A Study of Stress Analysis for a Residual Stress Model by Digital Photoelasticity

The stress analysis for a model with initial stresses, which we term a residual stress model, is performed by digital photoelasticity. The stresses applied on the residual stress model are obtained by analyzing both the initial stresses and the resultant stresses. The method used for analyzing the stresses applies the principle of superposition of the stress to photoelasticity, which is a well-known technique in the field of elasticity. In the digital photoelasticity technique used, the principal stress direction and the relative phase retardation  are analyzed by photoelastic techniques using linearly polarized light. This technique overcomes the phase difference error associated with a quarter-wave plate by employing incident light at three different wavelengths, and using an unwrapping technique that allows and  to be determined using the arctangent function. A residual stress model produced by a disk containing frozen stresses that was subjected to a diametral compressive load at an angle of 31 was used to experimentally test this method. The values of the stresses of the loaded disk model analyzed were in good agreement with corresponding theoretical values at all locations far from the loading points of the residual stress model. Introduction Photoelasticity is an effective method for measuring the directions of principal stresses [ and +(/2)] and the difference of principal stresses, which is related to the relative phase retardation , of a photoelastic model as visual patterns. The photoelastic model used is assumed to be free from the time-edge effect and from machining stresses, and initial stresses, which generate residual stresses. Developing such models is frequently a time-consuming, laborious process, and it requires a high degree of skill [1]. Since it is difficult to automate the production of models free from initial stresses, and it is desirable to perform stress analyses on models with initial stresses. Although such stress analyses are rare, they have been performed by determining the absolute value of  for the loaded model by measuring the initial stresses and two resultant stresses generated when two different loads are applied [2]. It is anticipated that the applied stresses in a model with initial stresses can be determined by applying digital photoelasticity [3-12]. Digital photoelasticity can be used to measure  and  by using circularly polarized light [5,6], linearly polarized light [7,8], elliptically polarized light[9,10] or white light[11,12] as the incident light. Applied Mechanics and Materials Online: 2008-07-11 ISSN: 1662-7482, Vols. 13-14, pp 59-64 doi:10.4028/www.scientific.net/AMM.13-14.59 © 2008 Trans Tech Publications, Switzerland This is an open access article under the CC-BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) The present paper describes a method for obtaining the applied stresses ( x- y,  xy) on a residual stress model from the values of  and for the initial and resultant stresses as measured by digital photoelasticity, where ( x, y,  xy) are the rectangular stress components of the applied stresses. In the present technique, the values of  and  are measured using linearly polarized light. The analytical method for determining the stresses applies the principle of superposition of stresses to photoelasticity. The main premise of the paper is the employment of digital photoelasticity to tackle the problem of stress analysis for the residual stress model. Photoelastic Effects In a two-dimensional stress system, the principal stresses ( 1,  2) and principal stress direction  are related to the rectangular stress components ( x,  y,  xy) by the following equations [1]:  1 2 = ( x y)/ cos2 = 2 xy / sin2 , (1)  = 0.5tan -1 {2 xy /( x y)} . (2) If light propagates through a model with principal stresses ( 1,  2) and principal stress direction , the relative phase retardation in optical lengths produced by double refraction is given by i= (2tCi)( 1 2) , (3) where i is the relative phase retardation at i (the wavelength of the light), Ci is the photoelastic sensitivity of the model at i, and t is the thickness. The relation is known as the stress-optic law. Stresses Applied on Residual Stress Model When stresses (  1,  2,   ) are applied onto the residual stress model with the principal stresses ( 0 1,  0 2) and principal stress direction  0 , a resultant stress field is produced in the model. Denoting the principal stresses of this resultant stress field by ( 1,   0 2), and the resultant principal stress direction by  , then the stress components of the residual stresses, the applied stresses, and the resultant stresses are given from Eq.(1). The applied stresses can be calculated by applying the principle of superposition of the stresses. Doing this gives the following expressions:   x  y = (  0 x  0 y)( 0 x 0 y) =( 1-  0 2)cos2  0 -( 1- 0 2)cos2 0 , (4)   xy =   0 xy  0 xy =0.5( 1-  0 2)sin2  0 -0.5( 1- 0 2)sin2 0 . (5) CAMERA y y x