O P T I C A L A S P E C T S O F T H R E E - D I M E N S I O N A LP H O T O E L A S T I C ! T Y . *

BY

RAYMOND D. M I N D L I N ,

Department of Civil Engineering, ColumbiaUniversity.

1. TWO-DIMENSIONAL PHOTOELASTICITY.

In two-dimensional photoelasticity a plate of transparentisotropic material is stressed by a uniplanar force systemcoplanar with the middle plane of the plate (Fig. I). Thematerial of the plate thereby becomes optically doublyrefracting and certain of the resulting optical properties aredetermined by examining their effects on p61arized lightpassed through the plate at normal incidence. The state ofstress in the plate is then calculated on the basis of experi-mentally determined relations between stress and opticaleffect carried out under similar conditions.

The state of stress, at a point in a plate loaded only byforces in its plane, is a particular case of a more generalstress condition; the most general being characterized by sixindependent quantities, namely, the magnitudes and thedirections of the three mutually perpendicular principalstresses. The three principal stresses will be denoted in thispaper by the symbols ~3, a2, at, and their relative magnitudeswill be taken as a3 > as > al.

In a plate loaded in its plane, one of the principal stressesis always normal to the plane of the plate and its magnitudeis zero, so that three of the six quantities are known a p r i o r i .The remaining three quantities are the magnitudes of thetwo principal stresses, lying in the plane of the plate, andtheir orientation. An important characteristic of such astress distribution is that, a t each point in the plate, oneprincipal plane of stress is parallel to the middle plane of the

* From a p a p e r presented at the Eighth Semi-Annual Meeting of the EasternPhotoelas t ic i ty Conference, December Io, I938, Co lumbia Univers i ty , New YorkC i t y .

349

3 5 0 R A Y M O N D D . ~IINDL1N. [J. 1:. I.

p l a t e . I t s h o u l d be n o t i c e d t h a t the two p r i n c i p a l s t r e s s e sin the p l a n e of the p l a t e m a y be a l a n d o2 or o2 a n d o3 or¢~ a n d ~1.

W h e n p o l a r i z e d l i g h t is p a s s e d n o r m a l l y t h r o u g h thep l a t e , the r e l a t i v e p h a s e difference of the two w a v e s , i n t o

J ' - ¢

f

FIG. I.

w h i c h the l i g h t is p h y s i c a l l y r e s o l v e d , is f o u n d t o be d i r e c t l yproport ional t o the difference b e t w e e n the two p r i n c i p a ls t r e s s e s in the p l a n e of the p l a t e . S i n c e two of the p r i n c i p a ls t r e s s e s v a n i s h on a free b o u n d a r y of the p l a t e , the r e l a t i v ep h a s e difference t h e r e is d i r e c t l y a m e a s u r e of the t h i r dp r i n c i p a l s t r e s s w h i c h is, of c o u r s e , p a r a l l e l t o the edge ofthe p l a t e .

A p r i l , ~942.] T H R E E - D I M E N S I O N A L PHOTOELASTICITY. 3 5 1

The m a x i m u m s h e a r i n g s t r e s s in the p l a t e is a q u a n t i t yof p r i m e e n g i n e e r i n g i m p o r t a n c e . I t is g i v e n by half thedifference b e t w e e n the a l g e b r a i c a l l y l a r g e s t a n d s m a l l e s tp r i n c i p a l s t r e s s e s . If , a t a n y p o i n t in the p l a t e , t h e twop r i n c i p a l s t r e s s e s in the p l a n e of the p l a t e are of o p p o s i t es ign (one t e n s i o n a n d the o t h e r c o m p r e s s i o n a s a t p o i n t C inFig. I) , the r e l a t i v e p h a s e difference is t h e r e a d i r e c t m e a s u r eof the m a x i m u m s h e a r i n g s t r e s s . I n g e n e r a l , h o w e v e r , i tc a n n o t be said t h a t the r e l a t i v e p h a s e difference m e a s u r e sthe m a x i m u m s h e a r i n g s t r e s s in the p l a t e . I t m e a s u r e s them a x i m u m s h e a r i n g s t r e s s on p l a n e s n o r m a l t o the p l a n e of thep l a t e a n d this will be the t r u e m a x i m u m only w h e n a2 = o.

E x p e r i m e n t s also show t h a t the p o l a r i z i n g axes of thet w o - d i m e n s i o n a l l y s t r e s s e d p l a t e c o i n c i d e with the axes ofp r i n c i p a l s t r e s s in the p l a n e of the p l a t e . H e n c e i t is as i m p l e m a t t e r t o d e t e r m i n e the d i r e c t i o n s of the p r i n c i p a ls t r e s s e s by o b s e r v i n g e x t i n c t i o n posit ions a s the p l a t e isr o t a t e d in i t s p l a n e b e t w e e n c r o s s e d n i c o l s .

