On the precision of integrated photoelasticity for hollow glassware

Opfics and Lasers in Engineer!ng 22 (1995) 121-135 Copyright 0 lYY5 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0143~8166/95/$950 014~8166(94)ooo19-0

ELSEVIER

On the Precision of Integrated Photoelasticity for Hollow Glassware H. Aben & J. Josepson Institute of Cybernetics,

Estonian

Academy of Sciences, 21 Akadeemia EE0026 Estonia

tee, Tallinn,

(Received 7 June 1994; accepted 27 June 1994)

ABSTRACT This paper analyses two sources of errors which may occur by measuring stress in hollow glassware using integrated photoelasticity. First, the mismatch of the refractive indexes of glass and of the immersion fluid is considered. Using real measurement data, stress distribution in three different glass articles is calculated assuming various mismatching of the refractive indexes. Secondly, bending of the light rays due to the stress-induced refractive index gradient has been calculated. It has been shown that in order to keep the error of surface stresses below 5%, the immersion fluid should match the glass with a precision of about 0401. In all three samples, bending of the light rays was negligible.

1 THE METHOD

OF TANGENTIAL

INCIDENCE

One of the applications of integrated photoelasticity is residual stress measurement in containers and in other axisymmetric hollow glassware.‘*2 For this, the specimen is placed in an immersion bath to avoid refraction of light, and observed in a polariscope (Fig. 1). By photoelastic measurements the wall of the specimen AB is usually scanned in the section of interest. This procedure is named tangential incidence. For the purpose of this study it is sufficient to consider the case when stress gradient in the axial direction z of the specimen is absent. In this case, the parameter of the isoclinic is zero and optical retardation is caused only by the axial stress cr, YZ

A(x) = 2C

I0

121

uz dy.

(1)

H. Aben, J. Josepson

122

Fig. 1.

Investigation

of a container

by tangential

incidence.

U, is expressed in the form

u* -

2

C2kP2k

(2)

k=O

where Czk are coefficients to be determined using measurement data, and p is the dimensionless radius. Inserting the expansion (2) into eqn (l), we obtain after integration

A(5) -= 2 czkC&(t) 2CR k=” where 5 = x/R, and G,=m

(4)

&

G2k=

t2G2k-2.

If optical retardation measurements are made on many light rays (on system of many values of 0, eqn (3) reveals an overdetermined equations for the determination of the coefficients Czk. Thus, the axial stress distribution has been determined. The radial (Us) and circumferential (ae) stresses can also be expressed in the form of power series a, = k=O U8 =

f$b2kp2k

k=O

(6) (7)

Precision of integrated photoelasticity for hollow glassware

123

Inserting the stress expansions (2), (6) and (7) into the sum rule2*3

(8)

a, = a, + ug

and into the equation of equilibrium UT - uo da,+_=

ap

0

it is also possible to calculate coefficients azk and a 2k-2(k+1)CZk -

b

= 2k

(9)

P b2k,2

1 (10)

2k+l

(11)

2(k + 1) C2k.

2 THE CASE OF MISMATCHING

IMMERSION

In the previous part of this paper it has been assumed that there is no refraction of light at the specimen-fluid interface. However, one has to deal with a certain mismatch of the immersion, i.e. the refractive index of the glass ng is different from that of the immersion fluid yl;. Let us assume that photoelastic measurements are carried out in a diffused light polariscope. Passing of light through the specimen in the case ni < n, is shown in Fig. 2. It is assumed that the registrating

D

Fig.

2.

P

Ill

Schematic representation of a ray of light in the case of mismatching immersion: (D) diffused light source; (P) polarizer.

H. Aben, J. Josepson

124

apparatus has zero aperture, i.e. optical information is obtained light rays which are parallel to the axis y of the polariscope. The following formulae are valid

from

sin (Y nR -=_ sin p IE;

(12)

r = R sin p = R ni sin (Y %

(13)

sin IY= rz R’

(14)

From eqns (13) and (14) we have

The latter formula shows that measurement data which are recorded at a distance x from the axis of the specimen are actually related to the light ray which passes the specimen at a distance r from the axis. It means that the observed optical picture should be compressed in the direction of the x axis with a coefficient ni/ng. If nj > n,, the observed optical picture should be extended with a coefficient n,/n,. From the condition sin (YI 1, it follows that the measurements can be carried out only in the region r max

Experience measurements


06)

-

has shown that it is possible to make photoelastic with considerable mismatch of the immersion.

