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Stress Analysis by ThreeDimensional Photoelastic Methods

Daniel C. Druckerand Raymond D. Mindlin

Citation: Journal of Applied Physics 11, 724 (1940); doi: 10.1063/1.1712728

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Stress Analysis by Three-Dimensional Photoelastic MethodsDANIEL C. DRUCKER AND RAYMOND D . MINDLINDepartment of Civil Engineering Columbia University New York New York

(Received June 13, 1940)This paper describes the results of an investigation andextension of the various proposed procedures and methods

of analysis for thE photoelasttc determination of threedimensional states of stress. Two important limitations ofthe previous developments are removed. A true extensionto three dimensions is given by the determination of theeffect on retardation of appreciable variation in theorientation of the secondary principal stresses in planesperpendicular to the wave normal (such variation being

INTRODUCTION

T \VO general practical methods have beenproposed, and to some extent used, for thephotoelastic determination of three-dimensionalstates of stress. In one, the stresses are frozenin the model and slices cut from it are analyzed;in the other, the phenomenon of scattering oflight is employed. The technique of measurement required is different for the two cases andtherefore they will be discussed separately. How-ever, both methods depend upon the effects ofstress on the propagation of light, and these arethe same regardless of the method of observation.In the previous developments of the theory, onlythose effects were considered which are present inordinary two-dimensional states of stress.

These effects in three-dimensional states ofstress differ considerably from those present inthe usual two-dimensional case. The rotationalong the wave normal of the axes of secondaryprincipal stress in planes perpendicular to thewave normal is the principal cause of the difference and has not been considered previously.

Knowledge of what occurs in general states ofstress permits broadening of the methods of observation to include entire planes rather thanconfining attention to a single point at a time.

WELLER'S METHODf one wishes to analyze a three-dimensional

state of stress in a model under actual load, the.. Publication assisted by the Ernest Kempton AdamsFund for Physical Research of Columbia University.1 R. D. Mindlin, A review of the photoelastic method ofstress analysis, J. App. Phys. 10, 229 (1939).

724

the rule rather than the exception for general states ofstress). Also, methods for the analysis of whole planes arepresented, thus avoiding the cumbersome and tediouspoint by point procedures that have been advanced. Inthe particular case of plane stress, these methods reduceto a purely optical technique for determining the principalstresses themselves, or more easily their sum, for the entiremodel.

only practical method of general applicationavailable at present is that proposed by Weller.This method is based upon the fact that in atransparent medium, light scattered at rightangles to the incident wave normal is planepolarized, the amplitude of the vibration reaching the eye of an observer being proportional tothe projection of the light vector on the planeperpendicular to the direction of observation.The result is similar to that produced by theanalyzer in a polariscope. (See Fig. 1.) f thelight incident at a point is plane polarized, theintensity of scattered light observed at rightangles to the wave normal will thus vary fromzero when looking along the direction of vibrationto a maximum when looking perpendicularly tothis direction.

When photoelastic materials are stressed theybecome doubly refracting and, for a given wavenormal, light can be transmitted only in vibra-tions along two mutually perpendicular axeslying in the planes perpendicular to the wavenormal. These directions, which we may designate as the axes of secondary principal stressare determined in the same manner as theprincipal directions for plane stress. (The normaland shearing stresses acting on the planes perpendicular to the wave normal need not beconsidered in measuring or calculating secondaryprincipal stresses as the former may be varied

2 R. Weller, A new method for p h o t ~ e l s t i c i t y in threedimensions," J. App. Phys. 10, 266 (1939). Three-dimensional photoelastic analysis by scattered light,Papers presented at the 9th meeting of the Eastern Photo-elasticity Conference, May 1939, p. 19.JOURNAL O ApPLIED PHYSICS



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at will without changing the latter, even thoughsuch variation changes the magnitudes and directions of the true principal stresses.) The. lightscattered from any point 0, Fig. 1 can therefore be considered as coming from two planepolarized vibrations (along p and q differing inphase by an amount depending upon the integrated optical properties of the material alongthe path from the point of entrance (0 ) into themodel to the point at which the scatteringoccurs. n general the two vibrations combine toform an elliptical vibration, the plane of theellipse being perpendicular to the wave normal.The intensity of the light scattered at rightangles to the wave normal is proportional to thesquare of the length of the projection (e.g.,or TT) of this ellipse on a plane perpendicular tothe direction of observation.f a whole plane of the model is illuminated bya sheet of parallel rays of polarized light formedby passage through a polarizer, collimator, anda slit, the light scattered in a given directionperpendicular to the wave normal will in generalvary in intensity from point to point in theplane depending upon the projections of theellipse of vibration associated with each point inthe plane, the alternate lighter and darkerregions forming a fringe pattern. This interference pattern will vary in distinctness and evenin form as the direction of observation is changed,but in general it may be said that the distancebetween fringes in the direction of travel of thelight is a function of the state of stress betweenthe points of minimum intensity.

