Whole-field measurement of three-dimensional stress by scattered-light photoelasticity with unpolarized light

In digital scattered-light photoelasticity with unpolarized light (DSLPUL), secondary principal stress direction ψ j and total relative phase retardation ρ jtot in a three-dimensional stressed model with rotation of the principal stress axes are obtained by measuring Stokes parameters of scattered light from optical slices. The present paper describes intelligibly the principle of DSLPUL, and then demonstrates that the ψ j and ρ jtot in a frozen stress sphere model are nondestructively measured over the entire field.


Introduction
Three-dimensional photoelasticity has encountered many difficulties for three-dimensional (3-D) stress analysis.This is mainly caused by the rotation of the principal stress directions along the light path.The optical phenomena occurring in a 3-D photoelastic model have been studied theoretically by many researchers [1][2][3][4], and it is well known that any 3-D photoelastic model can be reduced to an optically equivalent model consisting of a pure rotator and a linear retarder.These two sets of principal axes at the entrance and exit of the equivalent model are termed the primary and secondary characteristic directions, and the retardation is termed the characteristic phase retardation [2].
I have derived a SLPUL method for measuring ρ j and ψ j in a 3-D photoelastic model that has undergone principal stress rotation [17][18][19].Then a technique for automatic stress analysis of the method, digital scattered-light photoelasticity with unpolarized light (DSLPUL), is proposed and EPJ Web of Conferences developed, in which values of ψ j and the total relative phase retardation ρ jtot in a 3-D photoelastic model can be obtained over the entire field from the arctangent function, and the technique was validated through numerical simulations and experiments [21,22].
The present paper shows that the principle of DSLPUL is intelligibly described and the usefulness of the method is shown with nondestructive experiment results of the ρ jtot and ψ j in a frozen stress sphere model.

Relations between photoelastic parameters and Stokes parameters in three-dimensional photoelasticity
A schematic diagram of DSLPUL [17,20] is shown in Figure 1.In the figure, an unpolarized light impinges at a θ-deg angle to the x axis in the x-z plane at the point y j in the model.The resultant scattered light, S(y j ;θ), at the point y j is observed by an analysis system, which consists of a quarter-wave plate Q 2 , and a linear polarizer P 2 .
The quantity S(y j ;θ) is the Stokes vector of the scattered light, and is the linearly polarized light of the azimuth (the θ-deg axis) perpendicular to both the incident light direction and the observation direction [15].The state of polarization of the scattered light is then changed by the photoelastic model R(2ω j-1,0 ) M j-1,0 M j between points y j and y 0 .Here, M j is the Mueller matrix of the linear retarder with relative phase retardation ρ j and secondary principal stress directions (ψ j , ψ j +π/2) between the points y j and y j-1 .The matrices R(2ω j-1,0 ) and M j-1,0 are the Mueller matrices of a pure rotator with rotatory power ω j- 1,0 , and of a linear retarder with retardation ρ j-1,0 and principal axes of azimuths (ψ j-1,0 , ψ j-1,0 +π/2), respectively, for the optically equivalent model between point y j-1 and y 0 .The Stokes vector S(y j ,y 0 ;θ) of the observed light can therefore be expressed by the following matrix equation [4,21].
The values of ψ j and ρ j in the optical slice between the points y j and y j-1 are derived from Eqs.
3 Determination of ψ j and ρ jtotλi by use of Three Wavelengths
Here, the term [ ] 0.5 λi in the denominator represents the term at λ i .
The unwrapping of ρ jtotλi can be executed based on the continuity of ρ jtotλi .In evaluating the continuity of ρ jtotλi , the unwrapping process is performed by eliminating phase jumps for ρ jλi , and ρ jtotλi is determined by adding a constant value for connected ρ λi j .When ρ jtotλi at a point on the model is known, the constant value at the point is found by comparing ρ jtotλi with the connected ρ jλi .Furthermore, the ρ jtotλi at the point can be also determined from values of ρ jλ1 , ρ jλ2 , and ρ jλ3 [28].
In addition, α 2 is the azimuthal angle of the fast axis of P 2 and β 2 is the azimuthal angle of the transmission axis of Q 2 , while ∆ρ λi is the phase difference error by the mismach of Q 2 with respect to λ i .According to Eq. ( 22), the Stokes parameters at λ i are unaffected by ∆ρ λi of Q 2 due to λ i .

