4UG 29 l924 Thesis presented to the MASSACHUSETTS INSTITUTE OF TECHNOLOGY In partial fulfillment of the requirements for the degree of MASTER OF SCIENCE by WILLIAM PHELPS ALLIS Prepared under the direction of PROFESSOfl PAUL HEYXjAXS Signature redacted ,Signature redacted THE PHOTOELASTIC PROPERTIES OF CELLULOID, GLASS, AND FUSED QUARTZ 1 40273 ri PREFACE - In order to become a candidate for the degree of Master of Science it is necessary to have the degree of Bachelor of Science. As I had not yet presented a thesis in completion of the requirements for the latter degree I presented that part of this work relating to the production of uniform tension in glass (pp. 5 to 27). It is again included here for the sake of completeness, but should not be considered as counting towards the Master's degree. I wish to express here my thanks to the Massachusetts Institute of Technology for its liberality in supplying all the material that I needed for this work. I also wish to thank Professor Heymans and Messfrs T.Frost and W.Dietz for their many usefull suggestions and kind help. Signature redacted. Nancy, France December 5, 1923. A INTRODUCTION When an isotropic transparent body is stressed unequally in different directions it becomes doubly refracting. Let p, q, and r be the principal stresses, and let us consider two rays of light travelling along r, and polarized, respectively, in the planes r,p and r,q. These rays will travell at different speeds, the differencebeing proportional to p - q. After going through a given thickness of material one of these rays will have fallen behind the other, and the relative retardation will be proportional to p - q. Conversely we can obtain the difference p - q of the principal stresses in the material from a measurement of the relative retardation of two rays such as those described above. I will not go into the details of photoelasti- cimetry, it will suffice to say that models of individual pieces of ma.chineryor of complete structures, are made out of transparent materials, and then loaded as 2 the originals would be loaded in practice. Under suit- able conditions the stresses in the models are proportional to those in the original. By studying the dou- ble refraction produced in these models valuable infor- t mation is obtained as to the distribution of stresses in the originals. The success of the method depends to a great extent upon the choice of the proper material out of which to make the models. The investigations presented in this thesis were originally undertaken, at the suggestion of Professor Paul Heymans, to measure those properties of fused quartz which relate to its use in photoelasticime ry. Fused quartz is now being produced sufficiently homogenious and free from internal strains by the General Electric Co.at Lynn, Mass. As it is at present, owing to the small demand, very expensive, all the tests were first made on plate glass, which I obtained from the Standard Plate Glass Co. In order to avoid unnecessary breaking of glass, test pieces were cut out of camphor celluloid, produced by the Dupont Co., and used in setting up the apparatus I and in making the first trial tests. While the work was in progress a new kind of celluloid, deposited from a solution in naphtha, was obtained from Paris, and it seemed desirable to test it also. The tests were accordingly made on all four sub- stances, and this thesis is now presented as a comparison of their respective adaptability to photoelasticimetry. The stresses in the models can only be proportional to those in the originals if the deformations are similar. This requires that the materials obey Hooke's law and that they have the same value for Poisson's ratio. The latterfortunately, does not vary greatly from one material to anotherso that the error introduced by this factor is usually less that the accuracy of the.method. Its value lies in the determi- nation of p + q by lateral extensometer measurements. I therefore measured Poisson's ratio for the materials studied. I also took the stress-strain curves in order to determine Young's modulus and the limit of Hooke's 4 law. The stress-optical coefficient is of course a primary consideration, and the transparency is also important. The following pages contain a description of the measurement of these properties. 5 TESTS UNDER TENSION ATTEPT TO OBTAIN STRESS-STRAIN CURVE The maik difficulty in obtaining the stressstrain curve of glass, or any other brittle material, under tension, is in obtaining pure tension in a test piece up to the breaking point. Brittleness is due to a very small strain at rupture; only half a thousandth in glass. The yielding of materials when stressed tends to render the distribution of the stresses more uniform. Thus, with a test piece in a tension machine, the jaws of the machine at first make contact at only a few points. The material at these points yields, so that the areas of contact widen, and the pressure becomes more uniformly distributed. Also, if the test piece is not well centered it will bend,-and this bending will tend to render the distribution of the stress in the shaft more uniform. If the material of the test 6 piece is brittle two difficulties will be encountered: First, the test piece will tend to break where the jaws come in contact with the glass. Second, it will be found difficult to obtain uniform stress in the shaft. Method of holding test piece The ordinary types of holders are inapplicable to glass because they would crush it or because they would require it to be threaded. Attempts have been made to hold glass test pieces by boring