STUDY OF STRESSES BY THE METHOD OF PHOTOELASTICITY1

R. Mindlin, New York

Introduction.
I. Basic concepts concerning stresses and strains, and also some information from optics.
II. Materials, models, and loading machines.
III. Measurements of the difference of the principal stresses and of their directions.
IV. Direct determination of the principal stresses.
V. Special applications of the photoelasticity method.
VI. Bibliography for the period 1930–1938.

INTRODUCTION

Detailed information on the distribution of stresses in machine parts and elements is of great importance in many branches of engineering for design purposes. Mathematical methods of the theory of elasticity have been successfully applied to the solution of individual problems involving simple boundary conditions. However, in most cases, because of the inadequacy of the mathematical method, one must turn to experiment. The method of photoelasticity proves most useful in this case. This method is based on the fact that the optical properties of a transparent material in a state of stress can be measured and quantitatively related to the state of stress.

In applying the method of photoelasticity, the engineer selects a suitable transparent material (§ 12), makes from it a model (§ 13) of the same shape as the structure under investigation. The model is loaded (§ 14) in the same way as the structure, and the resulting optical effect (§ 8) is measured in various ways (§§ 15, 26–28). Most of these measurements are carried out by methods of physical optics; others have been developed especially for the method of photoelasticity. The optical properties of the model are reduced to stresses (§§ 9, 11, and 15–19), which are treated in accordance with the fundamental principles of the theory of elasticity and the theory of strength of materials (§§ 1–7 and 10). The distribution of stresses in the model is compared with the analogous stress system in the structure (§ 13), and the engineer uses this in design.

Optical observations are by no means always sufficient for a complete determination of the state of stress. To obtain complete information, additional methods were developed (§§ 20–31): optical, graphical, computational, or mechanical.

The methods of photoelasticity have been applied to the solution of a large number of problems in various branches of engineering and applied physics (§§ 32–38); a large number of studies on this topic have appeared in technical scientific journals. Many questions of the theory, technique, and application of this theory were

Fig. 1. Polariscope for photoelasticity investigations and the loading machine of Columbia University

described by E. G. Coker and L. N. Filon1 in the work Treatise on Photo-elasticity, published in 1931. Since that time the theory of the method has not changed; however, naturally, in a field of such great industrial importance, many innovations of a methodological character have appeared, and the range of application of the method has also expanded considerably.

The purpose of the present article is to give a survey of the basic principles of photoelasticity, to describe the most recent experimental technique, and also to note those areas in which this method has found application during the last ten years.

I. BASIC CONCEPTS OF STRESSES AND DEFORMATIONS, AND ALSO SOME INFORMATION FROM OPTICS

1. Definition of Stress

If a system of forces acts on a solid body, then one part of the body exerts an influence on another part adjacent to it. The interaction of neighboring parts of the body is accounted for by the concept of stress. For example, in a bar of cross section \(A\), under tension by a force \(P\) (Fig. 2), the part of the body on one side of the cross section \(n—n\) causes tensile stress in the other part of the body. The magnitude of the stress is determined by the ratio \(\frac{P}{A}\), and the direction of the stress is parallel to the axis of the bar. In the general case, the stress on a small area \(\delta A\) at a point \(O\) of the body (Fig. 3) is not necessarily normal to the section and, moreover, the magnitude and direction of the stresses may vary from point to point of the body. A complete analysis of stresses consists in determining the state of stress at each point of the body for an area passed through this point in an arbitrary manner.

Fig. 2. Prismatic bar under tension

Fig. 3. Stresses over an internal section

Consider an element of area \(\delta A\) passing through a point \(O\) in a body acted upon by a system of forces [Fig. 3(b)]. The normal \(N\) to the area serves to denote its orientation. The part of the body on one side of the area acts on the other part of the body with a force \(\delta P\); the direction of \(\delta P\) in the general case is inclined with respect to \(N\). If we decrease the area \(\delta A\) so that it always passes through the point \(O\), the magnitude \(\delta P\) will also tend to zero. The limit of the ratio \(\frac{\delta P}{\delta A}\), as \(\delta A\) tends to zero, is called the stress at the point \(O\) in the plane with normal \(N\); the limiting direction of \(\delta P\) is the direction of the stress. If the limiting direction does not coincide with \(N\), then the stress may be resolved into components normal and parallel to the section. The former are called normal stresses, the latter shear stresses.

2. Notation

Normal stresses are usually denoted by \(\sigma\), and shear stresses by \(\tau\). The subscripts on these letters indicate the directions of the stresses. Usually stresses are expressed in a Cartesian coordinate system. Normal stresses acting on planes perpendicular to the axes \(x\), \(y\), \(z\) are denoted respectively by \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\). A normal stress is called positive when it produces a state of tension in the part under consideration. In Fig. 4 let us consider a small rectangular parallelepiped with sides parallel to the coordinate axes. The normal stresses \(\sigma_x\), \(\sigma_y\), \(\sigma_z\) are shown in the figure by arrows drawn in the positive directions of the coordinate axes. To denote shear stresses it is necessary to use two subscripts. The first of the subscripts indicates the direction of the normal to the plane along which the shear stress acts; the second indicates the direction of action of the stress in that plane. Thus, for example, the notation \(\tau_{xy}\) means the shear stress acting in a plane normal to the \(x\)-axis and directed parallel to the \(y\)-axis. In Fig. 4 all shear stresses are drawn in the positive directions.

Fig. 4. Components of stress on the faces of a small rectangular parallelepiped

The quantities \(\sigma_x\), \(\sigma_y\), \(\sigma_z\), \(\tau_{yz}\), \(\tau_{zy}\), \(\tau_{zx}\), \(\tau_{xz}\), \(\tau_{xy}\), \(\tau_{yx}\) are called the components of stress. If we consider the moment of the stresses with respect to the \(x\)-axis, we find \(\tau_{zy}=\tau_{yz}\). Doing the same with respect to the other axes, we obtain \(\tau_{zx}=\tau_{xz}\) and \(\tau_{xy}=\tau_{yx}\). Thus, of the nine components of stress only six are independent.

3. Determination of Stress

In § 1 we saw that for a complete description of the state of stress it is necessary to know the stresses on any plane passing through each point in the material. Let \(O\) (Fig. 5) be the point at which we want to determine the stresses, and let \(N\) be the normal to an arbitrary plane passing through \(O\). Another plane with the same normal \(N\), but at a distance \(h\) from \(O\), will intersect the system of \(x\), \(y\), \(z\) axes at the points \(B\), \(C\), and \(D\). This plane is shown in the figure. Since \(h\) may be arbitrary, the two planes may be brought into coincidence, and the properties of these planes may be regarded as identical. In order to find the stresses in the plane perpendicular to \(N\), let us consider the equi-

equilibrium of the small tetrahedron \(OBCD\). The component of the normal stress in the plane \(BCD\) can be expressed in terms of the six stress components along the coordinate axes

\[ \sigma_N=\sigma_x l^2+\sigma_y m^2+\sigma_z n^2+2\tau_{yz}mn+2\tau_{zx}nl+2\tau_{xy}lm, \tag{1} \]

where \(l, m, n\) are the cosines of the angles between the normal \(N\) and the corresponding axes \(x, y, z\).

The resultant stress on \(BCD\) must, generally speaking, also have components parallel to this plane.

Fig. 5. Stresses in a small tetrahedron

Fig. 6. Stresses in a small rectangular parallelepiped with faces parallel to the principal stress planes

The component of the shear stresses in an arbitrary direction \(S\), determined by the direction cosines \(l', m', n'\), is equal to

\[ \tau_{NS}=\sigma_x ll' + \sigma_y mm' + \sigma_z nn' + \tau_{yz}(m'n+mn')+ \]

\[ +\tau_{zx}(n'l+nl')+\tau_{xy}(l'm+lm'). \tag{2} \]

Equations (1) and (2) show that the component of the normal stress in an arbitrary plane passing through \(O\), and the component of the shear stress in this plane in an arbitrary direction, are completely determined if the six stress components along the coordinate axes are known at this point.

4. Principal stresses and principal stress planes

If we consider the variation of \(\sigma_N\) according to equation (1) when the direction \(N\) is changed, we find that there always exist three mutually perpendicular directions for which \(\sigma_N\) has limiting values, i.e., either a maximum, or a minimum, or a minimaximum. These three directions are called the principal axes of stress; they are perpendicular to planes called the principal stress planes. It can also be found that

in the principal planes of stress the shear stresses are absent, i.e., the resultant stress in the principal plane has only a normal component.

Thus, near any point one can construct a small cube (Fig. 6) so that the stresses on its six faces are only normal. We shall denote these three principal stresses by $\sigma_3$, $\sigma_2$, $\sigma_1$ and agree that $\sigma_3>\sigma_2>\sigma_1$.

5. Small deformations and components of deformations

A small volume element having the shape of a rectangular parallelepiped is deformed in a stressed body, to a first approximation, into an oblique parallelepiped. In Fig. 7(a) an elementary volume before deformation is shown (the edges $PA$, $PB$, $PC$ are parallel to the corresponding coordinate axes $x$, $y$, $z$). After deformation the points $A$, $B$, $C$, $P$ will take new positions $A'$, $B'$, $C'$, $P'$ [Fig. 7(b)] such that the edges of the parallelepiped will no longer make right angles with one another, and will also assume new lengths $P'A'$, $P'B'$, $P'C'$. The change in length, referred to the unit length of an edge of the element, is called elongation and is denoted by $\varepsilon$ with a subscript indicating the direction of the edge.

Fig. 7. Deformation of a small rectangular parallelepiped

Thus,

\[ \frac{P'A'-PA}{PA}=\varepsilon_x \]

is the component of unit elongation in the direction of the $x$ axis.