Weller's method for a general state of stress isto fix attention on any point desired and rotatethe mode until the minimum distance betweenfringes straddling the point is obtained. f thestress field is perfectly uniform in the vicinity,this minimum distance gives the maximum principal stress difference at the point according tothe simple relation3

c 1)3 R. Weller and J. K. Bussey, "Photoeiastic analysis ofthree-dimensional stress systems using scattered light,"Tech. Note No. 737 N. A. C. A.

VOLUME 11 NOVEMBER 1940

where 113 and 111 are the algebraically largest andleast principal stresses, respectively, C is thestress optical coefficient in stress units per unitthickness per fringe, and d is the distance between fringes measured along the wave normal,i.e., along the direction of the intermediateprincipal stress

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EFFECT OF ROTATION OF AXES ON RETARDATION type4As we are interested in determining the effectof rotation of the axes of primary or secondaryprincipal stress, it is desirable to strip the

problem of all stress variables except orientation.The fundamental case to be considered is thusone in which the magnitudes of the stresses remain fixed while their orientations, in planesperpendicular to the wave normal, change at aconstant rate along the wave normal. In anycase we have to solve a problem of wave propaga-tion in a nonhomogeneous, nonisotropic medium.The mathematical formulation of such problemsgenerally leads to a system of simultaneous,linear, partial differential equations of the secondorder with variable coefficients.

The differential equations may be formulatedin terms of an elastic ether theory of light.From a general point of view, this theory is notthe most satisfactory one available; but it isvery convenient for photoelasticity as it is basedupon the ideas of stress, strain, and displacement,and upon the equations of motion with whichworkers in elasticity and photoelasticity arefamiliar. The resulting differential equations areexactly the same as those obtained from theelectromagnetic theory of light if the symbolsrepresenting displacement, density, and shearmodulus are interpreted as magnetic force, magnetic permeability, and reciprocal of dielectriccoefficient, respectively.

The equations of motion in terms of the stressesare

acr/ aT x / aTxz a2u- - - - = p ~ax ay az at 2

where cr,/, rv , r z , TXy , T z , T z are the etherstresses, u , v , w are the ether displacements,and is the density.

The ether stresses may be expressed in termsof the ether displacements and the elasticcoefficients of the ether by three equations of the726

av aw (av aw )cr/= - 2 G a - - - 2 G 2 ~ 2 E Iay az az ay 3)and three of the type

The elastic coefficients the Ei and Gi) arevariables for a nonhomogeneous medium. Theyare related to the principal wave velocities a, b cby equations of the type

5)pin which GI is a stationary value of the Gi .Finally, the principal velocities a, b c redetermined from the stress-optical laws5 bythree equations of the type

in which Vo is the velocity in the unstressedmedium and a and { are stress-optical coefficients, constant for a given material and wavelength, A, of light.

Considering a Cartesian coordinate systemx, y, z with the origin at entrance to the medium,the simplest state of stress that will satisfy theconditions of the present problem is a. simpletension, 11 = Ja, in the yz-plane inclined to thez axis at a variable angle P= (A/2)x, where A isa constant. f the wave normal is in the directionof the x axis, t:.=(21rcr/C)x=2kx is the relativeretardation in radians caused by the magnitudeof J alone. The ratio of rotation to retardationis PIt:. a constant which we shall call R12.Using these specifications for the state ofstress, and combining Eqs. 2), 3), 4), 5), and6) we find the following simultaneous differential

equations for v and w :; M. G. Lame, Leqons sur la Theorie Mathematique del Elasticite Paris, 1866), p. 231.6 E. G. Coker and L. N. G. Filon, Treatise on PhotoelasticityCambridge University Press, 1931), p. 198.