holes in the ends and passing pins through these holes and pulling on these pins. would break at the hole. In every case the test piece The surface of fracture would frequently have several times the area of the minimum cross section of the shaft. This is an indication that glass does not deviate much from Hooke's law even up to the breaking point. of the holes. High stresses arise at the surfaces If there were any deviation from Hook6's law the over-stressed material would yield, and in doing so would pass some of the stress on to the material 7 behind it. If there were enough material behind it the stress would be withstood at this point and rupture would occur in the shaft. If there is no deviation from Hooke's law the surface material will continue to bear the greater part of the load, until it breaks down. At the bottom of the crack thus formed there is at once a very high stress, and the material there fails. This repeats itself, the crack cutting its wiay through whatever thickness of material there may happen to be. The shape of test piece finally adopted is shown in figures 1 and 2. Figure 3 is a photograph of a test piece in the holders, ready to be put in the testing machine. Pins pass through the holes in the two holders and it is through these two pins that the tension is applied. The test pieces were ground by the Pynkham and Smith Co. of Boston. out of half inch steel plate. The holders were cut 1/4 inch pressure tubing slit down the side and slipped over the sides of the test piece prevented direct contact between glass and steel. I found it necessary to have the test piece and 9 83Piece ini Holde 1/ m4 11 holders fit pretty closely. If the angle of the sides of the head of the test piece was but slightly less than the corresponding angle in the holders, so that contact came near the lower end of the head, the fracture would be produced at the point of contact, as several of the test pieces photographed in figure 12 show. The angle of the sides of the head might be slightly greater than the corresponding angle of the holders without harm. Use of double refraction in centering The fracture of glass which has failed under uniform tension has the appearance of finely ground glass. If the stress is not uniform the glass will the stZesJ fail first there where AM is greatest. Usually the stress distribution at this point will be close enough to uniformity to give to this portion of the fracture the frosted appearance. But as soon as some of the material has failed, and a crack has been formed, there will be a concentration of the stress at the bottom of the crack. The high stress at the surface of the crack I 12 falling off very rapidly towards the interior of the material. The glass at the bottom of the crack will fail next, forming a deeper crack with high stress at the bottom of it. The glass here will fail, and so forth, the crack cutting its way through the material with a region of high stress always in front of it. The glass is then failing in a region in which the gradient of the stress is high; and it will produce the familiar "glassy" fracture. The true tensile strength of the glass is somewhere between the tension divided by the cross section, and the tension divided by that portion of the cross section which showgs the frosted appearance. To get the true tensile strength the glass must be broken under pure tension and the frosted appearance obtained over the whole fracture. In-order to verify, before fracture, whether the stress in the loest ri!eo wl niirformI made use of the property of glass to become double refracting, stressed. when Figure 7 shows the apparatus ueed. The light from an arc (a) was focused on a 13 nicol prism (d), which polarized it in a plane inclined at 45Q to the horizontal. it was then rendered parallel and directed on the test piece (g). (f) will be explaned later. The mirror system One of the colors of the beam incident on the test piece can be represented by the vector I, figure 4, vibrating in a plane perpendicular to the plane of polarization. This vector can be decomposed into its horizontal and vertical components I and lb. When the test piece is under tension, the direction of the stress being vertical, Ib will travel slower through the glass than I. 'If we consider the two rays corresponding to these two vectors at the same will instant, Iq will have gone farther than Ib, or lb have suffered a relative retardation with respect to Ia.. If we consider conditions at the same point in the glass, Ia and Ib will be out of phase. In either way of looking at it the phenomenon is proportional to .the stress and to the distance travelled through the glass. As Ia and Ib are out of phase at the farther surface of the glass, their resultant is not, in general, a vector 14 15 vibrating in a plane, but one whose extremity describes an elliptic helix (II,Fig. 4). The beam of light issu- ing from the test piece is elliptically polarized. I now focus it on a second nicol (i, Fig. 7) crossed with the first. through; Only vibrations perpendicular to I can get in this case represented by vector III. The length of this vector, that is the intensity of the light that gets through, depends upon the eccentricity of the ellipse II. The ray incident on the test piece is in rheality composed of a great number of vectors such as I, each corresponding to a different color and therefore vibrating with a different frequency. Upon pass- ing through the glass their components such as Io and Ib will have the same relative retardation, very nearly, and therefore quite different phase differences. The