R e g a r d i n g the i n d e p e n d e n t d e t e r m i n a t i o n of the m a g n i -t u d e s of the p r i n c i p a l s t r e s s e s in a t w o - d i m e n s i o n a l s t r e s ss y s t e m , t h e r e is an e x t e n s i v e l i t e r a t u r e , t o w h i c h r e f e r e n c e isn o t e s s e n t i a l for the p u r p o s e s of this p a p e r .

2. THI~E-DIMENSIONAL PHOTOELASTICITY.

T h e r e is now in the c o u r s e of d e v e l o p m e n t a t e c n h i q u efor o p t i c a l l y a n a l y z i n g t h r e e - d i m e n s i o n a l s t r e s s s y s t e m s . 1,~,3,4, ~In brief, the p r o c e s s c o n s i s t s in h e a t i n g a m o d e l made from

1 "A New P h o t o e l a s t i c Material ," b y A. G. So lak ian . Mechanical Engi-

neering, 57, 767-77I ( I935)-"Po la r i sa t ionsop t i sche Unte rsuchung rafimlicher Spannungs- u n d Deh-

nungszust~inde," b y G. Oppe l . Forschung auf dem Gebiete des Ingenieurwesens,7, N o . 5, 24o-248 (I936).

3 "Po la r i sa t ionsop t i sche Unte rsuchung des r~iumlichen Spannungszus t~ indesi m konvergenten Licht , " b y R. Hi l t scher . Forschung auf dem Gebiete des In-genieurwesens, 9, N o . 2, 91-1o8 (1938) .

* " D a s Kuns tharz Phenolformaldeheyd i n der Spannungsop t ik , " b y A.Kuske. Forschungauf dem Gebiete des Ingenieurwesens, 9, No. 3, I39-149 (I938).

5 "Pho toe l a s t i c Studies of Three-Dimensional Stress Problems," b y M.He t~ny i , presented a t the F i f t h Internat ional Cor /gress for App l i ed Mechanics ,Cambridge, Mass., September, I938 ; a l so " T h e Fundamentals of Three-Dimen-s ional Pho toe la s t i c i ty , " b y M. He t~ny i . Journal of Applied Mechanics, 5, N o . 4,A-I49-I55 (1938) .

352 RAYMOND D. I~INDL1N. [J. F. i.

a h a r d e n i n g r e s i n (for e x a m p l e , a p h e n o l f o r m a l d e h y d e r e s i nor a g l y c e r i n e p h t h a l l i c a n h y d r i d e r e s i n ) a b o v e a c e r t a i nc r i t i c a l t e m p e r a t u r e , a p p l y i n g l o a d s , c o o l i n g u n d e r l o a d , a n dthen r e m o v i n g the l o a d s . T h e r e s u l t i n g m a t e r i a l is o p t i c a l l yd o u b l y r e f r a c t i n g a n d i t s o p t i c a l p r o p e r t i e s are c l o s e l yc o n n e c t e d with the e l a s t i c s t a t e of s t r e s s p r e s e n t in t h em a t e r i a l w h e n it is in the l o a d e d c o n d i t i o n a t the e l e v a t e d

C

¢, ¢'

¢,(a)

FIG. 2.

t e m p e r a t u r e . O n e of the m o s t s t r i k i n g f e a t u r e s of thep r o c e s s is t h a t the o p t i c a l p r o p e r t i e s of the m a t e r i a l area p p a r e n t l y u n c h a n g e d by c u t t i n g t h e s p e c i m e n i n t o thins l i c e s . T h e photoelastic p r o p e r t i e s of h a r d e n i n g r e s i n s a te l e v a t e d t e m p e r a t u r e s h a v e been well u n d e r s t o o d as a r e s u l tof the w o r k of H e t ~ n y i (see foo tno te No. 5) a n d r e f e r e n c e t ohis p a p e r s s h o u l d be made for the d e t a i l s of the t h e r m a lp r o c e s s .