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i}2v {Fv a2V av 1- = H - - B c o s A x - B A sin A x at 2 ax2 ax 2 axa2w aw+B sin A x - - B A cos xax 2 ax

i)2w f)2w a2w aw- -= H - - B c o s A x - - -B A sin xat2 ax 2 ax 2 ax

a2v av+B sin A x - B A cos A x -ax2 axin which( a

f3 )B= fY,

7)

Equations 7) may be reduced to equationswith constant coefficients by referring the de-pendent variables to the axes of principal stress.Calling the displacements along the 2 and 3 axesperpendicular and parallel to the direction of (J)v and w, we find

a2v A 2D aw a2vF - - - -v I I A - = - -ax 2 4 ax at 2a2w A2F av a2wD - - - - - w - H Aax 2 4 ax at 2in which

F=vo 2+(3rr=H-B; D=vo2+af5=H+B.The independent variables may be separated

in the usual manner by settingV V(x)eiwtw= W(x)eiwt

and finally, the dependent variables V and Wmay be separated by differentiation and additionof the two resulting equations. This results inthe same fourth-order linear differential equationwith constant coefficients for both Vand W:

The exact solution of this equation is socumbersome that it is quite unusable. t canhowever be simplified by taking into account theorders of magnitude of the quantities involved.VOLUME 11, NOVEMBER, 1940

f the product 2k}.. is much less than unity mostof the terms in the solution disappear. Even inhighly stressed Bakelite, 2k A will be less than 0.01so that the simplification is justified, especiallyso as the square of the term appears and not theterm itself.

Considering only the forward traveling wavesand taking v M cos w at x 0, we then findv=[K sin kSx+M cos kSx] cos (wt- ;I1

+(L sin kSx) sin (wt- v ~ x ) , (11)where S = 1+ 2 t and K and L are cons an ts.Taking w=N cos w at x=O, the same equationwill hold for w except for constants. Thus thereare four constants to be determined. They canbe obtained in terms of M and N by substituting in Eqs. 8). When this is done, apparentlyeight relations exist among the four constants.Making the assumption above as to the magni-tude of the retardation and also a similar onefor the magnitude of A = 2\0/x and also forS= 1+R2)t, the eight relations reduce to fourwhich determine the constants uniquely. Even ifa rotation of axes of 3600 takes place within adistance as small as a millimeter, and if 2k is assmall as one-tenth of a fringe per inch, }..A and ASare small compared to unity and can be neglected,so that the assumptions are entirely justified.

v and w can then be expressed in terms of theirinitial amplitudes M and N as follows :

v = [ - ~ N sin kSx+M cos kSX]Xcos (wt--JHx)

C; sin kSX) sin (wt- ;Hx)w=[+ sin kSx+N cos kSX]

Xcos (wt--:jHx)+ : sin kSX) sin (wt- :;Hx)

12)

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assuming them to be in phase on entrance. I f pis set equal to zero these reduce to the expressionswhich would be obtained for simple tensionalong z. I f v and w have an arbitrary phaserelation on entrance, it is only necessary to adda phase constant e to the terms multiplied by Mand another phase constant e to the termsmultiplied by N. For example, if circularlypolarized light enters the model, 7r12 is added tothe phase of either the M or N terms dependingupon whether the polarization is right or left,and Mis then set equal to N. The term (wlvH)xappears in all of the phase angles with the samesign and therefore has no influence on therelative retardation.

From Eqs. (12) for the components of vibra-tion, or more clearly from the combined formv = - N sin kSx cos ( wt - ;1/

[M ( 1 ~ sin 2 kSX) ][ w tan kSX)] IXcos wt--It+arctan S I

v r(13)W= R M sin kSx cos w t - ~ x )S yH

[ ( R2 )N 1 S2 sin2 kSx ]Xcos [wt- ;Hx arc tan canskSX) ]

it may be seen that, wherever tan kSx=O, v andw a r e in phase. This requires kSx=n7r, orx=n7rlkS. The distance between points in phaseis thus d= 7r kS. Now 7r k is the spacing onewould find if only the magnitude of the stressesaffected the retardation, so that the factor l iis the effect of rotation. Hence the distancebetween points of equal phase is decreased, orconversely the retardation is increased by rota-tion. Intensity expressions for either plane orcircularly polarized light on entrance can beset up. From these it is found that minima ofintensity occur at intervals of 7rlkS for observation in any direction fixed in space or for a728

direction rotating with the axes of principalstress.The same results were obtained by a quitedifferent method of approach which is of valuebecause of its closer physical connection with

the problem. A series of parallel identical crystalplates (corresponding to constant magnitude ofprincipal stresses) each with its polarizing axesinclined at a constant angle to those of theprevious plate (uniform rotation of the axes ofprincipal stress) was considered. I f the numberof plates is made infinite while the total relativeretardation due to the magnitude of the stressesand the total rotation remain finite, the conditions of the problem are satisfied.