phase difference qis related to the relative retarda__ lion p by the formula 4 rr as can be readily verified. X The vectors representing the different wave lengths will 6ppcribe ellipses such 16 as II, but each with a different eccentricity. If the light incident on the test piece is white, that issuing from the analyzer-will appear colored, for some of the wave lengths will have been reduced in intensity more than others. The color depends on the value of p, which in turn depends upon the stresseand on the thickness oi glass through which the beam has passed. If a screen is placed in the focal plane of the image of the test piece, the image received on it will be colored. If the stress is uniform the color will be uniform. If the stress is not uniform the color at each point of the image will depend on the stresses at the points in the test piece through which the ray reaching that point of the image has passed. The colors of the image represent the stresses in the test piece integrated in the direction of the beam of light. An idea of the distribution of stresses in the test piece can thus be had from the colors of the image on the screen, (j), fig 7. 17 Procedure The first three tests were made in an Amsler, 5000 pound, tension and compression machine. It was found that the distribution of the stress in the test piece could be changed, while the load was on, by moving a nut, similar in function to (e) fig.10, by hitting it with a mallet. As soon as the tension was the nut was moved in such a direction as to make the - sufficient to produce colors in the image on the screen color more uniform. The first test piece used was not quite symmetrical (1, fig. 12). tension with it. I was unable to obtain uniform The fracture had a small frosted area barely visible to the unaided eye. The next test piece was more nearly symmetrical, and I obtained uniform stress up to 500 pounds, or 3600 lbs./ina. I was measuring the index of double refraction with this load on when the test piece broke, without warning. The effect of time on the condition under of glass this tension is, therefore, quite marked. I-- 4> t 'a b 19 The glass had been under tension fifteen to twenty minutes when it broke. The fracture was plane, perpendicular to the axis of the test piece, and with only some small chips along the edges missing. the frosted appearance. Almost half the fracture had This frosted area was symmet- rical about the short axis of the cross section, the direction of the beam of polarized light, but not about the long axis. This showed that the stress had been symmetrical about the shirt axis, as had been observed in the image on the screen. It also showed that the stress had not been symmetrical about the long axis. This could not have been observed in the image for, as was shown above, the colors depend on the integrated stresses along the short axis. In order to observe bending about the long axis it would be necessary to pass a beam of polarized light through the test piece in the direction of the long axis of the cross section. The test piece must be observed in two directions at right angles simultaneously. (%4 F A 21.. SI j 1 ~ 1 ~ ~1 4 ''t Eq |I / 41 I, Ir 9 21 I then designed several mirror systems which would divide the beam of light issuihg from the polarif zer into two, pass these through the test piece in two directions at right angles, and then combine them again to pass them together through the analyzer. My first attempt is shown in figure 5. It was never used, as I thought of a better design before finishing this one. The main objection to it was that the two images formed could not be focused on the same screen. The second design is shown in figure 6. Mir* rors (a) and (f) reflect half and transmit half the light incident on them. Mirrors (b), (c), (d), and (e) are surface silvered, that is they reflect on the side which is silvered. This device was used, but was not very satisfactory because there were too many mirrors to get out of adjustment. In the last design, which is shown in figures 7 and 8 together with the complete optical system used, mirrors (b) and (d), and (c) and (e) of the previous modelwre, respectively, combined into single mirrors. * Lo li .45 4:4 rrtTLs S w a In) n 77sr i 44 LJ~adiP 24 Also, (a) and (f)were fully silvered over half their surface. This throw, two semi-circular spots of light ed on ths screen which joixj along a common diameter, instead of two circular spots of half the brightness, as the previous model did. Besides the increased bright- ness, this produced a field more nearly the same shape as the image, and enabled the images to be brought closer together. This made it easier to pass both beams through the same analyzer. This mirror system is also shown in figure 9 with a test piece in place. The first glass tested while being observed with the aid of one of these mirror systems is shown in (3), figure 12. The nut (e)(Fig. 10) is free to move in all directions in a horizontal plane within the limits imposed upon rod (d) by the hole in plate (g). A similar condition of affairs existed in the Amsler machine, in which test piece (3) was tested. The limits of the hole were reached before uniformity of stress was attained. Although the stress was visibly not uniform I increased the tension until the test N A. Tenr.ion Mahine. 'S NNW to- Igjure 26 piece broke, at a tension of 910 pounds. This would have corresonded to a stress of 6300 lbs./inz had the stress been uniform. The test piece broke in several pieces, one of the fractures having a considerable area showing the frosted appearance. This fracture had a radial structure indicating the probable existance of some twisting. This form of stress cannot be observed optically with the system used. TQ 4etect it it would be necessary to pass a beam of polarized light lengthwise through the test piece. 