THE FAILURE OF MATERIALS AS a result of stress is both common and costly. One way to guard against such a failure is to test a sample object by applying enough stress to destroy it; when an identical object is put in service, the stresses are kept well below the destructive level. Another way is to make a model of the object out of plastic or some similar material and then to subject the model to photoelastic analysis, which yields a pictorial representation of the stresses in it.

Recently Frank R. Seufert of Cleveland showed me some examples of his photoelastic analysis of models of various objects. When someone is concerned about the stresses on an object, Seufert makes a model of it in Lexan or Tuffak plastic about an eighth of an inch thick. If the object is large, the model is a scaled-down version. Seufert mounts the model in a wood frame and undertakes to mimic the stress on the real object by stressing the model with rubber bands or machine screws. The screws work better because they can be tightened slowly and controlled more easily.

Seufert puts the model in front of a polarizing filter and illuminates it with light that passes from a 200-watt lamp through a ground-glass diffuser and then through the filter. Between the model and a 35-millimeter camera (a single-lens reflex model) he positions a long lens hood made out of cardboard. Attached to the camera is a Vivitar 2X teleconverter with its lenses removed. It serves as an extension tube to move the lens farther from the film; a different type of tube would serve as well. Seufert's setup includes a 135-mm. telephoto lens rated {t f2. 5. A second polarizing filter and a light blue filter (designated 80A) are mounted on the lens. The camera is held steady with the DD top piece from a tripod; the piece is attached to a block of plywood.

To make a photograph Seufert orients the first polarizing filter with its axis of polarization 45 degrees from the vertical. Then he puts his plastic model in front of it and turns on the lamp. As he sights through the camera he rotates the polarizing filter in front of the lens until a meaningful pattern is superposed on the image. of the plastic model. He records the scene with a cable release that also triggers an electronic flash positioned near the lamp. The extra light is needed for a good photograph.

The film is Fuji color negative rated at ASA 100; Seufert has glossy prints made from the negatives. The colored lines in the accompanying photographs reveal the stress patterns in the plastic models. Regions under stress stand out. They are the places where the object is most likely to fail.

The first polarizing filter in Seufert's setup polarizes the light. The passage of this kind of light through stressed plastic encodes information about the stresses. The second polarizing filter (the one mounted on the camera) makes the coded information visible to an observer. Figure 3 depicts the stresses on a tiny, thin element that is part of a model being investigated by photoelasticity. Each edge of the element is under stress because of forces pulling perpendicular to the edge. Each edge is also under shear as material on opposite sides of the element attempt to slide in opposite directions.

Such is the nature of the stresses on a randomly chosen element. The picture is simplified if a differently oriented element is chosen at this region in the material. The new element is special in that it is square and has two important axes called the principal stress axes. The advantage of considering such a specially oriented element is that its edges are not affected by shear. The only stress on an edge is the perpendicular one. The orientation of the principal stress axes is what is revealed in a photograph of a stressed model illuminated with polarized light.

To understand the interaction of polarized light with the principal stresses in the plastic it is necessary to understand the nature of polarized light. According to classical physics, light is a wave composed of oscillating electric and magnetic fields. It is a peculiar kind of wave in that nothing material participates in the wave motion. Water waves accord more with intuition because a wave involves oscillations of the water surface. Something material participates. In the wave concept of light, however, the oscillation is of immaterial electric and magnetic fields.

The electric components specify the polarization of light. An electric field is a vector (of a specified size and direction) assigned to a point to be examined, as is shown in Figure 4. The concept is useful when one considers how a charged particle might behave when it is placed at that point. It is also useful in forming a mental picture of light.

The illustration can be regarded as a snapshot of a light wave. Superposed on the ray, which indicates the direction of travel, are some of the electric vectors assigned to points along the ray. One point is singled out for scrutiny. At that point in the first snapshot the electric vector is long and points upward. A positive particle there would "feel" a strong force upward.

The light continues to move toward the right after the first snapshot is made. The electric field at the chosen point changes quickly, literally at the speed of light. A new snapshot reveals the change. The electric vector at the designated point is now downward. Since it is not one of the largest vectors in the illustration, the electric field is not as large as it could be and not as large as it will be immediately after this snapshot. A positive particle at this point would now "feel" a moderate downward force.