For small displacements, the change in the angle between two adjacent edges of the deformed element is called the shear component and is denoted by $\gamma$ with subscripts indicating the planes in which the edges previously lay. Thus, $\gamma_{xy}$ is the change in the angle between the edge $PA$, previously parallel to the $x$ axis, and the edge $PB$, previously parallel to the $y$ axis.

The six components of deformation—$\varepsilon_x$, $\varepsilon_y$, $\varepsilon_z$, $\gamma_{yz}$, $\gamma_{zx}$, $\gamma_{xy}$—are necessary and sufficient for the complete determination of the new shape and size of the element; these quantities determine the deformed state of the body.

From geometrical considerations, with the aid of Fig. 7, we find for small deformations

\[ \varepsilon_N=\varepsilon_x l^2+\varepsilon_y m^2+\varepsilon_z n^2+\gamma_{yz}mn+\gamma_{zx}nl+\gamma_{xy}lm, \tag{3} \]

where $\varepsilon_N$ is the deformation in the direction $PN$; $l$, $m$, $n$ are the direction cosines of the angles between $PN$ and the corresponding axes $x$, $y$, $z$.

6. Analogy between stresses and small strains

If in equation (3) we replace \(\varepsilon\) by \(\sigma\) and \(\gamma\) by \(2\tau\), we obtain equation (1), which gives the relation between the normal stress in the direction \(l, m, n\) and the components of stress along the axes. Likewise, if in equations (2) we replace \(\sigma\) by \(\varepsilon\) and \(\tau\) by \(\dfrac{\gamma}{2}\), we obtain

\[ \gamma_{NS}=2\varepsilon_x ll' + 2\varepsilon_y mm' + 2\varepsilon_z nn' +\gamma_{yz}(m'n+mn')+ \]
\[ +\gamma_{zx}(n'l+nl')+\gamma_{xy}(l'm+lm'), \tag{4} \]

i.e. we express the shear between the directions \(N\) and \(S\) through the strain components and the direction cosines \(l, m, n\) (direction \(N\)) and the direction cosines \(l', m', n'\) (direction \(S\)).

The complete analogy of the formulas for stresses and strains makes it possible to apply the conclusions drawn in § 4 also to displacements. It is only necessary to replace \(\varepsilon\) by \(\sigma\) and \(\dfrac{\gamma}{2}\) by \(\tau\). Thus, there exist three mutually perpendicular principal axes of strain \(\varepsilon_3, \varepsilon_2, \varepsilon_1\), i.e. three such directions for which the shear strains vanish.

7. Relations between stresses and strains

An elastic body possesses two specific properties. First, the strain components of an elastic body vary linearly as functions of the stresses; secondly, the strains disappear when the stresses are removed. If the body is isotropic, i.e. the properties of each of its points are identical, then the principal axes of stresses and strains coincide in direction. The relations between stresses and strains in this case are known:

\[ \left. \begin{aligned} \varepsilon_1 &= \frac{1}{E}\sigma_1-\frac{\nu}{E}(\sigma_2+\sigma_3),\\ \varepsilon_2 &= \frac{1}{E}\sigma_2-\frac{\nu}{E}(\sigma_3+\sigma_1),\\ \varepsilon_3 &= \frac{1}{E}\sigma_3-\frac{\nu}{E}(\sigma_1+\sigma_2) \end{aligned} \right\}, \tag{5} \]

where \(E\) is Young’s modulus, \(\nu\) is Poisson’s ratio; both are constant for a homogeneous material, i.e. the elastic constants are the same for different points of the body.

Fig. 8. Transformation of a sphere into an ellipsoid in a field of simple tension.

If the rod is subjected to an axial stress \(\sigma_3\), the resulting state of strain is obtained from equation (5):

\[ \varepsilon_3=\frac{\sigma_3}{E}, \qquad \varepsilon_2=-\frac{\nu\sigma_3}{E}, \qquad \varepsilon_1=-\frac{\nu\sigma_3}{E}. \]

If the \(z\)-axis is placed parallel to the axis of the bar [Fig. 8(a)] and the coordinates of some point before deformation are denoted by \(x, y, z\), and after deformation by \(x_1, y_1, z_1\), then

\[ x_1=x\left(1-\frac{\nu\sigma_3}{E}\right), \qquad y_1=y\left(1-\frac{\nu\sigma_3}{E}\right), \qquad z_1=z\left(1+\frac{\sigma_3}{E}\right). \tag{6} \]

Let us write the equation of a spherical surface of radius \(a\) in the undeformed bar:

\[ x^2+y^2+z^2=a^2. \]

Using equation (6), we see that the sphere is transformed into an ellipsoid [Fig. 8(b)]:

\[ \frac{x_1^2}{a^2\left(1-\frac{\nu\sigma_3}{E}\right)^2} + \frac{y_1^2}{a^2\left(1-\frac{\nu\sigma_3}{E}\right)^2} + \frac{z_1^2}{a^2\left(1+\frac{\sigma_3}{E}\right)^2} =1. \tag{7} \]

8. Propagation of Light in an Isotropic and Crystalline Medium

Wave theory describes light as a transverse wave disturbance. The wave front \(F\) (Fig. 9) of an optical disturbance propagating through a transparent medium is regarded as plane in the vicinity of the point \(O\) in the medium; the direction of the wave is characterized by the normal \(N\) to the front \(F\) at the point \(O\). In an isotropic medium the velocity of propagation of a light wave is the same in all directions, i.e., it does not depend on the direction of the normal \(N\) to the wave front. In a crystalline medium, in general, each wave normal corresponds to two velocities of propagation; both of these velocities depend on the direction of the normal to the wave front. This phenomenon is double refraction.

Fig. 9. Wave disturbances

Fig. 10. Propagation of a wave in a doubly refracting medium

Figure 10 shows two waves (\(W\) and \(W'\)) corresponding to the wave normal \(N\). Since both waves have different velo-

growth, then in the crystal there will exist two parallel wave fronts (\(F\) and \(F'\)) for each wave of this normal. Each of the waves is plane-polarized, i.e., the oscillations corresponding to the wave lie in one plane (the planes \(AOO'\) and \(A'O'O\)). The directions of oscillation (\(OA\) and \(O'A'\)) of both waves are perpendicular to each other and to the wave normal.

The optical properties of any point \(O\) of a crystalline medium can be described by the so-called ellipsoid of refractive indices (Fig. 11). The center of the ellipsoid lies at the point \(O\); the directions of the principal axes \(OA\), \(OB\), \(OC\) of the ellipsoid are fixed in the medium. If a wave propagates in the direction \(N\), then its optical characteristics are found with the aid of the index ellipsoid. A plane through the point \(O\), perpendicular to \(ON\), intersects the ellipsoid in the ellipse \(DE\). The semiaxes \(OD\) and \(OE\) of this ellipse are proportional to the refractive indices \(n_1\), \(n_2\) of the two waves, and the directions \(OD\) and \(OE\) will be the directions of oscillation.

Fig. 11. Index ellipsoid

If \(ON\) coincides with a principal axis of the ellipsoid, then \(n_1\) and \(n_2\) will be equal to \(n_a\), \(n_b\), \(n_c\). The latter are called the principal refractive indices; the directions \(OA\), \(OB\), \(OC\) will be the principal axes of optical symmetry, and the planes \(AOB\), \(BOC\), \(COA\)—the principal planes of optical symmetry. The optical axes and planes are analogous to the corresponding concepts of the theory of stresses and strains.

If light of wavelength \(\lambda\) passes normally through a crystalline plate of thickness \(d\), then both plane-polarized waves acquire phase differences \(\Delta_1\) and \(\Delta_2\) with respect to the unchanged wave:

\[ \left. \begin{aligned} \Delta_1 &= \frac{2\pi d}{\lambda}(n_1-n),\\ \Delta_2 &= \frac{2\pi d}{\lambda}(n_2-n) \end{aligned} \right\}, \tag{8} \]

where \(n\) is the refractive index of the medium outside the plate; \(\Delta_1\) and \(\Delta_2\) are measured in radians.

The relative phase difference is

\[ \Delta=\Delta_1-\Delta_2=\frac{2\pi d}{\lambda}(n_1-n_2). \tag{9} \]

9. The Photoelastic Effect

In 1816 Brewster found that stressed or deformed glass becomes birefringent. It was later established that the majority of transparent materials possess this property to a greater or lesser degree.

Maxwell104 discovered that a glass rod under simple tension (Fig. 8) corresponds to an ellipsoid of indices of the type of an elongated spheroid. He determined experimentally that the axis of rotation of the spheroid is parallel to the direction of tension and that the eccentricity is proportional to the load. This and analogous experiments show that the principal axes of stress coincide with the principal axes of optical symmetry and that the principal stresses are linearly related to the principal refractive indices. Consequently, the relation between stress and double refraction is the same as that between stresses and strains. The optical properties in an initially isotropic material after the application of stresses can be represented by an ellipsoid that would arise as the result of deformation of a sphere of radius \(n_0\), subjected to the action of a homogeneous stress field applied parallel to the principal axes.

Then, by analogy with equation (5), one may write

\[ \begin{aligned} n_a - n_0 &= C_1\sigma_1 + C_2(\sigma_2 + \sigma_3),\\ n_b - n_0 &= C_1\sigma_2 + C_2(\sigma_3 + \sigma_1),\\ n_c - n_0 &= C_1\sigma_3 + C_2(\sigma_1 + \sigma_2), \end{aligned} \tag{10} \]

where \(C_1\) and \(C_2\) are the optical stress coefficients. Equations (10) express the fundamental relation of the optical effect to stresses.