I t is therefore p o s s i b l e t o d e t e r m i n e the s t r e s s d i s t r i b u t i o nin an e n t i r e s p e c i m e n by d e t e r m i n i n g the o p t i c a l p r o p e r t i e s of

April , 1942.1 THREE-DIMENSIONAL PHOTOELASTICITY. 353

the slices. T h e m o s t i m p o r t a n t c h a r a c t e r i s t i c of the s l i c e s ,w h i c h d i s t i n g u i s h e s t h e m from the p l a t e s e m p l o y e d in two-d i m e n s i o n a l w o r k , is t h a t the o p t i c a l p r o p e r t i e s in g e n e r a lr e p r e s e n t a s t r e s s d i s t r i b u t i o n i n v o l v i n g both n o r m a l a n ds h e a r i n g s t r e s s e s on the p l a n e of the slice. F ig . 2(a) r e p r e -s e n t s a t h r e e - d i m e n s i o n a l m o d e l , a c t e d u p o n by a g e n e r a ls y s t e m of f o r c e s F1, F2 . . . , from w h i c h a s l i c e A B C D is cuta f t e r the m o d e l h a s been c a r r i e d t h r o u g h the t h e r m a l a n dl o a d i n g p r o c e s s p e r f e c t e d by H e t 6 n y i . Fig. 2(b) r e p r e s e n t sthe s l i c e w h i c h is t a k e n a s the x, y - p l a n e a n d s h o w s a s m a l le l e m e n t , E , in the s h a p e of a r e c t a n g u l a r parallelepipedw h o s e f a c e s are p a r a l l e l or p e r p e n d i c u l a r t o the p l a n e of theslice. T h e s t a t e of s t r e s s on the e l e m e n t is d e p i c t e d inF ig . 2 ( c ) . P l a n e bcgf, w h i c h is p a r a l l e l t o the p l a n e of theslice, is n o t a p r i n c i p a l p l a n e of s t r e s s , s i n c e it h a s s h e a r i n gc o m p o n e n t s 7z~ and rzu a c t i n g on it. T o o b t a i n the o r i e n t a -t ion of the e l e m e n t so t h a t only n o r m a l (principal) s t r e s s e sa c t on it, it w o u l d be n e c e s s a r y t o r o t a t e it u n t i l it r e a c h e s ap o s i t i o n s u c h as is i n d i c a t e d in Fig. 2 ( d ) .

T h e r e l a t i v e r e t a r d a t i o n for l i g h t p a s s e d n o r m a l l y t h r o u g hthe s l i c e in the u s u a l m a n n e r ( the z-direction in F ig . 2 ( c ) ) isa m e a s u r e of the difference b e t w e e n the secondary principalstresses, p a n d q, in the x, y - p l a n e w h e r e

fix "Jl- cryp, q - 4- ¢(cr~ -- au)2 + 4rxu2.

2

S i n c e p a n d q are r e s t r i c t e d t o lie in the x, y - p l a n e , they don o t , in g e n e r a l , c o r r e s p o n d t o ~ a n d ~2 or ~2 a n d a~ or ~3a n d ~1.

H e n c e , if t h e r e are s h e a r i n g s t r e s s e s on the p l a n e of theslice, the l a t t e r is n o t a p l a n e of p r i n c i p a l s t r e s s a n d ther e l a t i v e p h a s e difference, m e a s u r e d in the u s u a l m a n n e r , isproport ional , n o t t o a difference b e t w e e n p r i n c i p a l s t r e s s e sb u t t o the difference b e t w e e n the secondary principal stressesl y i n g in the p l a n e of the slice. T h e r e l a t i v e p h a s e differencefor the s l i c e will be r e l a t e d t o the p r i n c i p a l s t r e s s d i f f e r e n c et h r o u g h c e r t a i n a n g l e s w h i c h fix the o r i e n t a t i o n of the s l i c ewith r e s p e c t t o the p r i n c i p a l a x e s .

A u x i l i a r y m e a s u r e m e n t s ( t h a t is, m e a s u r e m e n t s s u p p l e -m e n t i n g t h o s e made in t w o - d i m e n s i o n a l photoelast ici ty) m u s t

354 RAYMOND D. MINDLIN. [J. F. I.

therefore be made to obtain the data requisite for convertingthe phase difference of the slice to maximum phase difference.The principles involved in this process are well known tostudents of crystal optics and there are several methodswidely used by petrologists for making the measurements.It is only necessary for workers in photoelasticity to a d a p tone of the petrological techniques to the requirements ofphotoelasticity and to ascertain the relations between theobserved effects and stress.

3. SOME PRINCIPLES OF CRYSTAL OPTICS.

The optical phenomena encountered in photoelasticityare adequately described by a wave theory of light. Of thewave theories which have been devised, the electromagnetictheory is, in many respects, the most complete. Chrono-logically, the electromagnetic theory displaced an elastice t h e r theory which, before it was abandoned, had beendeveloped to such a stage that it accurately explained all ofthe phenomena of crystal optics which iare now observed inphotoelasticity. Workers in photoelasticity are mostly civiland mechanical engineers who are, on the whole, morefamiliar with the stress equations of motion and the funda-mental ideas of stress and strain than with the principles ofelectricity and magnetism. Since the former are the basesof the elastic e t h e r theories, it seems likely that engineerswill more easily master the principles of crystal optics via anelastic e t h e r theory 6 than via the electromagnetic theory.