Experimental system and experimental results
In the experimental system, a beam having three wavelengths (λ 1 =632.8nm, λ 2 =514.5 nm, λ 3 =457.9nm) passes through P 1 , and an electro-optical modulation device EO orthogonally modulates the polarized light at 500 Hz.The light beam can therefore be approximately regarded as an unpolarized beam of light over the long exposure time of the CCD camera.The light beam is then converted into a light sheet, and is split into two light sheets having angles of incidents of 0° and 45°.The light scattered from the model passes through Q 2 of wavelength 515 nm and P 2 , and is imaged by the CCD camera.The phase Fig. 2. Isochromatics and isoclinics of a frozen stress sphere under diametral compression as measured using a crossed-plane polariscope.
A spherical frozen-stress model of diameter 2R = 50.1 mm as shown in Fig. 2 was immersed in a fluid that had a refractive index which closely matched that of the sphere.The model was optically sliced into four planes between the top point of the model and its meridian plane (see Fig. 3).Thus, the secondary principal axes of the optical slices have been rotated [25].The original 12 images were obtained for each wavelength λ i and for each cross section and were recorded using two incident unpolarized light sheets of 0° and 45°. Figure 4 shows the original images I j (45, 135, 135) λ1 measured from each section.The images I j (45, 135, 135) λ1 contain many speckles due to the dark current of the CCD camera head caused by long time exposures.
The image of ψ 4 was calculated from Eq. ( 21) using the three wavelengths, since ψ 4 obtained from Eq. ( 19) was not measured at positions for which ρ 4totλi =2πN λi .Figure 5 shows an image of the calculated ψ 4 , and it has abrupt jumps of π/4.The obtained ψ 4 compensates well for the influence of the positions of ρ 4totλi =2πN λi , although it contains many speckles in the scattered light.The values of ψ 4 were then unwrapped to extend the range to [-π/2, π/2] and the image of the unwrapped ψ 4 is shown in Fig. 6(a).The unwrapped value of ψ 4 was easily determined by just adding a constant π/4 or -π/4 to eliminate the abrupt jumps.In a similar manner, the images of unwrapped ψ j for the other slices are shown in Figs.6(b) -(d).
To obtain ρ 4 at λ 1 , ρ 4λ1 , the values of sin2ω 3,0 and cos2ω 3,0 were calculated using Eqs. ( 22 and ( 23) for the three wavelengths, respectively.The images of the calculated sin2ω 3,0 and cos2ω 3,0 are shown in Figs.7(a) and (b).Figures 7(a  14th International Conference on Experimental Mechanics everywhere except in the equatorial and meridian planes.This is a characteristic of the spherical frozen stress model [22].Then, ρ 4λ1 was calculated using Eq. (20)by using s 1 (y 3 ,y 4 ;0) λ1 , s 3 (y 3 ,y 4 ;0) λ1 , s 2 (y 3 ,y 4 ;45) λ1 , s 3 (y 3 ,y 4 ;45) λ1 , and the known values of unwrapped ψ 4 .Figure 8 show the image of ρ 4λ1 and the distribution of ρ 4λ1 exhibits abrupt jumps of π.Since the retardation of the model varies continuously, the values of ρ 4λ1 were made continuous by eliminating the phase jumps, either π or - π, from the center of the model.This connected ρ 4λ1 differs from ρ 4totλ1 by a constant value of 2π.Thus, the unwrapped ρ 4totλ1 was obtained by adding 2π to the connected ρ 4λ1 and is shown in Fig. 9(a) for the range [0,4π].In a similar manner, the images of the unwrapped ρ jtotλ1 for the other    slices' ρ jλ1 are shown in Figs. 9(b)-(d).The distributions of the unwrapped ρ jtotλ1 in the equatorial plane are shown in Figs.9(e-h).The images of the measured ρ jtotλ1 are clear, although speckles and the nonuniformity of the scattered light are visible.
These optical measurement results are sufficiently accurate compared with the mechanical slice models that was physically cut from the frozen stress sphere [19].

Conclusions
The principle of the digital scattered-light photoelasticity with unpolarized light is intelligibly presented, which can be used to nondestructively measure the secondary principal stress direction ψ j and the total relative phase retardation ρ jtot in a three-dimensional stressed model with rotation of the principal stress axes.In the present method, the unwrapping of ψ j and ρ jtotλi for the perfectly automatic stress analysis was obtained easily by using the phase unwrapping method that employs the arctangent function.The measured values of ψ j and ρ jλi are not affected by the phase difference error of the quarter-wave plate.The usefulness of the method then was shown for measurement of the ψ j and ρ jtotλi in a frozen stress sphere model.