The state of stress at any point can be calculated if it is possible to determine the principal refractive indices and the directions of the principal axes of optical symmetry. In the general case, experimental measurements present great difficulties; therefore the methods of photoelasticity have until now been limited for the most part to the investigation of such stress states for which, of the six unknown quantities (the magnitudes and directions of the principal stresses), three are known a priori. Let us consider this case in more detail.

10. Stresses in a Plane

In equation (1) the direction \(N\) was arbitrary, and we found three mutually perpendicular directions for which \(\sigma_N\) had a constant value. Suppose now that \(N\) is constrained so as always to lie in one plane. It can be shown that in this plane there exist two mutually perpendicular directions for which \(\sigma_N\) has a constant value and the tangential stresses vanish. These are the principal axes of stress—

stresses for the plane under consideration. They coincide with two principal stresses of the three-dimensional case if this plane is a plane of principal stresses. Principal stresses in a plane that does not coincide with a principal plane are called secondary principal stresses.

Consider a plane of the body lying in the coordinate plane \(x,y\). Let the direction \(N\) in this plane make an angle \(\theta\) with the positive direction of the \(x\)-axis; \(\theta\) is measured in positive units counterclockwise (from \(x\) to \(N\)) (Fig. 12). Then in equation (1) \(l=\cos\theta\), \(m=\sin\theta\), \(n=0\), and we have

\[ \sigma_N=\frac{1}{2}(\sigma_x+\sigma_y)+ \frac{1}{2}(\sigma_x-\sigma_y)\cos 2\theta+ \tau_{xy}\sin 2\theta . \tag{11} \]

Fig. 12. Stresses in a plane

Likewise, from equation (2), putting \(l'=-\sin\theta\), \(m'=\cos\theta\), \(n'=0\), we obtain

\[ \tau_{NS}=\frac{1}{2}(\sigma_y-\sigma_x)\sin 2\theta+ \tau_{xy}\cos 2\theta . \tag{12} \]

To find the directions in which the stress component is extremal, we differentiate equation (11) with respect to \(\theta\); equating the result to zero, we obtain

\[ (\sigma_x-\sigma_y)\sin 2\theta=2\tau_{xy}\cos 2\theta . \tag{13} \]

(13) is the equation for determining those values of \(\theta\) for which \(\sigma_N\) has an extremum. Note that equation (13) is the condition for the vanishing of \(\tau_{NS}\) [see equation (12)]. To find the roots of equation (13), substitute

\[ \cos 2\theta=\pm(1-\sin^2 2\theta)^{\frac{1}{2}} \]

and we obtain

\[ \sin 2\theta=\pm \frac{2\tau_{xy}} {\left[(\sigma_x-\sigma_y)^2+4\tau_{xy}^{\,2}\right]^{\frac{1}{2}}} \tag{14} \]

and

\[ \cos 2\theta=\pm \frac{\sigma_x-\sigma_y} {\left[(\sigma_x-\sigma_y)^2+4\tau_{xy}^{\,2}\right]^{\frac{1}{2}}}. \tag{15} \]

The upper sign in equations (14) and (15) corresponds to one value of \(\theta\), which we shall call \(\Phi\). The lower sign gives \(\Phi+\dfrac{\pi}{2}\); \(\Phi\) and \(\Phi+\dfrac{\pi}{2}\)

indicate the directions of the secondary principal stresses in the plane \(x, y\) and

\[ \left. \begin{aligned} \sin 2\Phi &= \frac{2\tau_{xy}}{\left[(\sigma_x-\sigma_y)^2+4\tau_{xy}^{\,2}\right]^{1/2}},\\[4pt] \cos 2\Phi &= \frac{\sigma_x-\sigma_y}{\left[(\sigma_x-\sigma_y)^2+4\tau_{xy}^{\,2}\right]^{1/2}}. \end{aligned} \right\} \tag{16} \]

To find the magnitudes of the secondary principal stresses, we substitute expression (16) into equation (11). Denoting by \(p\) the stress corresponding to \(\theta=\Phi\), and by \(q\) the stresses corresponding to the angle \(\Phi+\dfrac{\pi}{2}\), we have

\[ p=\frac{1}{2}(\sigma_x+\sigma_y) +\frac{1}{2}\left[(\sigma_x-\sigma_y)^2+4\tau_{xy}^{\,2}\right]^{1/2}, \tag{17} \]

\[ q=\frac{1}{2}(\sigma_x+\sigma_y) -\frac{1}{2}\left[(\sigma_x-\sigma_y)^2+4\tau_{xy}^{\,2}\right]^{1/2}. \tag{18} \]

The angle \(\Phi\) corresponds to the secondary principal stress of greatest absolute magnitude.

From equations (16), (17), and (18) we have

\[ \tau_{xy}= \frac{1}{2}\sin 2\Phi \left[(\sigma_x-\sigma_y)^2+4\tau_{xy}^{\,2}\right]^{1/2} =\frac{1}{2}(p-q)\sin 2\Phi. \tag{19} \]

The greatest shearing stress in the plane \(x, y\) will be maximal in the direction that bisects the angles between the secondary principal axes; it is equal to

\[ \tau_{\max} =\frac{1}{2}\left[(\sigma_x-\sigma_y)^2+4\tau_{xy}^{\,2}\right]^{1/2} =\frac{1}{2}(p-q). \tag{20} \]

The shearing stresses \(\tau_{xy}\) (19) vanish for \(\Phi=0,\ \dfrac{\pi}{2}\), so that the directions of the axes \(x, y\) are the directions in which the secondary principal stresses act. The stresses \(\tau_{xz}, \tau_{yz}\) disappear in the case of a plane state of stress. Such a stress arises in a thin plate under the action of forces lying in its plane. The stress components \(\sigma_z, \tau_{xz}\), and \(\tau_{yz}\) of such a plate of small thickness \(d\) vanish at \(z=\pm \dfrac{d}{2}\), if the mid-plane is taken as the plane \(x, y\).

If the thickness of the plate is small in comparison with the linear dimensions of the plate in the plane \(x, y\), it may be assumed that \(\sigma_z, \tau_{yz}\), and \(\tau_{zx}\) are equal to zero also inside the volume of the plate; the variation of \(\sigma_x, \sigma_y\), and \(\tau_{xy}\) with the coordinate \(z\) in this case is also neglected.

11. Optical stress coefficients in the case of two-dimensional photoelasticity

At each point of a plate loaded in its own plane, one of the principal stresses is equal to zero and is directed normally to the plate. One of the principal planes of stress therefore coincides with the plane of the plate. The two principal stresses in this plane may be \(\sigma_3\) and \(\sigma_2\), or \(\sigma_2\) and \(\sigma_1\), or \(\sigma_3\) and \(\sigma_1\); below we shall denote them respectively by \(p\) and \(q\).

Similarly, one of the principal planes of optical symmetry coincides with the plane of the plate. For light incident perpendicular to the plane of the plate, the indices \(n_1\) and \(n_2\) of the two resulting waves will be the principal indices; these may be \(n_c\) and \(n_b\), or \(n_b\) and \(n_a\), or \(n_c\) and \(n_a\).

Under this condition equations (10) reduce to

\[ \begin{aligned} n_1-n_0 &= C_1p + C_2q,\\ n_2-n_0 &= C_1q + C_2p. \end{aligned} \tag{21} \]

If the absolute phase retardations \(\Delta_1\) and \(\Delta_2\) of equation (8) are measured, then the principal stresses \(p\) and \(q\) can be calculated from equation (21). On the other hand, if the relative retardation \(\Delta\) of equation (9) is measured, then the difference of the principal stresses \(p-q\) can be calculated. From equations (9) and (21) we have

\[ \Delta = \frac{2\pi d C}{\lambda}(p-q), \tag{22} \]

where \(C=C_1-C_2\) is the relative optical stress coefficient.

The directions of the principal stresses are found by determining the directions of the principal axes of optical symmetry.

It should be noted that on the unloaded contour of the plate two of the three principal stresses vanish, namely, one normal to the plane of the plate and one normal to the contour. Moreover, the third principal stress acts parallel to the contour. At such a contour point, measurement of the relative phase difference [equation (22)] gives all the quantities necessary for determining the stresses.

The coefficient \(C\) in equation (22) has the dimension inverse to stress. It is usually expressed in \(10^{-13}\ \text{cm}^2/\text{dyne}\). Such a unit is called the brewster\(^5\). The value of \(C\) for a given material varies depending on the wavelength and on the temperature.

II. Materials, models, and loading frames

12. Photoelastic materials

The first step in investigating the properties of a body by the method of photoelasticity consists in selecting a suitable material for making the model. According to Filon\(^6\), Coker\(^43\), and Solakian\(^145\), the ideal…

STUDY OF STRESSES BY THE METHOD OF PHOTOELASTICITY

will be a material having: 1) a high optical coefficient, 2) good machinability, 3) a linear relationship between stresses and strains, 4) a high modulus of elasticity, 5) physical and optical homogeneity, 6) initial double refraction either absent or easily eliminated by annealing, 7) absence of creep, 8) absence of elastic aftereffect and residual double refraction, 9) absence of the “edge effect,” and 10) high transparency. The materials most often used are glass, celluloid (nitrocellulose), Bakelite BT-61-893 (glicerine p’tallic anhydride), phenolite, and trolon (phenol-formaldehyde resins). Glass was the first material used in optical experiments; it is difficult to machine and has a very small value of the coefficient \(C\); nevertheless, the remaining properties of glass make the use of this material desirable in precise experiments (see § 27). Celluloid, whose optical coefficient is five times greater than that of glass and which is readily machined, was the standard photoelastic material from 1906 to 1923. In that year Arakawa\(^{20}\) used a phenolic Bakelite resin for the method of photoelasticity, with an optical coefficient five times greater than that of celluloid. During the last fifteen years materials with still greater optical sensitivity have repeatedly appeared; they have successively displaced the less perfect materials.