3 THE RULER

For mixing the First, two fluids is (presumably) specimen glass. A preliminary

TEST

immersion fluid the following technique is often used.* should be chosen, the refractive index of one of which lower, and the other which is higher than that of the It should be possible to mix these fluids. immersion fluid is mixed. The specimen is immersed

Precision of integrated photoelasticity for hollow glassware

12.5

into a bath, and a ruler placed behind the bath under an angle of about 45” with the generant of the specimen, desirably in the part where the curvature of the generant is not big (Fig. 3). If, due to refraction, the image of the ruler inside the specimen is shifted upwards (Fig. 3(a)), then n, > ng. If the shift is in a downward direction (Fig. 3(b)), then ni < yt,. Now, one of the fluids is added to change the refractive index of the mixture, until the shift of the refracted image of the ruler is negligible. If, by photoelastic measurements, monochromatic light is used, the ruler test for matching the refractive indexes should be made at the same wavelength. For estimating the precision of the ruler test, the following experiment was carried out. In the case of a 0.5 litre colourless bottle the ruler test was performed 10 times, starting each time with a mismatching immersion fluid of different n,. Refractive index of the ‘matching’ immersion fluid was each time measured with a refractometer. The results of the ten tests were as follows: l-5209, 1.5208, 1.5209, 1.5210, 1.5210, 1.5214, 1.5212, 1.5210, l-5210, l-5209. The mean value of II, is l-5210, and the standard deviation is 1.73 X 10e4. This means that with a probability of 0.997, ~1,differs from n, by not more than 5-19 X 10-4. In the case of coloured glass the ruler test may give a less precise result. However, the mismatch of the immersion fluid determined with the ruler test may be considered to be about 0~001-0~002. It should be mentioned that measurement of the vertical shift h of the refracted image of the ruler (Fig. 3(b)) gives the possibility of estimating the mismatch of the immersion fluid quantitatively. For this, let us consider the passing of the outermost ray through the specimen, in the case nj < II, (Fig. 4). The following formulae are valid (17)

(18) p = 90” - r 2’

(19)

Assuming that An << 1, the angle y is small and we may take S’ = s in Fig. 4. From eqn (19) we have sinp=cos-Y=

2

(20)

H. Aben, J. Josepson

126

Fig. 3.

The ruler test.

Inserting eqn (20) into eqn (18), and taking Y = 45”, we obtain

(21)

AtI =+.

It can be shown that the formula (21) is also valid when n, > n,. Equation (21) allows estimation of the theoretical precision of the ruler test. Assuming that h is measured with a precision of 1 mm, and that s = 40 mm (typical for bottles), for n, = l-5000, An = 0.0001.

4 BENDING

OF LIGHT

RAYS

Until now we have assumed that light rays pass the specimen along straight lines. Since, due to stresses, the specimen is optically both anisotropic and inhomogeneous, a certain bending of light rays in the specimen takes place.

Fig. 4.

Passing of the outermost

ray through the specimen in the case n, < n,.

Precision of integrated photoelasticity for hollow glassware

Fig. 5.

In cylindrical coordinates

127

Bending of light rays.

the stress-optic

law can be written as2

II,=n,,+C,a,+C,(a,+a,) n,g= n,, + c, ug + C,(a, + aJ

(22)

n, = n, + c,(T, + C,(a, + a,> where C, and C, are the absolute stress-optic constants. Since components of the refractive index tensor are functions of the coordinates, the light rays do not pass the specimen along straight lines but bend according to gradient index optics.4*’ In addition, bending depends on the polarization of the light. For example, in Fig. 5, the angle cpZ denotes angle of deflection for the rays whose vibration direction is parallel to the z axis, ‘pZYdenotes the corresponding angle for rays with vibrations in the xy plane. Angles of deflection (pXyand (p3can be approximately expressed in the following way2a4

an,

1

% =

n,I -Gdy 1

cpz= -

(231

an,

I zdy

no

(24)

where 12,= n, cos20 + ne sin% Linear deflection k

(25)

is Ax=

I

qdy.