6300 lbs./in. is doubtless far below the true tensile 6trength of the glass. After this test I changed to the tension machine shown in figures 8, 9, and 10; which was designed by Mr. Kimball of the General Llectric Co. to. produce strictly uniforan tension. Other reasons for the change were that a device for moving nut (e) and a new mirror system could be made iore eaaily for this machine than for the Amaler. The machine needed, be ore it could be used, the two chucks (c),figure 10, so 1 made these. also made the device shown in figures 10 and 11 for adjusting Xx Ic 0 b c Te-t Piece 14olier s,. edjst C- AA Orass Nt Uprn sc/-Cw L ower 14e t~ft, r -d -5 W ir7" -7 krN 1w? 4 ~r 4- r ' - me- = 29 the tension.. By turning the screws kf), figure 10, nut ie) in moved in whatever direction is desired. This displaces rod (d) roughly parallel to itself, carrying with it chuck (c). Holder kb/ turns slightly on pin ki), changing the points of contact between ho.der ano. pin,and pin and chuck. There can be no couple in the plane uf the holder acting on the test piece, for this would produce rotation of the holder around the pin ki) Nor can there be a couple acting on the test piece in the vertical plane perpendicular to the holders, for the test piece is not held firmly in the holders and is free to turn in this plane. There may be a couple in the horizontal plane producing twisting, but if we neglect this, the forces acting on one end of the test piece can be reduced to a simple force. it will suf- fice that this resultant force pass through the center of the test piece in order that there be uniform stress over the middle cross section. By turning screws (f) it was shown that the points of contact between chuck, I I pin, and holder are changed. This changes the point of application of the resultant force, and it can be 31 made to epproach the center of the test piece. A ball bearing between rod id) and chuck (c) would eliminatd the possibility of twisting. Until this time I had not been troubled by failures occuring in the head, and had not realized the importance of the shape of this part of the test piece. The next four test pieces were defective in this respect, the sides of their heads making angles smaller than the corresponding angle in the holders. 7)p figure 12, show the result. (4), (5), and I tried holding test piece (Y) by' casting fusible alloy around the heads. This does Lot allow the test piece to move in the holders, and therefore bending in a plane perpendicular to the holders is not eliminated. The test piece broke as shown. the fracture having no frosted area. at all. Care was exercised in grinding the next test pieces to give the sides of the heads the proper slopes. I could not obtain uniform tension up to the breaking point, however, because, as in the Amsler machine, the hole in plate (g) was not big enough. .32 Time did not permit me to continue the tests, but I believe that, by the method used, uniform tension can be obtained up to the breaking point in a machine providing for sufficient adjustment. 33 TESTS UNDER TENSION TKE STRESS-OPTICAL COEFFICIENT The prkperty upon which the photoelastic method of stress analysis depends is the double refraction incident to stress. The amount of this double refraction is proportional to the stress, and for the same atress differs in different substances. The coef- ficient of proportionality between the double refraction and the stress is known as the stress-optical coefficient. It has the dimensions of the reciprocal of a stress, and is measured in square centimeters per gram weight. It is considered positive if, in a body under tension, the ray polarised in the plane containing the direction of the stress is retarded relatively to a ray polarised in the plane perpendicular to the stress. The stress-optical coefficient is usually negative, The stress-optical coefficient was measured 34 12 4 0 Babinet compensator Figure 13 - - 35 in test pieces that were used in the tension tests. Referring back to figure 4, 1 represents a ray polarised in a plane inclined at 45*. It can be considered as composed of two components I4 and 16 polarised, respecIn the tively, in the vertical and horizontal planes. test piece 14 travels faster than 16. Therefore in the ray issuibg from the test piece Ib is behind I.. The dimensions of the test piece being known, it remains to measure this relative retardation. Babinet compensator. This is done with a This instrument, figures 13 and 14, consists of a plate P and a wedge W cut out of crystals of quartz, a nicol prism N, and a lens L. The wedge can slide parallely to itself over the plate, its motion being controlLed by a graduated screw. The wedge and plate are placed with their crystal axes at right angles so that the double refraction of one is partially compensated for in the other. The wedge has the same thickness as the plate at the middle, so here the double refraction is completely compensated. The total relative retardation produced by klate and wedge varies uniformly with distance along the face of the I wedge, passing through zero at the middle. The are light (a, fig.7) is replaced by a sodium flame and the compensator is placed in the beam issuing from the test piece. Then, if the test piece is unstressed and the nicol N is crossed with the polarizer, the surface of the wedge