Carlton\(^{43}\) gives (see the table) the values of the relative optical stress coefficient (\(C\)) in brewsters (for \(\lambda = 5\,461\) Å)

Material	Optical stress coefficient in brewsters
Phenolite	56.8
Bakelite (USA)	50.2
Bakelite (England)	46.4
Orko (France)	39.3
Celluloid	12.8
Acetylcellulose	19.5
Polopas	17.5
Charmoid	20.6
Vinyllite	18.0
Glyptal	28—1,000
Glass	2.7
Rubber, soft	2,000
Rubber, hard	105

for thirteen materials used in the method of photoelasticity. These values are approximate; the exact values of \(C\) for each material depend on temperature, as well as on previous thermal and chemical treatment. The influence of temperature was studied for glass by Harris\(^{75}\), for celluloid by Harris and Seto\(^{76}\), for five grades of Bakelite and marbette by Lee and Armstrong\(^{96}\), for Bakelite by Hetényi\(^{77,78}\), and for trolon by Kuske\(^{93}\).

Carlton notes that the optical coefficient of the material depends strongly on the degree of polymerization of the synthetic resin. The principles underlying this phenomenon are most easily understood from the results of the work of Hetényi78 and Kuske93. Hetényi’s experiments on Bakelite BT-61-893 and the author’s interpretation of the process from the standpoint of the two-phase theory of hardening resins shed light on the nature of the initial and residual birefringence, the temperature effect, creep, elastic properties, and the edge effect.

Fig. 13. Time–deflection curves for Bakelite BT-61-893 at high temperatures78. \(M = 2.50\) dm/lb; \(\delta_{\max} = 57.0\) lb/dm\(^2\).

According to the two-phase theory, the hardened resin used in photoelastic experiments consists of two components, namely, an insoluble three-dimensional skeleton filled with a soluble matrix. Insolu-

Fig. 14. Stress–strain curve for Bakelite at \(115^\circ\)78.
1—\(\sigma\) actual; 2—\(\sigma\) nominal.

Fig. 15. Stress–strip-order curve for Bakelite at \(115^\circ\)78.
1—\(\dfrac{t'}{t}\sigma\) actual; 2—\(\sigma\) nominal.

The soluble part constitutes only a small part of the total volume. It was initially formed from the soluble part owing to a very slowly proceeding reaction (called polymerization).

At room temperature the soluble part has a high viscosity coefficient and takes up a large part of the load applied to the model; at high temperature the viscosity coefficient decreases so much that the soluble part flows and gradually transfers the load to the insoluble framework. The deformation under constant load approaches a limiting value determined by the degree of polymerization of the material and by its elastic properties. The higher the temperature, the more intense the creep process of the soluble part, and the sooner the limiting deformation is reached, as shown by Hetényi’s experimental curves (Fig. 13). At a temperature of 115° for Bakelite BT-61-893 the limiting deformation is reached almost instantaneously. The linear elasticity of the polymerized framework was shown by Hetényi in the stress—strain curves at 115° (Fig. 14). The linear change of the optical effect is shown in Fig. 15.

The material of this paragraph will be discussed further in § 34 from the standpoint of the three-dimensional problem of photoelasticity. An explanation of the photoelastic effect from the standpoint of atomic physics may be found in Müller’s work^115.

13. Model

In the last several years Bakelite BT-61-893 has become the standard material for models in the United States. It is manufactured in the form of rectangular plates measuring \(6 \times 12\) in., with thicknesses from \(1/4\) to 1 in. Usually the model is cut out of that part of the plate where there is no residual birefringence. Otherwise the material is annealed (for the latest data on annealing see Kuske^94).

The surfaces of the model are made plane-parallel by hand working with emery cloth, on a milling machine, or by working on an optical polishing machine (Fig. 16). The best results are obtained by the latter method, using cast metal wheels and a rotation speed of 250 rpm; for grinding, carborundum No. 150 is used successively, and then Nos. 240, 400, 600. The plate may be polished on the same machine if cloth is stretched over the wheel and used as a polish-

Fig. 16. Optical polishing machine

crocus polishing powder. In some laboratories the surface of the model is ground but not polished. Scratches from machining are removed from the optical image by immersing the model in a liquid of the same refractive index, or by lubricating the surface with oil, or by coating the surface with varnish. Fig. 17 shows the effectiveness of these four methods of treatment.

After machining

With the use of oil

With the use of varnish

Polished surface

Fig. 17. Comparative photographs of the patterns of the bands under various surface treatments (V. M. Bel and I. K. Bussey, California Institute of Technology)

The plate is cut according to the drawing with a hand or mechanical saw; the final cuts must be made very carefully. The choice of cutters used for the final trimming is determined by the individuality of the mechanic and by the configuration of the model. It is desirable: a) to avoid heating the material, b) to make sharp edges, c) to strive to have the cut surfaces strictly perpendicular to the plane of the plate. If these precautions are taken, the edges of the model in the image will be even and clean. In many laboratories straight edges are trimmed on a milling machine; sharp cutters lubricated with lard oil are used. Round holes are better drilled on a lathe than on a drilling machine. If the model has an irregular shape, good results can be achieved by careful hand filing.

The model should be examined for several hours after the mechanical treatment of the edges in order to avoid

Fig. 18.

(a) Photograph of the fringes of a model with a very small edge effect
(D. K. Drucker, Columbia University); (b) photograph of the same model
8 months later, showing the edge effect of the unloaded model;
(c) photograph showing the influence of the edge effect on the fringe pattern
in the loaded model.

“edge effect.” This little-studied phenomenon consists in an abnormally large value of the birefringence near the edge of the model, increasing with time. The process is accelerated at high temperatures and is retarded in the absence of air.

According to Hetényi,^78 the “edge effect” arises: a) owing to progressive polymerization of the material, b) from evaporation of bound or absorbed water, or c) from oxidation of the surface layers. The edge effect occurs immediately if the edges are cut with a blunt or insufficiently lubricated cutter, and also at a high cutting speed; all these are causes that create heating of the material. Photographs of the experimental results for a model with an edge effect and without it are shown in Fig. 18 (a), (b), (c) (see inset sheet I).

In studying a two-dimensional static problem under a load acting in its plane, the model is made geometrically similar to the original in all dimensions except thickness.

Michell’s theorem^5 shows that, in the case of a plane two-dimensional problem in the absence of body forces, the stresses do not depend on the elastic constants of the material, except in the case of a multiply connected plate with an equilibrating force on the internal contours. In the latter case, Filon^5 gave a method for obtaining corrections to the observed stresses, requiring the performance of a very complex additional photoelastic experiment.

Let us recall (§ 10) that the true state of plane stress is reproduced even in a thin plate only approximately; the smaller the thickness of the plate in comparison with the other linear dimensions, the better the approximation. It is impossible to create, with respect to the thickness and the linear dimensions, the conditions necessary for setting up a two-dimensional problem. If the plate has a hole whose radius of curvature is of the same order of magnitude as the thickness of the plate, then a plane state of stress will not occur. In this case one must expect that the stress-concentration coefficient obtained by the photoelastic method will not coincide with that obtained from the mathematical theory of elasticity.

If the thickness of the model satisfies the conditions of geometrical similarity to the original structure, then, apparently, the results of a study by the photoelastic method are closer to the truth than the mathematical solution of the two-dimensional problem of the theory of elasticity. It is impossible to verify this conclusion, since, except for limiting cases, the mathematical theory of elasticity has no solutions for plates with a circular hole.

14. Loading frames

The ideal design of a loading frame for applying forces to a model must: 1) be adapted for various—

ous methods of loading, 2) have a large free passage for the light beam, 3) possess the capability of applying forces up to 500 kg, 4) have suitable devices for measurements (precisely calibrated springs or levers, or both together), 5) provide for continuous increase and decrease of the load, 6) have the possibility of reproducing any load so that the experimenter can see the sequence of formation of the optical fringes, 7) provide for relative displacement of the loading frame and have an attachment for studying models of larger dimensions than the optical working field.

These requirements necessitated the design of special loading frames for studies in photoelasticity.

III. MEASUREMENTS OF THE DIFFERENCE OF THE PRINCIPAL STRESSES AND THEIR DIRECTIONS

15. Measurements of \(\Phi\) and \(p-q\)

Optical instruments and methods used for measuring the directions of the plane of polarization and the relative phase retardation serve this purpose; the former, as we saw in § 11, give the directions of the principal stresses, while the latter give the difference of the principal stresses. The basic parts of the polariscope used for the photoelastic method are:

1. Light source. Monochromatic and nonmonochromatic light are used. Most often a mercury lamp with a filter for the wavelength 5461 Å is employed. As nonmonochromatic light, the same mercury lamp without a filter is used, preferably an incandescent lamp.

Fig. 19. Schematic path of rays in a photoelastic polariscope with reflectors (California Institute of Technology)
(for designations see under Fig. 20)

2. Polarizer and analyzer. Prisms of Iceland spar are used (Nicol, Glan-Thompson, Glan, etc.), as are flat polaroids, and also reflecting or refracting glass plates set at the angle of complete polarization.

3. Method of observation. A screen (transparent or matte), a photographic plate, or an optical tube.

In order that a parallel beam of light fall on the model, a special system of lenses is used. For light fields of large diameter the lenses are expensive and in some installations40,72 are replaced by spherical reflectors (Fig. 19).

The usual arrangement of the principal parts is shown schematically in Fig. 20(a). The intensity of the monochromatic light passing through this system with crossed polarizer and ana-

...analyzer (neglecting losses due to reflection and absorption in the lenses):

\[ I=I_0\sin^2 2\Phi \sin^2\left(\frac{\Delta}{2}\right), \tag{23} \]

where \(\Phi\) is the angle formed by the plane of polarization of the faster of the waves that have passed through the model with the plane of polarization of the ray incident upon it, and \(\Delta\) is the relative phase retardation for some point of the model. Equation (23) is an expression for the intensity in the case of a “plane” polariscope (“plane” because the ray incident on the model is plane-polarized).

