Technical Note: Theoretical Analysis and Digital Photoelastic Measurement of Circular Disks Subjected to Partially Distributed Compressions

Abstract

-The disk in diametral compression has been quoted most frequently on developing conventional/digital photoelasticity to illustrate new theories and experimental techniques for several decades. Theoretically, the compression as a concentrated force is more conducive to analysis, but it is impossible to achieve such loading condition experimentally. The distributed compression on a finite area at rim is relatively closer to actual testing and it is complicated to seek an analytical solution. In this paper, we extend the work of Hondros to derive the full-field stress distribution of disk subjected to diametrai distributed compression in an explicit functional form. The principal stress difference and principal stress orientation related to isochromatic and isoclinic fringes, respectively, are also expressed in a simple closed form. The maximum shear stress for the whole disk and the validity of using isochromatic fringes to interpret the maximum shear stress are discussed in detail. The isoclinic fringes are compared with theory, and the fringe multiplication isochromatic is compared with simulated image. All of the comparisons are in good agreement with respect to full field. KEY WORDS--Disk, distributed compression, digital photoelasticity, isochromatics, isoclinics, principal stress Introduction The problem of a circular disk subjected to concentrated forces applied at its boundary is frequently used in classical elasticity either to illustrate theories or to estimate experimental data. The stress analysis of the circular disk has been discussed by many authors. 1-4 However, the actual loads in an experiment are not concentrated but distributed over finite portions of the disk. Sanford 5 proposed a method to conversely calculate the isochromatic fringe order from Frochet's full-field solution by employing a least-squares method. Berghaus 6 presented a method combining finiteelement results with photoelastic data to estimate the fullfield distribution of stress. Haake et al. 7 presented a photoelastic device containing a quarter-wave plate and a polarizer to generate six different angular orientations and came up with six images using a CCD. The experiment of a disk in diametral compression is easy to conduct. In addition, the analytical solution of the fullfield stress can be used as a theoretical reference to check the experimental results. However, the problem of the disk in distributed diametral compression is more difficult to analyze than that with concentrated forces. Hondros 8 obtained the full-field stresses in a series solution by using the series expansion technique and applied these solutions to evaluate the Young's modulus and Poisson's ratio of concrete by measuring strain. Instead of using a series solution for full-field stresses, in this paper we continue and extend the work of Hondros to successfully obtain an analytical solution in an explicit form which can provide a full-field sampling capability as well as in-depth theoretical comparisons. Concise mathematical forms are also provided for principal stress orientation and maximum shear stress. Instead of operating a traditional large-sided photoelasticscope, a compact optic system using a He-Ne laser as a light source is installed on an optic bench to obtain the experimental results for the testing of the disk in diametral distributed compression. Full-field comparisons of isochromatic/isoclinic fringes between testing and theory are presented. The validity of using isochromatic fringes to interpret the maximum shear stress for disks subjected to partially distributed compression is discussed in detail. Full Field Analytical Solutions for Disks in Partially Distributed Compressions Consider a circular disk loaded by two diametrically opposing distributed compressive forces located on the top and bottom of the disk. The geometric configuration of this problem is shown in Fig. 1. Hondros used a polar coordinate system to present the full-field stresses in a series solution and the results are expressed as follows: (YF 2p e~ + ~ 1 1 p2 n=l 0 2n-2 sin 2ne~ cos 2nO} (1) K-M. Hung was a Graduate Stttdent and C.-C. Ma (ccma@nm.edu.tw) & a Professor, Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China. K.-M. Hung now is an Associate Professor, Hwa-Hsia College of Technology and Commerce, Tripei, Taiwan, 23554, Republic of China. Original manuscript submitted." December 7, 2001. Final manuscript received." January 9, 2003. c ~ 0 = c~l 1 + 02 n=l p 2n-2 sin 2nc~ cos 2nO} (2) 216 9 VoL 43, No. 2, June 2003 9 2003 Society for Experimental Mechanics