Regardless of which theory is used, the conclusion isreached that light is propagated in a transparent, non-isotropic medium only in the form of plane polarized wavesand that, for each wave normal, there are in general two wavevelocities corresponding to two waves whose polarizationdirections are perpendicular to each other and to the wavenormal.

The wave velocities and directions of polarization may beconveniently described in terms of the geometry of Fresnel'sellipsoid. The semi-axes of a central section of the ellipsoid,perpendicular to a wave normal, give the reciprocals of the

6 See, for example, Lam6, "Th6orie de l'l~lasticit~," Paris, I866. ChaptersXVII to XXIV.

April, i942,] THREE-DIMENSIONAL PHOTOELASTICITY. 3 5 5

velocities for that wave normal; and the direction of a semi-axis of the section is the direction of polarization of the wavehaving the velocity given by the reciprocal of the other

z

C

d l b

\I \

II

~'~,~.I

\\

\N ,,

/

I/

! I

t!

FIG. 3.

semi-axis. The three principal semi-axes of the ellipsoid,as distinguished from the two semi-axes of an arbitrary.central section, thus represent the stationary values (maxi-mum, minimum and minimax) of wave velocity. These

356 RAYMOND D. MINDLIN. [J. F. I.

three axes are termed the principal axes of optical symmetryof the medium.

B

FI6. 4-

In Fig. 5, OA, OB, O C are the principal semi-axes of theellipsoid. If a, b, and c are the magnitudes of the principal"wave-velocities, then

OA = I / a , O B = I / b , O C = I / c .

April, I()42. ] THREE-DIMENSIONAL PHOTOELASTICITY. 357

T h e e l l i p s e pqpq is the intersection with the ellipsoid of ap l a n e p a s s i n g t h r o u g h the c e n t e r O, the n o r m a l t o the p l a n eb e i n g ON. T h e s e i m - a x e s Op a n d Oq of the e l l i p s e are ther e c i p r o c a l s of the w a v e velocities kl a n d k2, respectively, fora w a v e n o r m a l ON. Op is the p o l a r i z a t i o n d i r e c t i o n of k2,a n d Oq is the polarization d i r e c t i o n of kl. W e also have

a ->_k~ =>b >=k2 ~ c .

T h e r e are two p l a n e s (DOB and D'OB of Fig. 4) t h r o u g hthe i n t e r m e d i a t e p r i n c i p a l axis (OB) of the ellipsoid w h i c hi n t e r s e c t the ellipsoid in circles. S i n c e the s e m i - a x e s of eachof t h e s e s e c t i o n s are e q u a l , a w a v e f r o n t p a r a l l e l t o e i t h e rone has only one velocity. T h e n o r m a l s (OQ a n d OQ' inFig. 4) t o t h e s e p l a n e s are c a l l e d the o p t i c a x e s ; they lie inthe x, z - p l a n e a n d m a k e e q u a l a n g l e s f~ w i t h the z - a x i s . T h ea n g l e 2~ is t e r m e d the optic axial angle. F r o m the k n o w ng e o m e t r i c a l p r o p e r t i e s of the ellipsoid we h a v e

~ a2 - b2 ~ b2 - &sin ft = a2 & , cos ft = a2 - - C2 (I)

F u r t h e r m o r e , if 0 ( F i g . 3) is the a n g l e b e t w e e n the w a v en o r m a l ON a n d the o p t i c axis OQ, and if 0' is the a n g l eb e t w e e n the w a v e n o r m a l a n d the o p t i c axis OQ', we h a v e ,from the g e o m e t r y of the ellipsoid

k, 2 = ½[-a2 -t- & -k (a2 - c2) cos (0 - 0')~,1k2 ~ = ½E a ~ + c 2 + ( a 2 - c 2) c o s ( 0 + 0 ' ) ~ , [ ( 3 )

so t h a tkl 2 - k22 = (a2 - &) sin 0 sin 0' . (3)

I n Fig. 3, p l a n e pqpq is a c e n t r a l s e c t i o n of F r e s n e l ' sellipsoid d r a w n p a r a l l e l t o the w a v e f ron t . F o r l i g h t t ra-v e r s i n g the s l i c e a t n o r m a l i n c i d e n c e , this p l a n e is p a r a l l e l t othe p l a n e of the slice, a n d i t s n o r m a l , ON, is bo th the w a v en o r m a l a n d the n o r m a l t o the slice.