Considering any point of the model by means of equation (23), we see that if the plane of polarization of the incident ray is rotated (with crossed nicols), extinction of the light will be observed when \(\Phi\) is equal to \(m\frac{\pi}{2}\), where \(m\) is an integer. These values of \(\Phi\) determine the directions of the principal stresses \(p\) and \(q\) at each point; however, they do not indicate which of these two mutually perpendicular directions is the direction of the algebraically larger principal stress.

Restricting ourselves to the consideration of some point of the model, we see from equation (23) that in the case where the direction of the plane of polarization is at an angle of \(45^\circ\) with respect to the principal axes of stress, the intensity will be equal to:

\[ I=I_0\sin^2\left(\frac{\Delta}{2}\right). \tag{24} \]

Fig. 20. Schemes of the principal parts of a photoelastic polariscope:

(a) Polariscope with plane polarization.
(b) Polariscope with plane polarization and a compensator.
(c) Polariscope with plane polarization and a spectroscopic analyzer.
(d) Polariscope with circular polarization.
(e) Nernst bipolarizer.
(f) Reflecting polariscope.
\(L\)—light source, \(P\)—polarizer, \(A\)—analyzer, \(M\)—model, \(T\)—screen, \(C\)—compensator, \(S\)—prism, \(Q\)—quarter-wave plate, \(G\)—glass plate, \(R\)—reflector, \(H\)—half-wave plate, \(D\)—spherical reflector.

Since, according to (22), \(\Delta\) is directly proportional to \(p-q\), the intensity of monochromatic light will vary periodically from dark to light as the load on the model increases. Knowing the number of periods, one can calculate the difference of the principal stresses from equation (22), if the wavelength \(\lambda\), the thickness \(d\), and the optical stress coefficient are known. The latter is determined from an auxiliary experiment for

of a reference specimen, the value \(p-q\) of which is known. This method, if it is used to observe the entire stress field of the model, is called the fringe method.

If \(\Delta\) is small, then the fringe-counting method requires more accurate measurements. A number of other methods are also used: color comparison, the stretching-compensator method, the quarter-wave compensator method, the quartz wedge compensator method, and the spectroscopic analyzer method.

Color comparison

If we replace the source of monochromatic light by a source of white light, only one particular wavelength will be extinguished for a given value of \(p-q\); the remaining wavelengths will be transmitted partially or completely. For a given light source, polariscope, and model material, each \(p-q\) will have its own characteristic coloration, which can be determined by an auxiliary test of simple tension. The value of \(p-q\) at some point of the model is determined by comparing the color at that point with the color scale obtained by the auxiliary test of simple tension. A detailed analysis of the colors obtained was given by Baud and Wright\(^{27}\). Although this method does not have sufficient accuracy and is at present considered obsolete, the use of white light is nevertheless expedient in the following three cases. 1) Since at \(p-q=0\) light rays of all wavelengths are extinguished, this method provides a means of determining the position of those points of the model at which the principal stresses are equal. 2) Light rays of all wavelengths are extinguished when \(\Phi\) is a multiple of \(\frac{\pi}{2}\), which is important in observing isoclinics (see § 17). 3) For demonstration purposes, namely for obtaining large projections in a lecture hall, white light gives a brighter picture than sources of monochromatic light.

Stretching compensator

In the method proposed by Wertheim\(^{180}\) and further developed by Coker\(^{44}\), the stretching reference specimen is placed in the polariscope between the model and the analyzer [Fig. 20(\(b\))]. The reference specimen, made of the same material and of the same thickness as the model, is arranged so that its principal axes are parallel to the principal stress axes at the observed point of the model. It is desirable that the plane of polarization of the incident ray form an angle of \(45^\circ\) with the principal axes of the model, since, according to equation (23), this position gives maximum intensity. When the axis of the reference specimen is parallel to the direction of the algebraically smaller principal stress, then, if the stress in the reference specimen is gradually increased, the observed colors with a white-light source will change according to the color scale up to extinction. At this point the stress in the reference specimen will be equal to the value \(p-q\) of the model.

The indicated method serves two purposes: first, with its aid \(p - q\) is determined; second, the direction of the plane of polarization is found which corresponds to the algebraically greater or smaller principal stress.

Quarter-wave compensator

To determine fractional values of the relative phase retardation at points of the model, a birefringent quarter-wave plate is used as a compensator (the Sénarmont compensator). In this method the nicols are never crossed, but the analyzer and the quarter-wave plate are rotated independently in order to establish the maximum and minimum of intensity. From these positions the form and orientation of the elliptic vibration emerging from the model are determined. These data make it possible to obtain the required quantities \(\Phi\) and \(\lambda\). The quarter-wave plate is also used to determine the direction of the algebraically greatest and smallest principal stress. This is done in the same way as determining the major and minor axes of a birefringent plate, or determining the optical character of a crystal.

Quartz wedge compensator

Instead of changing the phase difference in the compensator by changing the magnitude of the double refraction (as in the tension compensator), the same effect can be obtained by changing the thickness. This is accomplished by moving a crystalline wedge (usually quartz) along the field by means of a micrometer screw.

Next the compensator (Babinet or Babinet–Soleil) is set so that its phase difference is equal and opposite to the phase difference at the point of the model. The Babinet–Soleil compensator, used with a half-wave plate, is a very precise instrument for measuring relative phase difference.

Spectroscopic analyzer

From equations (9) and (22) it follows that, instead of varying \((n_1 - n_2)\), \((p - q)\), and \(d\), one may change the wavelength \(\lambda\), as a result of which changes arise in the phase difference \(\Delta\) in the compensating instrument. This can be done by placing a spectrometer in the field of the polariscope [Fig. 20(c)] and observing the extinction of bands in the spectrum.

Large-field polariscopes

We have considered methods suitable chiefly for studying models point by point. In engineering problems it is often desirable to obtain a picture of the entire stress field. In some definite position of the model with respect to the plane

of the polarization extinction of the light is observed at all those points where the principal axes of stress are parallel or perpendicular to the plane of polarization. These points are joined by a dark line, called an isoclinic. In addition, dark bands are observed in monochromatic light with crossed nicols and pass through those points where the phase difference is a multiple of \(2\pi\). The latter curves are called isochromes; these are curves of equal differences of the principal stresses. Thus the field is covered by two systems of curves.

In our point method with plane-polarized light, we were able to avoid extinction for a given point by setting \(\Phi\) equal to \(45^\circ\), as a result of which the intensity satisfied equation (23); but in the stress field of a model, where the direction of the principal stresses is different at each point, this cannot be achieved for the entire field of the model. Usually the isoclinics can be eliminated if a \(1/4\)-wave plate is placed on both sides of the model, in such a way that the axes are inclined at an angle of \(45^\circ\) to the plane of the polarizer and analyzer [Fig. 20(d)]. Such an instrument is called a circular polariscope. The expression for the intensity is then given by equation (24), and in the field only the curves of constant differences of the principal stresses are visible.

If the two auxiliary plates do not exactly correspond to \(1/4\) of the wavelength of the monochromatic light being used, then, as Boas \(^{25}\) showed, the intensity is given by the expression

\[ I=I_0(1-\cos^2 2\Phi \cos^2 \Delta_0)\sin^2\left(\frac{\Delta}{2}\right), \tag{25} \]

where \(\Delta_0\) is the retardation of the corresponding phases of both plates. Equation (25) shows that the effect of incomplete correspondence of the phase difference in the plates to the given monochromatic light is manifested in the fact that the isoclinics do not disappear completely. Boas \(^{25}\), using equation (25), showed that in the case of white light in the circular polariscope the isochromes are slightly displaced. If the relative retardations of the auxiliary plates \(\Delta_0'\) and \(\Delta_0''\) not only differ from \(\dfrac{\pi}{2}\), but also from each other, then the intensity equation has the form \(^{112}\)

\[ I=I_0\left\{\sin^2\frac{\Delta}{2}\left[\cos(\Delta_0''-\Delta_0')-\cos^2 2\Phi \cos \Delta_0' \cos \Delta_0''\right]+\right. \]

\[ \left.+\sin^2\frac{\Delta_0''-\Delta_0'}{2}-\frac{1}{2}\sin \Delta \sin 2\Phi \sin(\Delta_0''-\Delta_0')\right\}. \tag{26} \]

(26) shows that the isochromes will also be displaced in monochromatic light \(^{113}\). All these effects, however, are small and influence the result of the investigation only in precision measurements.

The Herenberg bipolarizer [Fig. 20(e)]—a frequently used optical system—differs in that the polarized light passes through the model twice, whereby the difference

Fig. 21.

Isochromatics around a small hole in a plate under simple tension
(W. M. Murray, Massachusetts Institute of Technology)

Fig. 22.

165° isocline for the stress field around a small hole in a stretched plate

of the isoclinic curves, for which the angle \(\Phi\) will be constant, according to (16) has the form:

\[ \tg 2\Phi=\frac{2\tau_{xy}}{\sigma_x-\sigma_y}. \tag{30} \]

An isoclinic with parameter \(\Phi=\Phi_1\) is the geometric locus of those points of the model for which the direction \(p\) forms the angle \(\Phi_1\) with the \(x\)-axis; \(\Phi_1\) is at the same time the angle between the direction of polarization of the incident ray and the \(x\)-axis (Fig. 22, inset sheet II).