There are two essential parts to the solution.First, general expressions are obtained for thevibrations along each of the two crystal axes inany plate by a deductive-inductive process;second, the limits of these expressions, in usableform, are found as the number of plates becomesinfinite. Both of these steps are rather long andinvolved. A doubly infinite series must be set upand then summed. Series of Bessel functions ofthe zero and first order appear, which reduceeventually to the simple trigonometric terms ofEq. (12).

I t is of interest to note that the justifiableassumptions as to the magnitudes of the rotationand retardation found necessary in the previousmethod do not appear explicitly in this onethough obviously they must be contained in it.

From Eq. (12) we see that if the enteringvibration coincides with an initial direction ofprincipal stress, say the p direction, then thevibration along q at any point is given by

w= :M sin kSX) cos (wt- ; Hx . 14)I f the ratio, R12, of the rotation to the retarda-tion (due to the magnitude of stress) is small,even though the magnitude of the rotation itselfbe large, the intensity along q will be very small,the light will be approximately plane polarizedat all points. The light vector may be said tofollow along the direction of principal stress, i.e.,practically no light will be scattered in thisdirection. The eye of an observer moving alonga helix the intersection of a line along the

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FIG. 2. The light vector follows the axis of principalstress. Q q-o-entering direct ion of vibration of light coincides with a principal axis of stress.principal axis of stress with a cylinder having thepoints observed as its axis) will perceive a continuous dark band. (See Fig. 2.)Equation (14) shows that even if the ratio ofrotation to retardation is high, the intensityalong q is zero wherever the vibration is planepolarized. There is a following around of thelight vector, but between the points of planepolarization the intensity along q may be high.Fringes will therefore appear when there isrotation even though the entering vibrationcoincides with an initial direction of principalstress, while if there is no rotation, uniformintensity will prevail no matter what the direction of observation. Furthermore, the spacing ofthe fringes will be the same as the spacing whenthe entering light has components along bothp and q

The distance between fringes is decreased dueto the rotation of the axes of principal stress.As can be seen from Eqs. (12) and (13), planepolarization or circular polarization will reoccurat intervals given by:

1[' 1d= .k S (15)

The retardation is thus seen to be increased bythe factor S = 1 +R2) where R = 2 p/ f: , or thedistance between fringes decreased by 1/SThe difference between the secondary principalstresses in the general case is thus

C 1f , ) - j q = ~ . - ,d S (16)

VOLUME 11 NOVEMBER 1940

which is to be compared with Eq. (1). As Scanbe quite large, neglect of it may, in many cases,produce appreciable error, though on the safeside as the apparent stress will be higher thanthe true. f the total rotation is small, theformulas show that in practical cases it may beneglected as far as its effect on the distancebetween fringes is concerned. Usually one isinterested in regions of high stress where thefringes will be closely spaced. In such cases, thetotal rotation probably will not be very largeand the ratio of rotation to retardation probablysmall, though often appreciable in effect.While true extinction will occur only if the eyefollows the directions of principal stress in ahelix, it is nevertheless true that for a fixed direction of observation, minima will occur at thesame spacing as given in (15) and although thefringe pattern will not be as distinct it will yieldthe same results. The amount of rotation betweentwo fringes may be obtained by finding thedifference in the orientations for which eachappears darkest.

ANALYSIS OF GENERAL STATES OF STRESSHaving the interpretation of fringe spacing in

terms of stress magnitude and rotation of axes,it is possible to consider the procedures proposedfor making the necessary optical measurements.Weller s suggestion of universal rotation of themodel while under load would require a verycomplicated loading device. Also, the stress atonly one point at a time would be determined.The suggestion is therefore advanced that wholeplanes be investigated.

The state of stress at a point is specified bysix quantities, three normal stresses and threeshearing stresses across the planes on which thenormal stresses act. As the effect of an isotropicstress is only to change the initial index of refraction, it cannot be detected by a photoelasticpattern and therefore only the differences between the normal stresses plus the three shearingstresses, or the difference between the principalstresses and the planes on which they act, canbe found. Without a priori knowledge as to thestate of stress, five independent measurementsmust be made to determine the unknownquantities.