T h e p l a n e s of polarization are NOp a n d NOq. T h e s ep l a n e s b i s e c t , i n t e r n a l l y a n d externally, the d i h e d r a l a n g l e sb e t w e e n t h e p l a n e s c o n t a i n i n g the w a v e n o r m a l ON a n d theo p t i c axes OQ and 00_'. T h a t is, p l a n e s NOp a n d NOqb i s e c t the a n g l e s b e t w e e n p l a n e s NOQ a n d NOQ'.

358 RAVMO~'~) D. MINDL1N. IJ. V. I.

T h e m o s t g e n e r a l c r y s t a l is biaxial, t h a t is, t h e r e are twoo p t i c a x e s ; but, if b = c or b = a, the c r y s t a l b e c o m e s u n i a x i a l .C o n s i d e r i n g F r e s n e l ' s ellipsoid ( F i g . 3), with i t s g r e a t e s t axis2/c, m e a n axis 2/b a n d l e a s t axis 2/a, if we d e c r e a s e the m e a naxis u n t i l it e q u a l s the l e a s t a x i s , the ellipsoid d e g e n e r a t e st o a p r o l a t e s p h e r o i d for w h i c h the z-axis is bo th the axis ofr e v o l u t i o n and the o p t i c a x i s . T h e o p t i c a x i a l a n g l e (2~2)d e c r e a s e s t o zero so t h a t 0 a n d 0' are e q u a l . T h i s is theellipsoid for a positive u n i a x i a l c r y s t a l .

T o o b t a i n a negative u n i a x i a l c r y s t a l , i n c r e a s e the m e a naxis of F r e s n e l ' s ellipsoid u n t i l i t e q u a l s the g r e a t e s t axis 2/c.T h e ellipsoid d e g e n e r a t e s t o a n o b l a t e s p h e r o i d with thex - a x i s as both the axis of r e v o l u t i o n a n d the o p t i c a x i s .T h e o p t i c a x i a l a n g l e i n c r e a s e s t o ~r and 0 + O' = ~r.

T h e o p t i c a x i a l a n g l e , z~2, is d e f i n e d as t h a t a n g l e b e t w e e nthe o p t i c axes w h i c h is b i s e c t e d by the axis of l e a s t v e l o c i t y(z). W h e n 2~2 < r/2, the b i a x i a l c r y s t a l r e s e m b l e s , s o m e -w h a t , a p o s i t i v e u n i a x i a l c r y s t a l a n d the more a c u t e thea n g l e , the c l o s e r is the similarity. Such a c r y s t a l is c a l l e dpositive biaxial. W h e n 2f2 > ~-/2, the c r y s t a l r e s e m b l e s an e g a t i v e u n i a x i a l c r y s t a l a n d , the c l o s e r 2f~ a p p r o a c h e s ~r,the c l o s e r is the similarity. Such a c r y s t a l is negative biaxial.W h e n 2~2 = ~-/2, we h a v e , from Eq. (I)

a 2 - b2 = b~ - d. (4)

T h e r e l a t i v e p h a s e difference, ~i, of the two w a v e s , t r a v e l i n ga d i s t a n c e 0 t h r o u g h the c r y s t a l l i n e m e d i u m with velocitieskl and k.o is g i v e n by

27tO= T (n2 -- hi), (5)

where nl and n2 are the indices of refraction corresponding tokt and k~, respectively; i.e., if k is the velocity of light in vacuo,

k k- - .n x = = ( 6 )

Since the velocity differences are small in comparisonwith the velocities themselves, we may write, for Equations

Apri l , I942.] T H R E E - D I M E N S I O N A L PHOTOELASTICITY. 359

I ) , (3) a n d (4)

s i n Q ~no--nb ~b- -n~= - - , COS Q = - - - - ,na - - n c na n e

n2 - n l = (n~ - no) sin 0 sin 0' ,

n o - n h = n ~ - n ~ w h e n 2~2 = ~r/2.H e n c e

27rp= ~ - (no - no) sin 0 sin 0' .

(7)

(8)(9)

io)

4. STRESS-OPTICAL RELATIONS.

If the s c a l e of F r e s n e l ' s ellipsoid is a d j u s t e d so t h a t au n i t of l e n g t h r e p r e s e n t s the veloci ty of l i g h t in vacuo, t h es e m i - a x e s of a n y s e c t i o n r e p r e s e n t i n d i c e s of refraction.E x p e r i m e n t s i n d i c a t e t h a t the s a m e type of r e l a t i o n e x i s t sb e t w e e n s t r e s s a n d c h a n g e of i n d e x of r e f r a c t i o n a s b e t w e e ns t r e s s a n d s t r a i n d H e n c e , the o p t i c a l p r o p e r t i e s w h i c h a no r i g i n a l l y isotropic m a t e r i a l a s s u m e s on the a p p l i c a t i o n ofs t r e s s m a y be v i s u a l i z e d by i m a g i n i n g t h a t F r e s n e l ' s e l l i p s o i dis the r e s u l t of t h e d e f o r m a t i o n of a s p h e r e of r a d i u s nos u b j e c t e d t o p r i n c i p a l s t r e s s e s a p p l i e d p a r a l l e l t o the p r i n c i p a laxes of the ellipsoid. T h e m a t h e m a t i c a l s t a t e m e n t of thisp r i n c i p l e is