Fig. 23. Sketch of isoclinics around a small hole in a tensioned plate (a quarter of the field is shown)

Fig. 24. Construction of isostatics from isoclinics

If the plane of polarization is rotated into a new position \(\Phi=\Phi_2\) (while keeping the nicols crossed), the isoclinics move to a new position, along which \(p\) forms the angle \(\Phi_2\) with the \(x\)-axis. By revealing the isoclinics through a gradual change of the angle \(\Phi\) from \(0\) to \(\frac{\pi}{2}\), one can determine the directions of the principal axes of stress at all points of the model (Fig. 23).

18. Isostatics

This system of curves can be constructed graphically if the isoclinic curves are known. Isostatics, or stress trajectories, are curves whose tangent and normal at each point coincide with the directions of the principal stresses.

The method of constructing isostatics is shown in Fig. 24. Along each isoclinic crosses are drawn, the lines forming the crosses being drawn parallel to the directions of the principal stresses corresponding to each isoclinic. Then the isostatics are drawn by hand through the isoclinics so that each isostatic has, as its tangent, one line of the cross lying on each isoclinic.

A witty method of obtaining isostatics photographically was described by Bio[^31].

physicist Mabbu[^98] made use of the fact that in this case only one reflecting surface beyond the model is needed; he inserted a small photoelastic plate with a reflecting surface into a concrete structure and, by measuring the double refraction in the plate, was able to determine the stresses in the structure. Doubling the path difference in the Nörenberg compensator was used by many experimenters to increase the accuracy of measurements of small values of double refraction.

Another type of polarizer, known as the reflecting polariscope, is shown in Fig. 20(f). In this case a single polarizing device serves simultaneously as both polarizer and analyzer.

16. Isochromatics

We have seen that in a circular polariscope extinction of monochromatic light occurs when the phase retardation \(\Delta\) is a multiple of \(2\pi\). From equations (20) and (22) it follows that a monochromatic fringe, or isochromatic, is the locus of points for which

\[ (\sigma_x-\sigma_y)^2+4\tau_{xy}^{\,2}=(p-q)^2=\left(\frac{n\lambda}{dC}\right)^2 . \tag{27} \]

The number \(n\) is called the order of the fringe and is measured in relative units \(p-q\). For example, the difference of the principal stresses along a fringe of the 6th order is three times the difference of the principal stresses on a fringe of the 2nd order. The order of a fringe at any point corresponds to the number of cycles of intensity passing through this point under continuous loading of the model. In Fig. 21 (see insert sheet II) the isochromatics are shown near a small circular hole in a plate under its tension. With a white-light source all the isochromatics are colored, except the zero-order isochromatic, which will be dark. For \(n=0\), i.e., when the principal stresses \(p\) and \(q\) are equal, equation (27) gives

\[ \sigma_x-\sigma_y=0, \tag{28} \]

\[ \tau_{xy}=0. \tag{29} \]

Each of these equations represents a curve, and in the general case the two curves will intersect only at an isolated point.

Thus the zero isochromatic is usually observed as an isolated dark trace, known as an isotropic point.

An example of an isochromatic system with four isotropic points is given in Fig. 18(a).

17. Isoclinics

We have seen [equation (23)] that in a plane polariscope extinction of light occurs when \(\Phi\) is equal to \(m \frac{\pi}{2}\). Equa-

19. Points of isotropy

Substituting expressions (28) and (29) into equation (30), we find that at a point of isotropy the isoclinic parameter will be indeterminate. All isoclinics pass through the isotropic point, and it is very difficult to construct isostatics near such a point.

Philon[^59], Föppl and Neuber[^6], and von Mises[^57] showed that the form of the isostatics near a point of isotropy depends on the number and orientation of the directions for which the tangent to the isoclinic forms an angle with the \(x\)-axis equal to the isoclinic parameter. In the most frequently encountered cases, one, two, or three such asymptotic directions are observed. These typical patterns, with the corresponding isostatics, are shown in Fig. 25 (the asymptotic directions are drawn with heavy lines). They are all easily distinguishable from one another, with the possible exception of cases Ia and Ib. Case Ib differs from case Ia in that in the former the angle \(\psi\) between two asymptotic directions is less than

\[ \frac{\pi}{2} \]

(\(\Phi_a\) and \(\Phi_c\)), and the third direction (\(\Phi_b\)) passes inside this angle.

Fig. 25. Isotropic points of the first order:
(a) case Ia with three asymptotes, (b) case Ib—three asymptotes \(\psi < \dfrac{\pi}{2}\), (c) case Ic—two asymptotes, (d) case II—one asymptote

Fig. 26. Enlarged view of the isoclinics near point \(A\) of Fig. 23

To determine to which type a point of isotropy belongs, it is necessary to find the asymptotic directions. As an example of such an investigation, consider points of isotropy that occur on the contour of a circular hole in a wide plate under its tension. There are four such points, one of them at point \(A\) of Fig. 23. In Fig. 26 the isoclinics near point \(A\) are plotted in an enlarged view. Denoting by \(\zeta\) the angle which the isoclinic

at point \(A\), forms with the \(x\)-axis, we construct a curve with parameter \(\Phi\) of the isocline as abscissa and \(\zeta\) as ordinate (Fig. 27). The straight line \(\Phi=\zeta\) intersects the curves \(\zeta(\Phi)\) at points whose number is equal to the number of asymptotic directions (in this case, three). The intersections occur at \(\Phi=60^\circ,\ \Phi=120^\circ,\ \Phi=169^\circ 7'\).

Fig. 27. Determination of asymptotic directions from the isoclines of Fig. 26

Fig. 28. Stress trajectories from the isoclines of Fig. 23

Consequently, in the present case the isotropic point belongs to type Ia. The complete system of stress trajectories is shown in Fig. 28. von Mises \(^{174}\) and Föppel \(^{9}\) considered isotropic points with four asymptotic directions.

IV. DIRECT DETERMINATION OF THE PRINCIPAL STRESSES

20. Methods for obtaining the principal stresses

The technique described in § 15 makes it possible to determine the direction and the difference of the two principal stresses in the plane of the plate. As we have seen, on the unloaded contour of the plate the principal stresses normal to the contour vanish, so that the data obtained are sufficient to determine the principal stresses on the contour independently of one another. In most cases of interest to the engineer, the greatest stress in the model and in the specimen appears precisely on the free contour; in such cases it is sufficient to measure these stresses, and the method of photoelasticity gives a complete solution of the problem—further investigation is superfluous.

In a number of cases there is a need to determine the stresses at every point of the specimen, or likewise along the loaded contour of the specimen. To solve problems of this type, special methods were developed for the independent determination of each of the principal stresses; in §§ 21–31 the following of them will be described.

1. Filon’s method of graphical integration.
2. Neuber’s graphical method.
3. The lateral extensometer method.

1. Method of interference in an air layer.
2. Method of stress concentration.
3. Method of the Favors interferometer.
4. Method of the Fabry interferometer.
5. Method of Hiltscher’s convergent beam of light.
6. Membrane method.
7. Electrical method.
8. Computational method.

21. Filon’s Method

Consideration of the equilibrium conditions for a small curvilinear rectangle bounded by four neighboring isostats (Fig. 29) leads to the following two equilibrium equations,^5 which characterize the changes of the principal stresses along the isostats

\[ \left. \begin{aligned} \frac{\partial p}{\partial s_1} &= -\,\frac{p-q}{\rho_2},\\ \frac{\partial q}{\partial s_2} &= -\,\frac{p-q}{\rho_1}. \end{aligned} \right\} \tag{31} \]

where \(s_1\) and \(s_2\) are orthogonal curvilinear coordinates measured respectively along the isostats \(p\) and \(q\), and \(\rho_1\) and \(\rho_2\) are the corresponding radii of curvature of these isostats (Fig. 29). Maxwell noted,^104 that these equations, when the isochromatics and isostatics are known, are sufficient for determining \(p\) and \(q\) throughout the entire

Fig. 29. Curvilinear quadrilateral formed by four neighboring isostats

Fig. 30. Filon’s method of integration along isostats

plate. The numerator in the right-hand side of the equations is known from measurements of the optical phase difference, while the denominator may be found by measurements of the isostats. The equations can be integrated graphically, starting from a point of the boundary where the stresses are known.

The quantities $\rho_1$ and $\rho_2$ are practically determinable with a low degree of accuracy; therefore equations in the form (32) are inconvenient for graphical integration.

Filon$^{58}$ proposed using, for practical computations, equations (31) in the form

\[ p=p_0+\int_{\Phi_0}^{\Phi} (p-q)\operatorname{ctg}\gamma\,d\Phi, \tag{32} \]

\[ q=q_0-\int_{\Phi_0}^{\Phi} (p-q)\operatorname{ctg}\gamma\,d\Phi, \]

where $\gamma$ is the angle between the isocline and the isostatic line, as shown in Fig. 30; $\Phi$, as before, is the angle between the isostatic line $p$ and an arbitrary direction $x$; $p_0$ and $q_0$ are the known values of $p$ and $q$ at the initial point of integration. The order of computation is as follows: for a series of points $A_0, A_1, A_2,\ldots$, lying on one isostatic line [for example, $p$ (Fig. 30)], the values of $p-q$, $\gamma$, and $\Phi$ are determined by one of the methods described in § 15. Then a graph is constructed of the variation of the quantity $(p-q)\operatorname{ctg}\gamma$ as a function of $\Phi$ (Fig. 31). For example, the area bounded by the curve and the ordinates $\Phi_0$ and $\Phi_2$ is equal to

\[ \int_{\Phi_0}^{\Phi_2}(p-q)\operatorname{ctg}\gamma\,d\Phi . \]

Fig. 31. Graphical integration
for

\[ \int_{\Phi_0}^{\Phi_2}(p-q)\operatorname{ctg}\gamma\,d\Phi \]

Adding the value of this integral to the quantity $p_0$ (the value of the stress $p$ at point $A_0$), we find the stress $p$ at point $A_2$. Since $p-q$ is known, $q$ will thereby also be determined. A similar calculation is carried out for all points of the isostatic line.