As light sweeps past the point, the electric vectors oscillate in direction and strength. Do not, however, be misled by the pictorial representation. The vectors are not material. A ray has no vectors sticking out of it like thorns on a rose stem. The electric vectors are bits of imagination stuck on a line called a ray.

Fictitious though the electric vectors are, they are almost essential to an understanding of the polarization of light. In the two snapshots of light the electric vectors are all flat in the plane of the page. Light from most common sources has vectors that are not as limited in direction. The vectors are necessarily perpendicular to a ray of the light, but they can point in any direction in a plane perpendicular to the ray. Such light is said to be unpolarized.

If light passes through a polarizing filter, the oscillation of the vectors is strictly limited to a single axis. Such light is said to be linearly polarized. (Some call it plane-polarized light.) If unpolarized light from an electric light bulb strikes a filter, the emerging light has its electric vectors confined to a single axis in a plane perpendicular to the ray. The direction of polarization of the light is the orientation of that axis. If the axis is vertical, the light is said to be vertically polarized.

Polarizing filters work by a process of elimination. The filter incorporates molecular chains that can be visualized as long, parallel absorbers. As light reaches them the electric vectors that oscillate parallel to them are eliminated; the perpendicular vectors pass through.

For example, if the filter's long molecules are stretched out horizontally (parallel to the x axis), only the horizontal components of the electric vectors in the light are eliminated. Figure 5 presents a shorthand notation in which two double vectors represent unpolarized light; the filter eliminates the horizontal one and passes the vertical one. The result is vertically polarized light.

Normally the orientation of the molecules in a filter is not specified. Instead the filter is assigned a polarization axis perpendicular to the length of the molecules. This fictitious axis is parallel to the polarization of the emerging light.

Suppose a ray of vertically polarized light encounters a second polarizing filter. Will the light pass through? That depends on the polarization axis of the filter. If the axis is vertical, it is parallel to the polarization of the light approaching it. All the light is transmitted. If the axis is horizontal, none of the light is transmitted.

When light moving through air at a speed of 3 X 10⁸ meters per second enters any transparent material, its effective speed decreases. The reason is that the light interacts with the molecules in its path. At each encounter with a molecule the light is absorbed; after a brief delay it is reemitted. Between molecules its speed is 3 X 10⁸ meters per second, the value it has in a vacuum. Since it is intermittently delayed, however, it takes longer to travel through the material than it does to travel an equal distance in a vacuum. The light is said to be traveling slower in the material.

This effect was measured indirectly long before anyone knew about molecules. In order to tabulate the effect each transparent material was assigned the number called the index of refraction. A glass with an index of 1.6 transmits light slower than one with an index of 1.5. (In either case the actual time of transmission is so incredibly short that the difference is immaterial in everyday life.)

In 1816 David Brewster discovered how the index of refraction can aid in the analysis of stress in a transparent material. When he applied a stress to a sheet of glass illuminated with linearly polarized light, he found a change in the index of refraction of the glass. Moreover, the index then depended on how the light was polarized.

To follow Brewster's experiments visualize a vertical sheet of glass uniformly compressed by forces applied at the top and bottom in such a way that the principal stress axes are vertical and horizontal. When the light illuminating the glass is polarized vertically, it encounters a smaller index of refraction and moves with higher velocity than it does when it is polarized horizontally If the glass is under tension (stretched by forces at the top and bottom), the results are just the opposite. A material of this type, with a speed of transmission that depends on the polarization of the light, is said to be birefringent, or doubly refracting.

A simple example will demonstrate how birefringence can aid in stress analysis. A vertical sheet of plastic is uniformly compressed to make the principal stress axes vertical and horizontal. The way polarized light interacts with the stressed plastic depends on whether the polarization of the light is aligned initially with one of those axes. You can create such an alignment by illuminating the plastic through a polarizing filter with its axis vertical. On the opposite side place another polarizing filter with its axis horizontal. The first filter is often called the polarizer, the second one the analyzer; the two are said to be crossed.

The light passes through the sheet of plastic at whatever speed the stress allows. The light emerging from the plastic is still vertically polarized and is therefore stopped by the analyzer. An observer looking through the analyzer sees darkness. In the experimental setup transmission is eliminated whenever the plastic is illuminated with light polarized parallel to a principal stress axis.