no - no = Cl0.1 + C2(0.2 + 0.3)),|nb - - n0 : 010"2 "3i-- 02(0"3 + 0.1), t (I I )

0 . 'n , - no = Cl0.~ + C2(0.1 + 2),j

in w h i c h C1 a n d C2 are c o n s t a n t s , n , > nb > na a n d 0.3 > 0.2> ¢1. T h e axes a, b, c, a n d I, 2, 3 a n d x, y, z a r e c o i n c i d e n tin the o r d e r n a m e d , i.e., a, I a n d x are coincident, etc . If C1is positive, a u n i d i r e c t i o n a l t e n s i o n (0.2 = 0.1 = o) will e l o n g a t ethe s p h e r e t o a p r o l a t e s p h e r o i d and so p r o d u c e a p o s i t i v eu n i a x i a l c r y s t a l . If , c o m b i n e d with this t e n s i o n , a c o m p r e s -sion is a p p l i e d a t r i g h t a n g l e s , a p o s i t i v e b i a x i a l c r y s t a l isp r o d u c e d w h e n the a b s o l u t e m a g n i t u d e of the c o m p r e s s i o n isless than the a b s o l u t e m a g n i t u d e of the tension. W h e n them a g n i t u d e s are r e v e r s e d , a n e g a t i v e b i a x i a l c r y s t a l r e s u l t s .

T "On the Equi l ibr ium of Elas t ic Solids," by J . C. Maxwell . Transactionso f the Roya l Soc i e t y o f Ed inburgh , 2o, 87-12o (1853).

360 R A Y M O N D D . M I N D L I N . [J. F. I.

W h e n we h a v e a s i m p l e s h e a r , i .e., w h e n a b s o l u t e m a g n i t u d eof the t e n s i o n a n d c o m p r e s s i o n are e q u a l , we f i n d , s u b s t i t u t i n g0.8 = - 0.1 a n d ~2 = o in Eq . ( I I ) a n d u s i n g Eq . (9) , t h a t theo p t i c axes i n t e r s e c t a t r i g h t a n g l e s .

It is a p p a r e n t t h a t a k n o w l e d g e of the t h r e e p r i n c i p a li n d i c e s of r e f r a c t i o n is n e c e s s a r y for the c a l c u l a t i o n of thet h r e e p r i n c i p a l s t r e s s e s from E q u a t i o n s (I I). T h e d i f f e r e n c e sb e t w e e n r e f r a c t i v e i n d i c e s are g e n e r a l l y e a s i e r t o m e a s u r et h a n the i n d i c e s t h e m s e l v e s . T h e s e differences are p r o p o r -t i o n a l t o the p r i n c i p a l s t r e s s differences. T h u s from E q u a -t i o n s (I 1)

no - na = C(0. - i),I

. o - = - " ( 1 2 )

m - n c = C ( ~ 2 - ~ , ~ ) , j

w h e r e C = C1 - C2.F r o m E q u a t i o n s (I2) ' , (IO) a n d (7),

9

o'3 - - ¢ I 2"rrpC s i n 0 s i n 0 'X~ sin2 f~

0"2 - - 0 . , = - - ' ( 1 3 )2 ~ - p C sin 0 sin O'

X~ cos2 ft J0.3 - cr2 = 2 7 r o C s i n 0 s i n 0 ' "

Of the q u a n t i t i e s on the r i g h t in E q u a t i o n s (13), ~ is ap h a s e r e t a r d a t i o n w h i c h m a y be m e a s u r e d w i t h a c o m p e n -s a t o r . T h e w a v e l e n g t h , X, l e n g t h , o, a n d s t r e s s o p t i c a lcoefficient C are d e t e r m i n e d in the u s u a l m a n n e r . H e n c e ,if we can m e a s u r e the a n g l e s 0 a n d O' b e t w e e n the w a v en o r m a l a n d the o p t i c a x e s , we c a n d e t e r m i n e the m a x i m u ms h e a r i n g s t r e s s ½(0.a - - 0.1). If , in a d d i t i o n , we can m e a s u r ethe o p t i c a x i a l a n g l e 2f~, we c a n d e t e r m i n e all t h r e e p r i n c i p a ls t r e s s differences.