as seen through the lens L will be crossed by dark bands. The central one corresponds to the middle of the wedge where there is no total relative retardation, and the others to parts of the wedge where the 37 total relative retardation is a multiple of the wave length. In these places the vectors I and I are in phase when they issue from the wedge and have no component III. When the test piece is stressed these bands move, parallelly to themselves, to parts of the wedge where the relative retardation produced in the compensator is equal in magnitude but of opposite sign to that produced in the test piece. The bands can be brought back to their original positions by moving the wedge by means of the screw. The relative retardation in the test piece is then proportional to the turns of the screw. It remains only to determine the factor of pro- portionality between the graduations on the screw and the relative retardation in centimeters.- This is done by turning the screw until the bands have moved a distance equal to the separation between two successive bands. The number of graduations passed on the screw corresponds to a relative retardation of one wave length of the D-line of sodium. The relative retarda- tion measured with the compensator divided by the stress 38 andIthe thikness of the test piece is the stressoptical coefficient. I measured the stress-optical coefficient of the celluloids, of glass, and of fused quarts in this way. For tensions of about 200 pounds, at which these measurements were made, I was able to obtain, as nearly as could be determined, uniform tension. are given in the table on page 67. The results 39 TESTS UNDER BE-DING POISSON'S RATIO Let us consider a rectangular bar under uniform bending; that is, subjected to forces whose bending moment is uniform over the length considered. Such a system of forces is shown in figure 15. The bending moment being uniform, so will be the curvature, and the bar will be bent in an arc of a circle. The 40 upper half of the bar will be under tension, and the lower half under compression. In accordance with the property of matter of contracting laterally when it is under tension and of expanding laterally when it is under compression, and of which Poisson's ratio is a measure, the upper half of the bar will contract laterally and the lower half will expand laterally. These lateral deformations can be derived by multiplying the longitudinal deformations by Poisson's ratio. They will therefore be proportional to distance from the neutral layer, and will give to the bar a lateral curvature of opposite direction to the longitudinal curvature. (Fig. 16) anticlastic. The surfaces of the bar will be This double curvature can be clearly observed on a glass bar. The upper surface is ground "plano" and an optical flat is placed upon it and illuminateo. normally with monochromatic light. Where "Plano" is a term used by opticians to designate a surface which has been ground mechanically as flat as possible, but without the repeated hand grinding and testing which is necessary in producing a "flat" surface. I I 41 the distance between the lower surface of the flat and the upper surface of the bar can be expressed by (+ n n being an integer and X the wave length of the light used, the light reflected from the lower surface of the flat and that reflected from the upper surface of the' bar will interfere. Where the distance is n& they will reinforce each other. Interference bands will thus be formed and will follow the curves obtained by cutting 42 the upper surface of the bar by equidistant planes parallel to the flat (Fig. 17). These bands will resemble a family of hyperbolae. (Fig. 18) Let us make the assumptions usually made in the theory of beams that the bar 'is long enough so that the effects of the compressions produced at the points of application of the forces can be neglected, and let us consider a longitudinal section of the bar (Fig. 19). 43 Let d be the thickness of the bar, 1 the length considered, r the radius of the neutral axis. The portion of the upper surface which fqrmerly had the length 1 now 1 + el -. I whence the elongation r + has the length 1 + el, and we have, from the figure: e = L rI 2r 44 M 45 I will chose as coordinate axes the tangents in the longitudinal and transverse planes and the normal to the upper surface of the bar. When the bar is bent, the equation of the intersection of the upper surface and the xz-plane is: X2 + (r + whence and A22a z)2 (r + d)2 x 2 + Z2 =(2r + d)z r = X 2 +z 2 2z _ d 2 The measurement is taken in an interference band so z can be obtained by counting the interference bands. x can be measured directly. In the measurements made .x was about half an inch and z less than 15 wave lengths of the sodium D-line. z2 is therefore entirely negligible in comparison with x 2 and A, about 1/8 inch, 2 is also negligible in comparison with A . The expression for r then simplifies to x2 r 7=zg- 46 e = dxz x whence Similarly the lateral contraction is given by: c = d/2r' where r' is the radius of curvature in a transverse r'Y=11 2z' section. c = whence Poisson's ratio is 1= E = z'x 2 in which x, z, y', and z' can be measured. With the measurement of Poisson's ratio in view, simultaneously with the work on pure tension, I designed an apparatus for producing uniform bendin,. The bars used were 1/4 by 1 1/4 inches in cross section by 4 inches lorA. The objection may be raised that the width of the bar is not small in comparison with the length and, therefore, the deviations from the theory given above arising where the knif6 edges come in