic set-ups. The optical alignment need not be very precisely because the digital image processing technique is powerful enough to obtain the experimental data. We can see that these devices and equipment are relatively low in cost and easy to align.

The set-up of our system is presented in Fig.2(C). We can arrive at the following relation between the caustic diameter and the stress intensity factor:

When we change our experiment arrangement, we use a standard specimen and loading to obtain the constant R. However the rate of crack growth is often very large, so that it is difficult to analyze this in an experiment of dynamic fracture. Here we try to use a critical loading value in order to obtain a slow crack growth rate.

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