The sources of inaccuracy of the method lie in the difficulty of exact measurement of the angles $\Phi$ and $\gamma$. The accuracy of measurement depends on the form of the isoclines, while isoclines are usually observed in the polariscope as broad and blurred bands. Specializations of this method for certain particular cases of stress determination were carried out by Filon$^{5}$, Frocht$^{61}$, Bod$^{26}$, and Frocht$^{69}$.

22. Neuber’s graphical method

Proceeding from another variant of equations (31), Neuber$^{125}$ arrived at a method that makes it possible to determine $p$ and $q$ by a graphical construction not involving the operation of integration. This

method makes it possible to construct a family of curves that are the locus of points with equal \(p+q\). Such curves are called isopachs. (The meaning of the term will be explained in the next paragraph.) From the quantities \(p+q\) and \(p-q\), the stresses \(p, q\) are determined for each point.

In Fig. 32, \(O1\) and \(O2\) represent the directions of the isostatics \(p\) and \(q\) passing through the point \(O\). These are the same directions along which \(ds_1\) and \(ds_2\) were measured in Fig. 29. Let us denote the direction of the isopach passing through the point \(O\) by \(O3\), and the direction of the normal to it by \(O4\). The direction of the isochrome passing through the point \(O\) will be \(O5\), and the normal to it \(O6\). Finally, the direction \(O7\) is the direction of the isocline passing through \(O\), and \(O8\) is the normal to it. As before, \(\Phi\) is the angle which the isostatic \(p\) (\(O1\)) makes with the \(x\)-axis, and \(\gamma\) is the angle between the isocline and the isostatic \(p\). The angles which the isopach and the isochrome make with the isostatic \(p\) will be denoted respectively by \(\xi\) and \(\eta\).

Fig. 32. Diagram used in determining stresses by Neuber’s method. \(1\)—isostatic direction, \(3\)—isopachic direction, \(5\)—isochromatic direction, \(7\)—isoclinal direction.

From Figs. 29 and 32 we find the following identities (putting \(p+q=P\) and \(p-q=Q\)):

\[ \frac{1}{\rho_1}=\frac{\partial \Phi}{\partial s_1} =-\sin\gamma\,\frac{\partial \Phi}{\partial s_8}; \qquad \frac{\partial P}{\partial s_1} =-\sin\xi\,\frac{\partial P}{\partial s_4}; \qquad \frac{\partial Q}{\partial s_1} =-\sin\eta\,\frac{\partial Q}{\partial s_6}. \]

\[ \frac{1}{\rho_2}=\frac{\partial \Phi}{\partial s_2} =\cos\gamma\,\frac{\partial \Phi}{\partial s_8}; \qquad \frac{\partial P}{\partial s_2} =\cos\xi\,\frac{\partial P}{\partial s_4}; \qquad \frac{\partial Q}{\partial s_2} =\cos\eta\,\frac{\partial Q}{\partial s_6}. \]

Using equation (31), we obtain:

\[ \left. \begin{aligned} \frac{\partial P}{\partial s_4}\sin\xi &= -\frac{\partial Q}{\partial s_6}\sin\eta +2Q\,\frac{\partial \Phi}{\partial s_8}\cos\gamma,\\ \frac{\partial P}{\partial s_4}\cos\xi &= \frac{\partial Q}{\partial s_6}\cos\eta +2Q\,\frac{\partial \Phi}{\partial s_8}\sin\gamma \end{aligned} \right\}. \tag{33} \]

This is the form of the equilibrium equations according to Neuber.

Let the stress increment in passing from one isochrome to another be \(\Delta Q\), so that the parameters of the isochromes will be \(Q=0,\Delta Q,2\Delta Q,\ldots\). We choose the same magnitude of increment for the isopachs, so that \(\Delta P=\Delta Q\). Let the parameter of the isocline in passing from one curve to the neighboring one be equal to \(\Delta\Phi\). Denote by \(a\) the distance between two neighboring isopachs, by \(b\) the distance between two adjacent isochromes, and by \(c\) the distance between isoclines; \(a,b\), and \(c\) are measured respectively in the directions \(4,6,8\).

Then equations (33) can be written in the approximate form

\[ \begin{aligned} \frac{\Delta P}{a}\sin \xi &= -\,\frac{\Delta Q}{b}\sin \eta + 2Q\,\frac{\Delta\Phi}{c}\cos \gamma,\\ \frac{\Delta P}{a}\cos \xi &= \frac{\Delta Q}{b}\cos \eta + 2Q\,\frac{\Delta\Phi}{c}\sin \gamma . \end{aligned} \tag{34} \]

The quantities on the right-hand side of equations (34) can be measured from the systems of isoclinics, isochromatics, and isostatics; putting \(\Delta P\) equal to \(\Delta Q\), we have only two unknown quantities, \(\xi\) and \(a\), and two equations for determining them. The angle \(\xi\) gives the direction of the isopachs at the given point. Knowing these directions for a number of points of the plate, one can construct the isopachs in the same way as isostatic curves are constructed from the system of isoclinics.

On the free contour \(P=Q\), if the contour belongs to line 1, and \(P=-Q\), if the contour belongs to line 2. The order of an isopach is determined by the order of the corresponding isochromatic, which meets it on the free contour. Neuber gives a somewhat modified method for the case when the isopach does not intersect the contour.

As in Filon’s method, the accuracy of the method depends on the accuracy with which the angles \(\Phi\), \(\gamma\), and \(\gamma_0\) can be measured.

23. The lateral extensometer method

Mesnager\(^{16}\) observed that the deformation in the direction normal to the plate is proportional to the sum of the principal stresses, so that, by measuring the change in thickness at some point of the plate, it is possible to determine \(p+q\). Consequently, \(p+q\) is constant along a line of constant thickness; the term introduced by Filon, “isopach,” now becomes clear: it means a line of equal thickness.

Fig. 33. Mesnager interferometric frame

If the plate lies in the plane \(x,y\), then \(\sigma_z=0\), and the deformation in the direction normal to the plate, according to equations (5), (17), and (18), is equal to

\[ \varepsilon_z=-\left(\frac{\nu}{E}\right)(p+q). \tag{35} \]

Denoting by \(d\) the thickness of the plate and by \(\Delta d\) the change in thickness, we have

\[ p+q=-\,\Delta d\,\frac{E}{\nu d}. \tag{36} \]

To get an idea of the magnitude of \(\Delta d\), let us suppose that the model is made of bakelite, for which one may take \(E=600\,000\ \text{lb/in.}^2\) and \(\nu=0.24\). Further, let \(p+q=1\,000\ \text{lb/in.}^2\) and \(d=0.25\ \text{mm}\). Then \(\Delta d=0.0001\ \text{mm}\). Consequently, we must have an extensometer that would measure quantities of the order of one hundred-thousandth of an inch.

Menage constructed an interferometric frame for measuring \(\Delta d\). It consists of two rods \(AB\) and \(CD\) (Fig. 33), connected to one another at the ends \(A\) and \(C\) by a flexible steel plate. Two contacts at intermediate points \(E\) and \(F\) grip the model so that, if the thickness of the model changes, this change alters to an even greater degree the distance between the ends \(B\) and \(D\) of the rods. The relative displacement is measured by observing the displacement of interference fringes in the air layer between two optical plates fastened at \(B\) and \(D\).

Fig. 34. Westinghouse lateral extensometer

Fig. 35. Coker lateral extensometer

When the \(5461\,\text{Å}\) line of a mercury arc is used, the displacement of a fringe corresponds to a movement of \(0.00001075\) dm.

An instrument based on the same principle, but of a different design, was made by Voze[^175].

The Westinghouse research laboratory uses the Gugenberger extensometer shown in Fig. 34.

In Coker’s lateral extensometer[^45] (Fig. 35), the points of the fork actuate a mechanical lever, which in turn rotates a small concave mirror. The trace of the beam of light reflected by the mirror measures \(\Delta d\). The instrument is mounted on a movable support, so that the change in thickness along any line on the model can be determined in this direction before and after loading.

The three instruments mentioned above are more advanced than Coker’s instrument.

Methods of direct measurement of \(p+q\) are the most valuable, since: 1) the measurements are independent of the magnitudes \(p-q\) and \(\Phi\), 2) measurements at one point of the model are independent of measurements at other points, 3) the instruments are inexpensive.

24. Interference Method of the Air Layer

Maris \(^{101}\) proposed, for determining \(\Delta d\), observing interference fringes in a thin layer of air formed by the surface of the model itself and by a certain auxiliary optical plate. The method was further developed by Thesard \(^{157}\) and Frocht \(^{66,70}\), so that it became possible to observe and photograph a whole series of isopachic curves.

Interference fringes are the loci of points of constant thickness of the air layer and, consequently, the loci of points of constant thickness of the plate (if the auxiliary optical plate is correctly oriented with respect to the model). Successive fringes will represent lines of isopachs of equal increment \(p+q\). Thesard \(^{157}\) described the following method of orienting the auxiliary plate. Three points not lying on one straight line are chosen on the free edges of the model. The values \((Q_1, Q_2, Q_3)\) of \(p-q\) at these points are known from the isochromatics, so that the values \((P_1, P_2, P_3)\) of \(p+q\) are also known at these points. The auxiliary plate is fixed in such a way that the orders of the isopachs at these points are in the ratio \(P_1 : P_2 : P_3 = Q_1 : Q_2 : Q_3\) (with accuracy up to the algebraic sign, as indicated in § 15). This method can be applied only in the case when the surface of the model is sufficiently well polished (in the absence of load, deviations from planarity must be of the same order as the errors in measuring \(\Delta d\)). If the polishing of the model is not sufficiently good, then it is first necessary to examine the fringe pattern of the unloaded model in order to take into account the irregularity of its surface. Thesard recommends making observations from both sides of the model. The system of fringes determines the inclination of the perpendiculars to the surface of the model at a given point with respect to the plane passing through the three edge points.