contact with the glass, and annul the lateral curvature, 47 are not negligible even in the middle section where the measurements were made. I at first used a narrower bar. This gave me only one interference band in which I could measure y', and that, being near the center where the slope of the surface is small, was too broad, so that the error in estimating the middle was quite large in comparison with the distance from the center of the bar. The error in C was even larger because this distance figures in the square. wider bar it was tOO I#&e U tus also. When I changed to a change the bending appara- To justify the results I took measurements near the center as well as near the edges and,also,for different values of the load. The values for Poisson's ratio computed from these different measurements were all in good agreement. The bar was supported on two knife edges,and two other knife edges were made to bear down on the extremities of the bar. Several devices were designed and constructed to hold the knives, without being used, because it was found difficult to hold the knives properly parallel. The last device constructed, and the one actually used, is shown disassembled in figure o7F ~ - K L ~ L~4~ 49 20. The same with some minor improvements is outlined in figure 21. A wire passes around rod (c) on which hangs a weight W. Notch (b), in which the wire passes, is in the middle of the bar, so that the forces transmitted'to each of the nuts (d) is the same. These forces are transmitted to the upper plate (h) by the rods (e). The upper knives (j), attached to the upper plate by the cleats (i), are equidistant from the rods (e), and hence each must exert a force W/2 plus half the weight of the system (c d g h i j m) on the bar tested (k). Bar (k) rests on the two lower knife edges. These are equidistant from the holes in the lower plate (a) through which the rods (e) pass, and are hence equidistant from the upper knife edges. They therefore each exert an upward force on the bar equal to the downward forces exerted by the upper knife edges. The weights of the bar (k) and the flat (1) are negligible. Buffers (f) receive the upper plate when the bar (k) breaks. A glass plate km) inclined at 45* reflects the light from a sodium flame down normally to the bar. The reflected light passes up through kim) again and into a 411 TIu T C7 51 microscope. This microscope,is shown in figure 21, is mounted on a bench and is moved by means of a screw graduated in thousandths of a millimeter. measured with it directly. x and y are The distance actually measured is from the nth band on one side of the center of the interference pattern to the nth band on the other, that is 2x and 2y. The flat does not touch the center of the bar, but rests on its edges, on account of the lateral curvature. The distance of the flat from the center of the bar increases with the lateral curvature as W is increased. W can be adjusted so that the center of the interference pattern appears dark. It is then known that the distance of the flat from the bar at the center is a whole number of half wave lengths plus X/4. The nth band from the center, not including the one passing through the center, is n half wave lengths above or below this level. Substituting this Therefore z n>/2 e d xn a Fi ~I1 i I I -4 [' iLOfr 22 N 53 c C And Poisson's ratio e e n n (2y )2 2x x The values obtained by this method are given in the table on page 67. In order to have comparable values, the same method was used in measuring Poisson's ratio for the celluloids. Here interference plenomena cannot be produced, but the curvature is sufficient to be measured directly with a Geneta Lens Measurer. This instrwuent is shaped like a watch, ,and has three steel pointe projecting in the same plane. The outer ones are fixed. The middle one is connected, th1rough a system of levers, to a pointer moving on a dial. When the outer points rest on a surface the middle one is forced in an amount The pointer varying with the curvature of the surface. then indicates the curvature in diopters. One diopter is a curvature whose radius is half a meter. If0r is the curvature in diopters the radius of curvature is 1/2 a- 54 Substituting in the formulae given above e d cr c = dcr' andt 55 TESTS UNDER BENDING STRESS-STRAIN CURVE I have given a formula for the elongation of The stress at the the upper surface of a bent bar. upper surface, assuming that the moduli of elasticity in tension and in compression are the same, and that Hooke's law holds, is given by f = M/S where M is the bending moment and S the section modulus. For a rectangular cross section S = If g, d 6 is the distance between the upper knife edges and g, that between the lower knife edges, and if the rods (e), fig. 19, pass accurately through the centers of the holes in (a), then 2 2 where V is the weight of the system (c d g h i j i). If the rods (e) are not quite centered in the holes in 56 (a) M remains practically unchanged, for let them be Let Ua and U& be off-center by an amount e6g, fig 23. the weights supported by the lower knife edges. U6 + Ua = W + V Taking moments about the center of the bar, half way between the lower knife edges 4 - ( g) - = - -U4= U + ag) - Ubi (f (W + V, 4 Ub = W +L 2 (1 + 2) 92 At the middle of the rod IL W + V( 2 2 + g) - U6 a = 2 IL 2 W+ V 2 ( 2 +4g)- ( 2 2 +4g) -L. g, 2 M, instead of being constant over the rod, varies uniformly, passing by the same value at the middle of the bar that it would have had were the upper plate properly centered. As readings were taken only near therniddle, the small variation allowed by