Next rotate each filter by 45 degrees in the same direction. The plastic sheet now receives light polarized at 45 degrees from the vertical. The course of light through the sheet is a bit harder to follow because the polarization must be considered in two components parallel to the principal stress axes. The two components travel through the plastic at different speeds because the indexes of refraction along the two axes differ.

When the components emerge from the plastic, they in effect recombine. The light is then likely to have a new direction of polarization. Whether or not the light will pass through the analyzer depends on how the polarization has been changed.

To determine the change one must consider how the wavelength of the light is altered by the plastic. For simplicity assume that the light is of a single wavelength. When light passes from air into a transparent material, its wavelength shortens. The larger the index of refraction, the shorter the wavelength of the light in the material.

Since stressed plastic has a different index for each principal stress axis, the extent to which the wavelength shortens depends on the polarization of the light. If the light is vertically polarized, its wavelength is divided by the index of refraction associated with the vertical principal stress axis. If the light is horizontally polarized, the wavelength is divided by the index associated with the horizontal axis. When the polarization lies between the two axes, it must be considered in components The vertical component is shortened in wavelength by one amount and the horizontal component by a different amount.

As the two components proceed through the plastic with their different wavelengths they oscillate a different number of times. For example, the component polarized parallel to the vertical stress axis might have 1,000 oscillations (wavelengths); the other component, being of shorter wavelength, might have an additional oscillation, for a total of 1,001. The two components begin their trip exactly in step and emerge again in step even though one of them has an additional oscillation.

When one mathematically recombines the two emerging components, the electric vectors of the light are found to be oscillating exactly as they were before they entered the plastic. The polarization is tilted from the vertical by 45 degrees. Since the polarizing filters are crossed, the light reaching the analyzer is blocked.

If instead one component has an additional half oscillation, recombination yields a polarization rotated by 90 degrees. This new polarization is parallel to the polarization axis of the analyzer and so the light passes through.

Any intermediate result is also possible. Recombination is then harder to visualize. The components do not yield linearly polarized light. Instead the polarization rotates continuously about the ray of light. The electric vector rotates from upward to downward and then back to upward. Light with a rotating polarization is said to be elliptically polarized. (In the special case where the maximum size of the electric vector remains the same throughout the rotation, the light is said to be circularly polarized.) Elliptically polarized light is partly transmitted by the analyzer. The component of polarization that is parallel to the polarization axis of the analyzer is passed whereas the other component is blocked.

The point of this discussion is that what the observer sees through the analyzer depends on the angle between the polarization of the light and a principal stress axis in the plastic. Alignment of the polarization with an axis yields darkness at the analyzer. If in misalignment the plastic releases light of unchanged polarization, the analyzer again blocks it. In other instances at least some light gets through the analyzer.

The example is simple in that the stress is uniform and the principal stress axes are vertical and horizontal at any point in the plastic sheet. If the plastic is subjected to various stresses, the axes will be oriented differently for each point in it. The purpose of photoelasticity is to discover how they are oriented; in this way one might be able to locate a section in the model that might fail under the stress. When a complex and unknown arrangement of stresses is applied to the plastic, the model as seen through the analyzer has a superposed pattern of the dark and bright lines called fringes.

A dark fringe marks points within the plastic from which the emerging light is polarized in exactly the way that keeps it from going through the analyzer. The reason is either that the light passing through one of these dark points is polarized parallel to one of the principal axes or that it has two components (one parallel to each principal stress axis there) that recombine to yield the same polarization the light had when it entered the plastic. In either case the analyzer blocks the light.

Usually the first condition creates most of the dark fringes. A fringe caused in this way is called an isoclinic. The pattern of isoclinics reveals the orientation of the principal stress axes within the plastic.

To map the principal axes you photograph the plastic through the analyzer at a particular orientation of the two polarizing filters. Suppose the axis of the first filter is vertical and the axis of the analyzer is horizontal. The plastic is thus illuminated with vertically polarized light. The isoclinics photographed through the analyzer mark the points on the plastic that have one of their principal stress axes vertical. (Since the stress axes are at right angles to each other, the other axis must then be horizontal.) The pattern in the photograph is traced on paper, and the principal axes are drawn superposed on several points on the isoclinics.

Next the filters are rotated 10 degrees, say, and another photograph is made. The isoclinics mark the points with a principal stress axis tilted from the vertical by 10 degrees. These isoclinics are added to the trace and again the principal axes are sketched in at a few places. After a few more photographs have been made the trace shows how the principal stress axes are oriented at many places in the plastic.