5. SPECIAL CASES.

T h e r e a r e c e r t a i n s p e c i a l c a s e s for w h i c h it is n o t n e c e s s a r yt o m e a s u r e all of the t h r e e a n g l e s O, O' a n d f~; and o t h e r s forw h i c h the m e a s u r e m e n t of t h e s e a n g l e s will y i e l d m o r ei n f o r m a t i o n than s i m p l y the differences b e t w e e n p r i n c i p a ls t r e s s e s .

M o s t i m p o r t a n t is the case w h e n the s t r e s s d i s t r i b u t i o nh a s a p l a n e of s y m m e t r y . This will o c c u r w h e n both the

April, 1942.] THREE-DIMENSIONAL PHOTOELASTICITY. 36I

m o d e l a n d the l o a d i n g s y s t e m are g e o m e t r i c a l l y s y m m e t r i c a la b o u t a p l a n e . T h e p l a n e of s y m m e t r y is a p r i n c i p a l p l a n eof s t r e s s so t h a t a s l i c e w h i c h c o n t a i n s this p l a n e will have ar e l a t i v e r e t a r d a t i o n p r o p o r t i o n a l t o one of the t h r e e p r i n c i p a ls t r e s s differences. T h e p l a n e w h i c h c o n t a i n s the o p t i c axeswill be e i t h e r p a r a l l e l or n o r m a l t o the p l a n e of s y m m e t r y .I n the l a t t e r c a s e , the b i s e c t o r of the o p t i c a x i a l a n g l e ise i t h e r p a r a l l e l or p e r p e n d i c u l a r t o the p l a n e of the slice.H e n c e we have 0 = 0' = 7r/2 or 0 = 0' = f~ or 0 = 0' = 7r/2- f~, d e p e n d i n g on w h i c h p r i n c i p a l p l a n e of o p t i c a l s y m m e t r y

c o i n c i d e s with the p l a n e of the slice. H e n c e , if we m e a s u r e0 a n d a in a d d i t i o n t o ~, we m a y c a l c u l a t e the t h r e e p r i n c i p a ls t r e s s differences from E q u a t i o n s ( I 3 ) .

The t w o - d i m e n s i o n a l s t r e s s s y s t e m o r d i n a r i l y o b t a i n e d int w o - d i m e n s i o n a l photoelast ic i ty is a s p e c i a l form of the caseof s y m m e t r y a b o u t a p l a n e ; for , in a p l a t e in a s t a t e of p l a n es t r e s s , the p l a n e of the p l a t e is a p r i n c i p a l p l a n e of s t r e s s a n d ,in a d d i t i o n , the p r i n c i p a l s t r e s s n o r m a l t o the .p la te h a s zerom a g n i t u d e . A s in the more g e n e r a l case of s y m m e t r y a b o u ta p l a n e , we m e a s u r e 0 a n d f~ (0 a n d 0' b e i n g e q u a l ) b u t now,s i n c e one of the p r i n c i p a l s t r e s s e s is z e r o , the o t h e r two m a ybe d e t e r m i n e d s e p a r a t e l y from E q u a t i o n s ( I 3 ) . This m e t h o dfor the i n d e p e n d e n t d e t e r m i n a t i o n of the two p r i n c i p a ls t r e s s e s in t w o - d i m e n s i o n a l photoelast ic i ty was f i r s t d e s c r i b e dby H i l t s c h e r ?

H i l t s c h e rz also o b s e r v e d t h a t , s i n c e one of the p r i n c i p a ls t r e s s e s is zero on a free b o u n d a r y in a t h r e e - d i m e n s i o n a ls t r e s s field, the o t h e r two p r i n c i p a l s t r e s s e s m a y be d e t e r m i n e dif 0 a n d f~ are k n o w n .

I n a p r i s m a t i c or c y l i n d r i c a l b a r s u b j e c t e d t o St . V e n a n ttorsion, s one of the p r i n c i p a l s t r e s s e s a t a n y p o i n t in the b a ris zero a n d the o t h e r two are e q u a l in m a g n i t u d e a n d o p p o s i t ein s i g n . T h e zero p r i n c i p a l s t r e s s d i r e c t i o n is p e r p e n d i c u l a rt o a g e n e r a t o r of the b a r a n d the o t h e r two p r i n c i p a l s t r e s sd i r e c t i o n s m a k e a n g l e s of 45 ° with the g e n e r a t o r . T h i s s t a t eof s t r e s s c o r r e s p o n d s t o a b i a x i a l c r y s t a l with a n o p t i c a x i a la n g l e of r / 2 , the o p t i c axes b e i n g p a r a l l e l a n d p e r p e n d i c u l a rt o the g e n e r a t o r . A s l i c e of the b a r n o r m a l t o a g e n e r a t o r

~"Theory of Elasticity," by S. Timoshenko. McGraw-Hill Book Co.,New York, 1934. P. 228.