(26)

H. Aben, J. Josepson

128

Inserting expressions (22) into eqns (23) and (24), we have w&) 4Cl

= 2 [&k(z+k +

qwk)

+ bx(qL

+ Kk)

+ Gkqh

+

K&J1

k-l

(27) F

= I

2

[&,k

+

b2k)

+

C2kjK2k

(28)

k=l

where K2k

=

ktG2k~-

2

LZk = [G2k__2+ (k - 1)[3G2k_4 Mzk = (k - 1)(&-&2 - !?Gz&

(29) (30) (31)

q = G/C,.

Equations (27) and (28) permit calculation of the angles of bending if the stress distribution in the specimen is known. Linear deflection can be calculated using eqn (26). If these calculations show that bending of light is considerable, an algorithm should be elaborated which takes into account the curved light path. Such algorithms have been elaborated in refraction tomography.(‘.’ Bending of light rays has been considered here as a disturbing phenomenon. At the same time, measurement of the angles ql,, and cpI gives supplementary information about the stress field. This has been used in integrated gradient photoelasticity.

5 EXPERIMENTS To estimate the influence of the aforementioned sources of errors, stress distribution was experimentally determined using matching immersion fluids in three different glass articles. One of the specimens was a tempered tumbler with high compressive stresses on the internal and external surfaces. The second specimen was a tempered salad bowl. Due to large diameter, the light path in the salad bowl was comparatively long and therefore possible bending of light could be large. The third specimen was an annealed bottle, manufactured using a common container production process. Thus, these three specimens were different but typical hollow glass articles. Measurement data for different mismatching immersion fluids was generated on the computer, and corresponding stress distributions

129

Precision of integrated photoelasticity for hollow glassware R = 38.2 -

t=2.1

A

t+31.2-r

Fig. 6.

Geometry

were calculated. Finally, bending using eqns (26)-(28). 5.1 Specimen l-a

of the tumbler.

of the light rays was estimated

by

tumbler

The geometry of the strongly tempered tumbler is shown in Fig. 6. The matching immersion fluid found by the ruler test was ni = 1.520. A mixture of different silicon oils was used as immersion fluid. Axial stress distribution was determined using the algorithm described in Section 1. Using computer-generated data for different mismatching immersion fluids, corresponding stresses were calculated. Axial stress distribution in section 01 for three values of n, are shown in Fig. 7. It is seen that while the general distribution of the stress remains

30 0 gf

-30

2_ N-60

D

-90

3 21

-1201 I"\ .I501 26

Fig. 7.

Axial

I

28

I

30 r (mm

I 32

I

34

stress u, distribution in section 01 of the tumbler: 2 - n, = 1.520 (matching), 3 - n, = 1.525.

l-

n, = 1.515,

H. A ben, J. Josepson

130

Fig. 8.

Dependence of the errors in the internal (ACT,)and external (Age) surface stress in the tumbler, on the mismatch of the immersion fluid.

the same, the stresses on the surfaces depend on the value of ni. This dependence is shown in Fig. 8. Finally, Fig. 9 shows the distribution of the angles of deflection and linear deflection along the coordinate x. By calculating bending of rays, the following values of the stress-optic constants were used:9 C, = -0.8 TPa-‘, C, = -3.4 TPa-‘.

2-

-6 -

’ IntAnal boundary

Fig. 9.

Angular

Extkrnal boundary

(cp*,, cp,) and linear (AX,,, AX,) deflection tumbler.

of the light rays in the

Precision of integrnted photoelasticitJ for hollow glassware

Fig. 10.

5.2 Specimen

2-a

Geometry

131

of the salad bowl.

salad bowl

The geometry of the salad bowl is shown in Fig. 10. The matching immersion fluid was n, = 1.511. Axial stress distributions in section 01 for three values of n, are shown in Fig. 11. Dependence of the error in determining surface stresses on the mismatch of the immersion fluid is shown in Fig. 12. Deflection of the light rays is shown in Fig. 13. 5.3 Specimen 3-a

O-5 litre bottle

The geometry of the bottle is shown in Fig. 14. The matching immersion fluid was n, = 1.514. Axial stress distribution in section 01 for three values of n, are shown in Fig. 15. Dependence of the error in determining surface stresses on the mismatch of the immersion fluid is shown in Fig. 16. Deflection of light rays was much less than in the

50

-100 105

I 106

I 107

I 108

I I09

r (mm)

Fig. 11.