Now you add to the map lines that connect points having equal stress values. For example, a line would be drawn from a principal axis at one point to another principal axis at a nearby point. It is likely to be a curved line. Although a certain amount of guesswork is inescapable here, the result is a rough map of the lines of principal stress.

The dark fringes resulting from recombination are called isochromatics. They are usually masked by the isoclinics, but they can be employed to assign values to the principal stresses revealed by the isoclinics. To make use of them you must first eliminate the isoclinics.

You can do it by putting two additional filters, called quarter-wave plates, in the path of the light. One plate goes between the first polarizer and the plastic and the other one goes between the plastic and the analyzer. The function of a quarter-wave plate is to transform linearly polarized light into circularly polarized light.

The plate works something like the stressed plastic being analyzed. The plate is birefringent, that is, it has two orthogonal axes ("fast" and "slow") that pass light at different speeds. Suppose the first polarizer yields light that is polarized vertically. The first quarter-wave plate is set with its fast axis tilted from the vertical by 45 degrees. The light reaching the plate thus has components along both the fast and the slow axes. As the two components pass through the plate they oscillate a different number of times. The polarization of the light emerging from the plate depends on that difference. Since the plates are designed to make one component complete a quarter of an oscillation more than the other component (hence the name quarter-wave plate), circularly polarized light emerges.

The second quarter-wave plate is arranged with its fast axis perpendicular to that of the first one. The function of the second plate is to subtract the quarter-wavelength difference imparted to the two components of light by the first plate. The net result may seem hardly worth the trouble, but a useful end is gained. When the circularly polarized light passes through the stressed plastic, the polarization cannot be strictly parallel to a principal stress a is. Therefore the isoclinics resulting from such an alignment are eliminated. What one receives through the analyzer is a pattern of the isochromatics.

The advantage of this pattern is that the fringes are related to the size of the stresses in the plastic. At any given point in the plastic the difference between the principal stresses determines the polarization of the light. Thus the stress difference determines whether the point ends up as part of a dark fringe or as part of a bright one. In principle the stress values for any point in the plastic can be computed by examining the isochromatics.

Since the isochromatic pattern depends on the shortening of the wavelengths of the two components of light transmitted through any point in the plastic, the pattern depends on the wavelength of the light illuminating the plastic. With white light. each color creates its own pattern. At one point in the plastic one of the colors might leave with the same polarization it had when it entered. That color is eliminated by the analyzer. The other colors passing through the same point are changed somewhat in polarization and so are at least partly transmitted by the analyzer. The observer sees that point on the plastic as being neither white nor dark but a color. The composite of the points generating the same color forms a pattern of colored isochromatics.

The actual color the observer will see at a particular point is difficult to predict. It depends on how the various colors are altered in polarization. It also depends on the perception of the observer or the color response of the film in a camera. Color prediction becomes even more complicated when the stress differences at each point in the plastic increase. When the differences are large enough, the colors begin to wash out into white.

The photographs made by Seufert are from experiments where linearly polarized light is sent through stressed plastic. Both the isoclinics (which are only bright and dark) and the isochromatics (which are color-dependent) are included in the photographs.

The two photographs in Figure 1 show plastic models with concentrations of stress in their concave regions. The isochromatics are closely spaced there, indicating that the stress differences at points in those regions vary considerably. Since the models are under compression from forces on both the left and the right, the concave regions are under compression. In the top photograph the convex region is being stretched. The more uniform distribution of color in the straight sections indicates that the distribution of stress is more uniform there than it is in the curved sections.

The polarizing filters and quarter-wave plates for photographs of this type can be obtained from the Edmund Scientific Co. (101 East Gloucester Pike, Barrington, N.J. 08007) or Jerryco, Inc. (601 Linden Place, Evanston, Ill. 60202). Readers wanting to write to Seufert may do so at 2050 West Boulevard, Cleveland, Ohio 44102.

Bibliography

PHOTOELASTICITY: PRINCIPLES & METHODS. H. T. Jessop and F. C. Harris. Dover Publications Inc., 1960.

POLARIZED LIGHT. William A. Shurcliff and Stanley S. Ballard. D. Van Nostrand Company, Inc., 1964.

PHOTO-ELASTIC ANALYSIS. A. W. Hendry. Pergamon Press, 1966.

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