362 t~AYMOND D. MINDLIN. [J. F. I.

therefore gives zero phase difference for light at normalincidence; but a slice at any o t h e r ang l e to a generator willgive a phase difference from which we may calculate theprincipal stresses. To effect this calculation, it is necessary toto know the angles (0 and 0') which the wave normal makeswith the opt ic axes. One of these will be constant and equalto the angle between the generator and the normal to theslice, but the o t h e r will vary over the section. For eachpoint in the slice, we need therefore measure only the latterang l e (0') and the phase difference ~. Then Equations (13)give the principal stresses directly, provided the wave-lengthk, the slice thickness p, and the stress optical coefficient Care known. It was observed by Het6nyi 5 that a 45° sliceexhibits the maximum number of fringes, but it should benoted that the fringes do not represent loci of constantprincipal stress difference. It is necessary to know the angle0' for each point in the 45° slice. This angle may be de-termined in t.he following manner: Let O be a point in the45° slice, at which 0' is to be found. The normal to the sliceat O and the line through O parallel to a generator of the bardetermine a plane which intersects the plane of the slice in aline O A . Let O B be a polarizing axis of the slice at O (de-termined from the isoclinics in the usual manner) and letangle A O B = a. Then it may be shown that

sin 0' = (I + cos2 2a)112.

Similar formulas hold for slices cut at o t h e r angles or viewedobliquely.

6. OPTICAL TECHNIQUES.

There are three standard methods for measuring n~nb and n0 but none of them appears to be suitable for thecurrent development in three-dimensional photoelasticity.

In Stokes' method,9 a plane parallel p la te of the crystalis placed over a rectangular grid of ruled lines and the latterare observed through a microscope. On account of thedouble refraction in the plate, each family of parallel linesin the grid will appear in focus at two positions of the micro-scope objective. The three indices can be calculated from

9"Analytical Theory of Light," by J. Walker, Cambridge UniversityPress, 19o4. Pp. 225-230.

April, 1942. ] THREE-DIMENSIONAL PHOTOELASTICITY. 363

the relative focus positions. However, the photoelasticdouble refraction is so weak that this method does not appearfeasible.

The prism method 10 would require the cutting of a prismfrom the model for each point at which the stress is desired.

The total reflection methods n are very accurate but theymeasure the optical properties of a surface layer of thespecimen. In a photoelastic slice the surface properties arenot a reliable indication of the internal optical characteristics.

For determining the differences between principal re-fractive indices in a doubly refracting plate, we have seenthat it is necessary to measure the relative phase retardationof the plate, the optic axial angle (2~2) and the angles (0 and0') between the optic axes and the plate normal. Themeasurement of the three angles is ordinarily effected by oneof two methods; one involving the use of the universalrotating stage12 and the other involving observations withconverging polarized light.13 The latter is the method usedby Hiltscher? On account of the very weak double refractionof photoelastic materials, and the necessity of using thinslices, the converging light method is likely to be inaccurateif not impossible in many cases. A combination of theuniversal rotating stage for measuring the three angles 0, 0'and ~ with a wedge compensator for measuring ~ seems tooffer good possibilities for the basis of an optical techniquefor three-dimensional photoelasticity.Note added i n proof:

Since this paper was written, another photoelastic method for analyzing three-dimensional systems has been developed by R. Weller. Reports of recent workin this field may be found in the following papers:

R. Weller, Journa l o f App l i ed Phys i c s , Vol. IO, p. 266 (1939).R. Weller and J. K. Bussey, Tech. Note No . 737, Na t . Adv. Comm. f o r Aero-

naut ics .(1939).R. Weller and B. Fried, B u l l . No . Io6, Ohio S t a t e Univ. Eng. Exp. Sta. (I94o).D. C. Drucker and R. D. Mindlin, Journa l o f App l i ed Phys i c s , Vol. II, p. 724

(194o).P. R. Rosenberg, Proc. I3 th Semi -annua l Eastern Photoelasticity Conference,

Mass. Inst. of Technology, Cambridge, Mass., June, I941.

10j . Walker, op. cit., pp. 2 3 o - 2 4 2 .11j . Walker, op. cit., pp. 2 3 0 - 2 4 2 .1~"Manual of Petrographic Methods," by A. Johannsen. McGraw-Hill

Book Co., New York, 1918. PP. 487-507.a3A. Johannsen, op. cit., pp. 449-486. See also reference No. 3.