Axial

stress

u, distribution in section 2 -- n, = 1.511 (matching).

01 of the salad bowl: 3 - n, = 1.516.

1 - n, = 1.506,

H. Aben, J. Josepson

132

-2o-

Fig.

12.

Fig. 13.

Dependence of the errors in the internal (Aa,) and external (ACT<)surface stress in the salad bowl, on the mismatch of the immersion fluid.

Angular

(rpx,, rp,) and linear (AX,, AX,) deflection salad bowl.

of the light rays in the

Precision of integrated photoelasticity for hollow glassware

i i R = 33.5

133

8

I I

Fig. 14.

33

stress

34

j

-fig

I

1

Geometry

35 r(m)

tr of the bottle.

36

31

CT, distribution in section 01 of the bottle: 2 -n, = 1.514 (matching), 3 - n, = 1.519.

1 - n, = 1.509,

Fig. 15.

Axial

Fig. 16.

Dependence of the errors in the internal (ACT,)and external (ACTS)surface stress in the bottle, on the mismatch of the immersion fluid.

H. Aben, .I. Josepson

134

former cases: maximum angular deflection was 0.14 min, and maximum linear deflection 3.1 X 10m4 mm.

6 CONCLUSIONS Figures 8, 12, and 16 show that by taking measurements with nonmatching immersion fluid one obtains erroneous values for the surface stresses (which are the most important ones). In Section 3 it was shown that by using the ruler test the value of II, can be determined with a precision of about O*OOl-O-002.According to Figs 8, 12, and 16 the error in surface stresses remains below 5%. Consequently, one should be careful about choosing the immersion fluid. However, the simple ruler test gives satisfactory results. As for the bending of the light rays, Figs 9 and 13 show that angular bending is obviously negligible. Maximum linear deflection is less than 0.02 mm. Since the precision of determination of the coordinate of the measurement axis is mostly lower than 0.05 mm, in the three specimens investigated experimentally, bending of the light rays does not cause additional measurement errors. While this investigation included both annealed (the bottle) and tempered (the tumbler, the salad bowl) glassware, it may be expected that the same influence of sources of errors will be present in other axisymmetric hollow glassware.

ACKNOWLEDGEMENT The financial support of the Estonian acknowledged.

Science Foundation

is gratefully

REFERENCES 1. Aben, H., Idnurm, S., Josepson, J., Kell, K.-J. & Puro, A., Integrated photoelasticity for residual stress in glass specimens of complicated shape. In Second Inter. Con& on Photomechanics and Speckle Metrology: Speckle Techniques,

Birefringence

Methods,

and Applications

to Solid Mechanics,

ed. Fu-Pen Chiang. SPIE 1554 A, 1991, pp. 298-309. 2. Aben, H. & Guillemet, C., Photoelasticity of Glass. Springer Verlag, Berlin, 1993. 3. O’Rourke, R. C., Three-dimensional photoelasticity. J. Appl. Phys., 22 (1951).

872-78.

Precision of integrated photoelasticit?l for hollow glassware

135

4.

Marchand, E. W., Gradient in&x Optics. Academic Press, NY, 1978. 5. Hecker, F. W. & Pindera, J. T., Influence of stress gradient on direction of light propagation in photoelastic specimens. VDI-Berichte, 313 (1978) 745-54. 6.

Lira, I. H. & Vest, C. M., Refraction correction in holographic interferometry and tomography of transparent objects. Appl. Opt., 26 (1987) 3919-28.

7.

Vest, Ch. M., Tomography for properties of materials that bend rays. A tutorial. Appl. Opt., 24 (1985), 4089-94. 8. Aben, H., Integrated photoelasticity as tensor field tomography. In Proc. Int. Symp. on Photoelasticity. Springer Verlag, Tokyo, 1986, pp. 243-50. 9. Berezina, E. N., Photoelastic constants of optical glasses. Sou. J. Opt. Technol.,

40(2)

(1970)

38-39.