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Let a plane be chosen. Consider some line inthe plane as a reference direction. f a narrowbeam of circularly polarized light is passed inthe plane so that the wave normal is first in thedirection of the reference line, then at one angleto it and next at another, and photographs aretaken of the light scattered at right angles tothe wave normals, three independent measurements will have been made for each point inthe plane. Call the reference line the x axis, theplane the y plane. The fringe spacings obtainedfor three positions of the wave normal do notcompletely determine the state of stress. Twomore independent measurements are required toget U x - U y , Uy -U z , T xy , Tyz, T xz . Two positions ofthe wave normal in the xz plane other than alongx will supply the additional information required.If, however, the orientations of the secondaryprincipal stresses can be determined or areknown, one need work only in the y plane, thethree fringe spacings and the orientations aresufficient.

The general problem of finding the orientationof the secondary or primary principal stresses israther complicated. f the rotation ratio is smallat all points, then plane polarized light enteringalong a direction of principal stress will indicatethe directions of principal stress all along thewave normal. Only when the entering light isvibrating in such a direction will a continuousdark band appear. f the ratio is large, no suchreliable way is available. A continuous dark bandwill not exist, even the entering direction of theprincipal stresses cannot be determined. Dis-

0 1 U2 ~ C ~ + ~ + ~ ) .3 dx d ~ d ~

tinctness of fringe patterns cannot be used as acriterion; minimum distinctness does not necessarily occur when the observer is looking along aprincipal stress direction, and in general it wouldbe difficult to determine such a minimum withany degree of exactness.

Fortunately, in the majority of cases the ratioof rotation to retardation will be small and thedirections of the principal stresses can probablybe determined.

ANALYSIS OF PLANE STRESSIn plane stress, the magnitude and direction

of a principal stress is known at every point,(u. = 0 acts perpendicular to the plane of themodel). There are thus only three quantities tobe determined, the principal stresses in the plane,U1> U2, and their orientation.Consider rectangular axes x, y in the plate(Fig. 3). UY the normal stress in the y direction,is determined from the fringe spacing when lightis passed in the x direction, and n when thewave normal is at two other known orientationswith respect to the x axis. Taking these positionssymmetrically about x, at some angle Y to it,and calling the distance between fringes obtainedalong these directions dx d< we have U y = Cld x U = CI U = C The three normal stressesdetermine the principal stresses and their orien ta -tion in the plane. f Y is taken as 60, the formulas become rather simple. They are entirelyanalogous to those developed for the equiangularstrain rosette.6

17)

6 R. D. Mindlin, The equiangular strain-roset te, Civ. Eng. 8, 546 (1938).730 JOURN L O PPLIED PHYSICS



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a

FIG. 3. Analysis of plane stress. Fringe spacings obtainedfrom three positions of the wave normal in the planedetermine the principal stresses and their orientation.Similar formulas result for ,),=45.

As iTx iTy=iTl iT2, two measurements at rightangles are sufficient to determine the sum of theprincipal stresses. It will be more accurate fortwo-dimensional work to take an isochromaticpicture in the usual manner and then find thesum of the principal stresses as above, usingthe additional result as a check on the accuracyof measurement. For the principal stresses themselves, or for their sum and difference, care willhave to be taken with the sign of the distancebetween fringes. f called positive when due totension, it is negative when due to compression.The sign will often be apparent, if it is not,white light may be employed, the color sequencewill show whether the stress is of the same oropposite sign as the stress at the previous point,and following back to a known point, as forexample one on the boundary, the sign can bedetermined. Due to dispersion, it may be necessary to reduce the load for this determination.The isoclinics can also be used for determinationof the sign. In the case of plane stress, the variation of stress through the thickness is small anda wide beam of light can be employed. The widerbeam will give a much greater intensity ofscattered light and may therefore be desirable.

I t should be noted that a sheet of photoelasticmaterial may appear free from initial birefringence when observed in a circular polariscopeand yet have a fairly high equivalent stressVOLUME 11, NOVEMBER, 1940

perpendicular to the plane, which may make itunsuitable for this proposed method.A two-dimensional state of stress also exists atan unloaded plane boundary. An isochromaticpicture can be obtained to a certain extent butit would have little accuracy. The principalstresses and their orientation can be found bythe method and formulas, (17), given for planestress.A plane of symmetry or a principal plane in athree-dimensional case is very similar to theplane stress case. The quantities found from thethree measurements at 60 will be the differencesbetween the principal stresses and their orientations, which is all that one can expect to obtain.Calling the normal stress on the plane iT z, justsubstitute iTl - iT z for iTl and fJ 2 - fJ z for fJ 2 in (17).

