Experimental Method Of Caustics For Civil And Mechanical Engineering Students

The ever-increasing industrial demand for more sophisticated structural and machine components requires a good understanding of the concepts of stress, strain, and behavior of materials. In particular, stress concentrations are a major concern for engineers. Experimental investigations provide the required tool to understand the development of stress concentrations. This paper proposes the use of the optical method of caustics to study the development of stress concentration around circular holes. The goal of these demonstrations is to improve the students’ comprehension in mechanics of materials as well as in structural and machine design. Introduction At the sophomore level, students in a civil and mechanical engineering programs are introduced to the concepts of stress and strain in a solid body through the Strength of Materials course. In the first Machine Design course, junior mechanical engineering students learn to get a stress concentration factors (SCF) for practical problems from a chart. A senior civil engineering student utilizes similar charts to extract SCF in a steel design course. The theory usually deals with infinite members. Kirsch developed the theoretical stress distribution in the vicinity of a circular hole in an infinite elastic isotropic plate. This theory predicts a stress-concentration factor (SCF) of 3.0 for the hole with the maximum tensile and compressive stresses being 0 and 90 degrees from the horizontal axis of the hole, respectively. In the field of stress concentrations, the limited established theory does not give an insight for the understanding of the development of stresses in the vicinity of a discontinuity. Thus, experimental work is required to enhance the learning such as stress concentrations. The solution for the circular hole in a finite-width plate under uniaxial tension was published by Howland in 1930. In more resent years, experimental solutions have been obtained for a wide variety of hole shapes under different loading conditions. Electrical-strain gages are widely used devices to measure the strains in stressed members. However, the averaging effect of a strain gage is problematic in regard to measuring the strain in the vicinity of discontinuties. It is important that the students visualize the nature of the quantities being computed. Therefore, the enhancement of the student’s overall understanding of the concept of stress concentrations is P ge 10612.1 Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition Copyright © 2005, American Society for Engineering Education discussed in this paper. This is accomplished by utilizing the experimental method of caustics. The determination of SCF is beyond the scope of this paper. Optical method of caustics The method of caustics is relatively new as it was developed in the last 30 years. The methods of transmitted and reflected caustics in various investigations have proven to be a powerful method to measure stress intensity factor at a crack tip in static and dynamic fracture mechanics problems. In 1991, the use of the method by undergraduate students was suggested to extract mode I stress intensity factor. The optical method of caustics is a technique based on geometrical optics. The method is accurate, simple and economical because the optical bench has relatively few components. The overwhelming majority of studies that utilize the method of caustics use specimens that are made of Plexiglas; it is assumed that Plexiglas is an optically and mechanically isotropic material at room temperature. The accuracy of the proposed experiments derives from the fact that the physical stress models must obey the practical laws of physics. The principle of the method is simple in concept. The formation of the caustics image is dependent on the stresses in a structural member or machine component. Therefore, it is an ideal method to be used for when there is a SCF since high stress gradients produce large deflection of the light rays and an image with distinguishing characteristics. The advantage of caustics relative to other optical experimental techniques is that the same equipment can be used in either a reflection or transmission arrangement. Setting The proposed experiments introduce the students to a practical experimental technique of caustics for the understanding of stress concentrations in the vicinity of a hole in a plate. The determination of SCF is beyond the scope of this paper. The objectives of the experiments that use specimens in uniaxial tension are to show the students the following: • The development of stress in a member. (ABET outcomes e, k) • The region where the theoretical axial stress equation is valid. (ABET outcome a) • Design of experiments in the stress analysis field. (ABET outcome b) • The importance of understanding the optics laws and physics in the civil and mechanical engineering fields. (ABET outcomes e, k) • Availability of experimental stress analysis techniques that are not part of a curriculum. (ABET outcome i) Equipment setup and calibration The schematic and suggested arrangements of the optical system for the experimental transmitted and reflected caustics are shown in figures 1 and 2, respectively. Briefly, a monochromatic and coherent light beam emitted from a point source He-Ne laser, which was widened by spatial lens, impinges normally on the specimen. The light beam has to fulfill only one, very important requirement, the light beam has to be parallel. The cost of equipment is less than $2000. To achieve this property, the light source must have the essential features of a point source. Divergent light is used primarily to enlarge the caustic image. The direct recording of the P ge 10612.2 Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition Copyright © 2005, American Society for Engineering Education caustics image is possible in transmission arrangements as well as in reflection arrangements. This was accomplished by projecting the reflected light rays on screens. In the reflected case the light beam was slightly tilted with respect to the normal axis to the specimen in order to separate the reflected beam from the impinging beam. The rotation of the model produced a light beam that was not perpendicular to the specimen. This rotation created only a translation of the caustic without effecting the size, shape and relative position of the caustics. However, a rotation of the screen distorts the caustic image. Therefore, the screen should be always parallel to the model. The live caustic image can be captured by a camera. Fig. 1 Experimental transmitted caustic setup The magnification factor (M) can be determined by using the following formula: plane image in the length ing correspond plane reference in the length any = M (1) However, if the screen is not parallel to the model, an error in the evaluation of magnification factor is obtained. This can be eliminated by using the well known divergent light magnification factor law: i i o Z Z Z M + = (2) P ge 10612.3 Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition Copyright © 2005, American Society for Engineering Education Fig. 2 Schematic reflected caustic experimental setup Where Zi is the distance between the divergent light source and the model, and Zo is the distance between the model and screen. The difference between the calculated magnification factor from equations 1 and 2 indicates the extent of the errors. The main error is that the screen is not parallel to the model and can be easily corrected. Stresses The optical method of caustics is particularly convenient for the study of singular stress fields. The stress singularity of the elastic field is transformed into optical one represented by a highly illuminated surface that contains the necessary information for determining the applied stress in this study. The sum of in-plane principal stresses F1 and F2, in the vicinity of a hole of radius a, in terms of the polar coordinates, r and 2, is given by Kirsch's solution as: σ σ 1 2 + = = + θ σ σr σ θ 2 2 4 2 2