the holes in (a) could not affect the results. We can therefore write: 57 6 ( bd +V) 2 (g, 2 2 , bd (w+v) which gives the stress in terms of quantities which can be measured. In measuring the factors in the formula for e 4 new difficulty enters. Readings to be used in calcu- lating Poisson's ratio were taken when the center of the interference pattern was dark, and the position of the center of the bar was known, plus or minus a half a wave length. In taking a stress.strain curve,read- ings were also taken when the center was dark, but this 58 did not give a sufficient number of points. For inter- mediate points it was assumed that, between successive readings taken when the center was dark, the flat was raised proportionally to the increments in weight. z is thus a multiple of a half wave length minus a z = (n where W, and W. frZ) - fraction of a half wave length. are the preceeding and succeeding weights which produce a dark center, and n is the number of bands counted from the center. The formula used in calculating the stress is correct only if the substance obeys Hooke's law. The formula for the strain on the other hand holds, for a sufficiently long bar, even aboe the elastic limit, for the only assumption made in establishing it is that a cross section through the bar remains plane after the bar is bent. This is true, for there is no more reason for the cross section to bend one way than the other as the bending moment is the same on both sides of it. Let us consider the strain e as the independent 59 variable. Below the limit of Hooke's law W varies linearly with e. When the limit is reached for any part of the bar, tkis part will not, for a given strain e, support its proportional share of the stress. W will therefore fall below its linear value in terms of e. f, as calculated by the formula given above, varies linearly with W. Therefore,after the limit of Hooke's law is reached,the stress-strain curve calculated by this formula will bend down. The true stress-strain curve will bend down even more, but the important point is that the limit of Hooke's law will be indicated on the curve calculated by the above formulae. As the curves for glass and for fused quartz, figure 25, calculated by these formulae did not deviate from a straight line, these substances obey Hooke's law up to rupture; at least if they are not left loaded too long. A time effect was quite noticeable at the higher loads, and present, though less pronounced, at quite small loads. The interference bands would creep across the field of the microscope, very slowly, but fast enough so that successive readings on the same YT4 4 62 band had a tendency to vary in the same direction. The actual change in the value of e during a measurement was too small to show on the curves, and it is probable that two curves in which the loads were left on different lengths of time would differ more markedly in the position of the breaking point than in the slope. The measurements for each of the curves given took about three hours to perform. Curves for celluloid, figure 24, were also taken by this method, using a Geneve Lens 2easurem. Be Because of the greater curvatures, and therefore greater departures from ideal conditions assumed in the theory, these curves are much less reliable than those for glass and for fused quarts. in neither case were the curves continued till rupture. ihey were discontinued when the curvature had reached tne maximum allowed by the apparatus. PHOTOMETRIC MEASUPMeJNTS COEFFICIENT OF ABSORPTION When light travels through a material medium it is progressively decreased in intensity by absorp- tion. The amount absorbeddiin a small length, dt, is a constant fraction of the total intensity i. As di is negative we must Write di = - kidt The factor k is called the coefficient of absorption. It has the dimensions of an inverse length, and is measured in centimeters 1 . The above equation inte- i = i ekt grates into When light passes from one medium to another there is a loss by reflection. The fraction lost at the front and back faces of a slab of material is given by Fresnel's formula (n n + 11)2, where n is the index of refraction. The light which gets through a slab of material is then i = cci = i.ekt (1 .(n - 1) 2 n + 1 1 2 ekt 4n (P n + 17 64 The instrument used, a Lummer-Brodhun photometer, of figure 26, consists of a graduated rod, at each end which is an electric light, Moveable on the rod is a box in the middle of which is a screen, and in the opposite sides of which are holes of equal rizo. Both sides of the screen can be observed simultaneously through a telescope. A slab of the material whose holes, absorption is sought is placed over one of the until both and the box moved along the graduated bod Let x, be sides of the screen appear equally bright. placed the distance read on the rod. The slab is then 65 over the other hole and a balance again obtained. x. be this reading. Let If S and S' are the strengths of the lights and 1 their distance, then, if both holes are free, the intensities of illumination on the two sides of the screen are respectively - x and St (l - x)z When a slab is placed over one of the holes, the illumination of this side of the screen is reduced in the ratio cx. This is compensated for by moving the box nearer one of the lights. Writing that the illumination of both sides of the screen was the same when the readings were taken Cw = (1-X, x S x Dividing A S' - C=2 x, whence OL= But Cn )2 (1- 1 X)2 - n + x ek e 66 e =23026 log(l - x, ) - + xZ X, Xf' 4n 4n 2. - xz n + kt = log (1 k - xx n ekt = 1 x1 - X, hence log x, + log x7 - log(l - xZ) + log 4n - 2 log(n + 1) In this formula everything was measured except n. For this I assumed the following values: Fused quartz n = 1.47 Glass n = 1.51 For the celluloids the terms in n were omitted because n was not known. For camphor celluloid these terms are certainly negligible besides the high absorption. For Camphor celluloid they would only produce a small difference. . The results are given in the table on page67 L TABLE OF RESULTS m0 // .C+ 10 0p0 WS6. 