THE FREEZING METHODThe preceding development has all beenapplied specifically to the determination of the

state of stress from the patterns produced bythe light scattered from a plane. t applies alsowith but few modifications to the freezingmethod. In this method7. 8 the model is heated,loaded, then cooled under load. Most of thebirefringence present when the model is underload at the elevated temperature remains aftercooling, and it is not disturbed by careful sawing.Thin slices are cut from the model and analyzed.When the slice is placed in a polariscope, anintegrated effect through the thickness is obtained. f the stress in the neighborhood of apoint is constant, the fringe order appearingthere when the slice is in a circular polariscope is

(fJ p-fJ q)tn = - - ~ ~ - ~ - (18)

where, as before, fJ p and fJ q are the secondaryprincipal stresses in the plane perpendicular tothe wave normal (in general not the plane of theslice), is the thickness of the slice in the directionof the wave normal, and C is the stress-opticalcoefficient. The difference between the secondary

7 G. Oppel, Polarisationsoptische Untersuchung raum-licher Spannungs- und Dehnungszustande, Forschung aufdem Gebiete des Ingenieurwesens 7, 240 (1936).8 M. Hetenyi, The application of hardening resins inthree-dimensional photoelastic studies, J. App. Phys. 10,295 (1939).



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principal stresses is therefore determined directlyfrom the fringe order, their orientation can beobtained in the same way one obtains the iso-clinics for a model under plane stress. f thefringe orders and the orientations are obtainedfor three different directions of the wave normal,all information possible as to the state of stressis determined. The method is entirely analogousto that indicated in the section on Analysis ofGeneral States of Stress and applies withpossibly even more accuracy because of thegreater ease of determination of the directions ofthe secondary principal stresses. Little difficultyis caused by a small stress gradient provided thedirections of j p and j q remain unchanged alongthe wave normal.

Rotation of the slice about only one axis(can be taken in the slice) is required, and onlythree positions need be investigated in thisrotation. Immersion in a liquid of approximatelythe same refractive index will eliminate re-flections.

f the p and q axes do rotate, it should bepossible to find the amount of this rotation in aplane polariscope if the stress is sufficiently high,and generally one is interested only in the highlystressed portions. As shown in the precedingwork, plane polarized light entering along aprincipal stress direction will rotate with theprincipal stress axes and emerge practically planepolarized. Independent rotation of the polarizerand analyzer will find the orientations and theamount of rotation. The correction factor tothe fringe order to find the stress is, as before,1 S. The difference between the secondaryprincipal stresses is given by

n 1jp- jq=_._- .t (19)

732

The corrected values for the differences betweenthe secondary principal stresses and the orienta-tions found as above should be used referred tothe central plane of the slice.

Hiltscher 9 proposed analyzing the slice cut outfrom the model in a microscope point by pointusing convergent polarized light as one does incrystallographic investigations. f rotation of theprincipal axes through the slice is at all appreci-able, the proposal does not appear practicable.

CONCLUSIOKThe investigation of the effect of rotation of

the principal axes in planes perpendicular to thewave normal led to two important results:

(1) Rotation increases the relative phase re-tardation by a factor S=(1+(2cp/Ll)2) which attimes may be quite large. f the ratio 2cp l isnot constant, the ratio of the increment in 2cp tothe increment in d between fringes can be lIsedin regions of high stress.

(2) At high stress, the light vector rotates withthe principal stress directions, plane polarizedlight entering along a direction of principal stresswiII remain practically plane polarized in thedirection of this stress through the entire model.

The method of working in planes rather thanpoint by point should be of value for generalstates of stress when a large portion of the modelis to be analyzed. In the particular case of planestress it yields a purely optical method of deter-mining the sum of the principal stresses, or theprincipal stresses themselves, over the entiremodel.

9 R. Hiltscher, Polarisationsoptische Untersuchung desraumlichen Spannungszustandes im konvergenten Licht;Forschung auf dem Gebiete des Ingenieurwesens 9, 9(1938).

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