72 0 t m noe A7e Naphtha Kg/cm2 Fused Quartz C+ 0 C .214 H.3 lbs/in lbs/ fin 2 m"C+H2 .1 8703- m 1 x 300k 1.87 n2 .195 0C0 4 -r&r0 428 2.505 - -- 428 10 000 000 6 090 A 6 tress-stra pls strai ht at 1 ast up 185. - 320AD 3020 0(0 2.8 522 68.86 7800 cm 1- 1x1 429 .lu i312.17 274 1 . ---W 25j 00 30 P- 25 000 f; 0fl _ x 9 I- 5. 000 Z6 Glass _ 32P .0 Y-/7-P2 728 000 _ t-4-2-H 4 C 0 (I 1iI 25 000 Celluloid _ 0 totoCD none lbs/in 2zahh Units 0 0 R .108 -0_ _ 3 3 71 _O__ _ this 1 p-t. 7.4 _ 7 1 - 0t 68 CONCLUSIONS In measuring a stress by the photoelastic method, the relative retardation of the two rays is They are made to pass through not actually measured. a comparison member which is stressed until its double refraction exactly compensates that of the member studied. Then, if the thicknesses are the same, the stress in the model and in the comparison member are the same. The comparison member is usually subjected to tension, which is measured directly on a spring balance. Where there is compensation the image of the member appears dark. The precision of a measurement depends on how small a variation in stress will produce a noticeable departure from darkness. Let p be the relative retardation in the beam issuing from the model, and let ap be the smallest variation in retardation which occasions a noticeable departure from darkness. The relative error ist6p/p. bp is independent of p for p is compensated and therefore 69 4? is always from a total relative retardation equal to zero. 4 P being independent of p the only way to reduce the relative error is to increase p. But P = tp - q) Ct in which p and q are the principal stresses, C the stress-optical coefficient, and t the thickness of the member examined. possible. These quantities must be as great as p and q are limited by the limit of Hooke's law, for above this limit the proportionality between the stresses in tne model and in the original no longer holds. The greater the thickness t used the less light gets through. Two materials should be compared in thicknesses which let through the same fraction of the light incident on them. proportional to l/k. and we can write f Such thicknesses are is therefore limited by LC/k, p k/LC. LC/k can be taken as a measure of the precision which can be attained when using models made of a substance having these constants as properties. Its value for the four substances considered is tabulated in the last 70 columin of the table on page 67. Measured by this factor naphtha-celluloid is far superior to any of the other three. Other considerations ented, however, such as the internal strains present in the material. Non- crystalline substances have to be used, and these are either super-cooled liquids or colloids deposited from solutions. in the first case internal strains are produced in cooling. They are functions of the coeffi- cient of thermal expansion, the teaparature of fusion or of softening, the thermal conductivity,and the specific heat. The approximate values of these functions for glass and for fused quartz are given in the table below Property Coef. of exp. X Fusion point F Thermal cond. Specific heat 4cf Fused quartz 9 x 10-8 1658 3.3 x 10.15 1.49 x 10- Glass 7 x 10-6 500 2.9 x 10-3 .20 35. x 104 t71 The product xT is also given because, other thipgs being equal, the strains would be proportional to this product. This product is favorable to quartz as is also the higher conductivity and the lower specfic heat. Similarly, uneven drying produces internal stresses in the celluloids. They are functions of the contraction of the celluloid on drying and of the rate of diffusion of the liquid through the celluloid. The internal stresses in glass and in fused quartz can be largely eliminated by prolonged annealing. To eliminate the internal stresses in the celluloids by slow cring would require much too long. For thicknes- ses above an inch the time of drying would extend into years. Even the lesser thicknesses do not attain A permanent state but are continually "ageing",, and this has to be taken account of. The internal stresses in naphtha-celluloid appear to be particularly high, producing first order blue in a thickness of half an inch. Another consideration is the ease of construction of the model. Celluloid has the great advantage 72 of being easy to machine. the shape desired. Glass has to be ground to Fused quartz may, because of its better thermal properties, be cast to approximately the shape desires and then the surface ground off. From the above it appears that fused quartz has every advantage over the glass tested. Naphtha- celluloid is very much better than camphor-celluloid, except for the very high internal stresses. The rela- tive merits of naphtha-celluloid and fused quartz are more difficult to decidc. They depend mainly on whether the high internal stresses ii the celluloid are sufficient to aniiul the higher accuracy that could otherwise be attained with it. I consider naphtha- celluloid the best on account of the exceptionally high value of the function LC/k and because of the greater ease in making the models out of this substance.