The technique of this method must be carefully developed; however, the effort expended is compensated by the results of the investigation, namely by obtaining a complete system of isopachic curves.

25. Method of Stress Concentrations

This method, proposed by Thesard \(^{10}\) and Baud \(^{1}\), makes use of Kirsch’s solution \(^{5}\) for stress concentrations at a small circular hole in a homogeneous tensile field. In Fig. 36(a) (if \(p\) is the tensile stress of the plate far from the hole), the stresses at \(A\) and \(A'\) will be equal to \(3p\), while at \(B\) and \(B'\) they will be equal to \(p\). If another field with tensile stress \(q\), at right angles to \(p\), is superposed on this field [Fig. 36(b)], then the stresses at \(A\) and \(A'\) will be

\[ P_N = 3p - q \quad \text{and} \quad P_m = -p + 3q. \]

$P_M$ and $P_m$ are, respectively, the maximum and minimum stresses near the hole. Then

\[ p=\frac{1}{8}(P_m+3P_M) \]

\[ q=\frac{1}{8}(P_M+3P_m) \tag{37} \]

To determine $p$ and $q$ at a point of the photoelastic plate, a small hole is drilled at this point, and from the isochromatics $P_M$ and $P_m$ are determined on the contour of the hole (Fig. 21); $p$ and $q$ are then calculated from equation (37).

Fig. 36. Diagram used in determining stresses by the concentration method

Fig. 37. Favre interferometric polariscope

This method has two difficulties: 1) the possibility of drilling a hole without an “edge effect” is open to doubt; 2) in light of Boze’s investigations (§ 13), the correct values of $P_m$ and $P_M$ are obtained only when the diameter of the hole is of the same order as the thickness of the plate.

26. Favre’s Interferometric Method

As we saw in § 11, measurements of the absolute magnitude of the phase retardation of two waves polarized in the directions $p$ and $q$ make it possible to determine $p$ and $q$ directly. Favre[^53][^54][^56] constructed, for such measurements, an interferometer of the Mach–Zehnder type. In this instrument (Fig. 37), light from a light source $S$ is polarized by prism $P$ and is split into two beams $T$ and $R$ by the half-silvered plate $M_1$. The transmitted beam $T$ is reflected from the totally reflecting mirror $M_2$, passes through the model $M$, and is reflected into the viewing tube by the half-silvered plate $M_3$. The other beam $R$ is reflected from $M_4$, passes through $M_3$, and then combines with the first beam. A half-wave plate $H_1$ is used to align the direction of polarization with one of the directions of the principal stresses for the given point of the model. A second half-wave plate $H_2$ rotates the plane of polarization back to its former position. The glass plate $C$ is rotated so as to bring the phases of the beams $T$ and $R$, which travel along different paths, into agreement when the model is not loaded. When the plane of polariza-

coincides, for example, with \(p\), the model is loaded and the phase of ray \(T_p\) changes with respect to the phase of ray \(R\) for two reasons. The first is that the refractive index of a wave polarized in the direction \(p\) changes from \(n_0\) to \(n_1\), and the second is that the thickness of the model changes by the amount \(\Delta d\). The phase difference between \(T_p\) and \(R\) will be

\[ \Delta_1=\left(\frac{2\pi}{\lambda}\right)\left[d(n_1-n_0)+(n_0-1)\Delta d\right]; \tag{38} \]

the notation is the same as in the preceding paragraph. \(\Delta_1\) is measured by rotating the glass plate \(C\) until \(T_p\) and \(R\) coincide in phase. In the same way the plane of polarization \(T\) is brought into coincidence with \(q\); by an analogous experiment we measure the difference between \(T_q\) and \(R\); it is equal to

\[ \Delta_2=\left(\frac{2\pi}{\lambda}\right)\left[d(n_2-n_0)+(n_0-1)\Delta d\right]. \tag{39} \]

Substituting into equations (38) and (39) the values \((n_1-n_0)\), \((n_2-n_0)\), and \(\Delta d\), given by equations (21) and (36), we find

\[ \begin{aligned} \Delta_1&=adp+bdq\\ \Delta_2&=bdp+adq \end{aligned} \Biggr\}, \tag{40} \]

where

\[ \begin{aligned} a&=\frac{2\pi}{\lambda}\left[C_1-\frac{\nu}{E}(n_0-1)\right]\\ b&=\frac{2\pi}{\lambda}\left[C_2-\frac{\nu}{E}(n_0-1)\right] \end{aligned} \Biggr\}. \tag{41} \]

The constants \(a\) and \(b\) depend on the material of the model and on the wavelength of the light source; they can be determined in angular units of rotation of the glass plate \(C\) when testing some standard specimen in tension.

Fig. 38. Fabbry apparatus (I. G. A. Brat, American Bureau of Orders)

Knowing \(\Delta_1, \Delta_2, d, a, b\), we can solve equation (40) for \(p\) and \(q\). Such measurements are made from point to point over the entire plate. By this method one can obtain accurate values if: 1) the surfaces of the model are sufficiently plane and parallel to one another (the material of the model is glass), 2) the polarization interferometer is constructed with all precautions, 3) the room temperature is kept constant. A photograph of the interferometer at the American Bureau of Orders is shown in Fig. 38.

Tank[^151] supplemented the Fabry interferometer method so that it can be adapted for measuring \(p-q\) and \(p+q\) from point to point of the model.

27. Fabry’s Interferometric Method

In this method[^52] both surfaces of the model are silvered (50%) and serve as interferometric surfaces. A beam of parallel light passes through the model normally to its surface. The interference that takes place between the rays that have passed through the plate and the rays that have undergone two reflections (Fig. 39) is observed through an analyzer. The light ray entering the loaded model is split into two waves \(p\) and \(q\), plane-polarized in two mutually perpendicular directions.

Fig. 39. Fabry’s method

Each of these waves is partly transmitted (\(p_1\) and \(q_1\)) and partly reflected (\(p_2\) and \(q_2\)) from the second surface. The reflected wave (\(p_2\) and \(q_2\)) undergoes a second reflection (from the first surface) and then leaves the model in the first direction together with \(p_1\) and \(q_1\). By rotating the analyzer, one can extinguish one pair of rays, for example, \(q_1\) and \(q_2\), since both these rays are polarized in directions perpendicular to \(p_1\) and \(p_2\); in this position, interference will occur between \(p_1\) and \(p_2\). By rotating the analyzer through \(90^\circ\), one can observe interference between \(q_1\) and \(q_2\).

This method requires analysis at each point, since the directions of the planes of polarization \(p\) and \(q\) are different at different points of the model. In the Fabry interferometer method the surfaces of the model must be plane and parallel to one another.

28. Hiltscher’s Convergent-Light Method

Returning to our consideration of the index ellipsoid (§ 8 and Fig. 11), we see that there exist two planes \(BOD\) and \(BOD'\) (in Fig. 40) containing the mean principal axis \(OB\), which intersect the ellipsoid in circles. The normals \(OQ\) and \(OQ'\) to these planes are called optical axes; the angle between them \(2\Omega\) is called the angle between the optical axes. If the differences between the principal refractive indices \(n_a\), \(n_b\), \(n_c\) are small, as is the case in photoelasticity, then it can be shown, proceeding from the geometrical properties of the ellipsoid, that

\[ \sin^2 \Omega=\frac{n_a-n_b}{n_a-n_c}, \qquad \cos^2 \Omega=\frac{n_b-n_c}{n_a-n_c}. \tag{42} \]

Let us also recall (§ 8) that any radius \(ON\) in Fig. 11 represents a possible direction of the normal to a wave; the plane,

passing through \(O\) perpendicular to \(ON\), intersects the ellipsoid in an ellipse \(DE\), whose semiaxes \(OD\) and \(OE\) are proportional to the refractive indices \(n_1\) and \(n_2\) of the two waves passing through the birefringent medium. Now, if \(\theta_1\) (Fig. 11) is the angle between the wave normal and the optical axis \(OQ\), and if \(\theta_2\) is the angle between the wave normal and the optical axis \(OQ'\), then it can be shown, again from the geometrical properties of the ellipsoid, that

\[ n_1-n_2=(n_a-n_c)\sin\theta_1\sin\theta_2, \tag{43} \]

if the differences between the indices are small.

Fig. 40. Index ellipsoid showing circular sections

From equation (10) we have

\[ \begin{aligned} n_c-n_a&=C(\sigma_3-\sigma_1)\\ n_a-n_b&=C(\sigma_1-\sigma_2)\\ n_b-n_c&=C(\sigma_2-\sigma_3) \end{aligned} \Bigg\}, \tag{44} \]

so that, according to (42), (43), (44), and (9), we have

\[ \begin{aligned} \sigma_3-\sigma_1&=\frac{\lambda\Delta}{2\pi d C}\, \frac{1}{\sin\theta_1\sin\theta_2}\\ \sigma_2-\sigma_1&=\frac{\lambda\Delta}{2\pi d C}\, \frac{\sin^2\Omega}{\sin\theta_1\sin\theta_2}\\ \sigma_3-\sigma_2&=\frac{\lambda\Delta}{2\pi d C}\, \frac{\cos^2\Omega}{\sin\theta_1\sin\theta_2} \end{aligned} \Bigg\}. \tag{45} \]

We have just seen how one can measure the two quantities appearing on the right-hand side of equation (45), apart from \(\Omega\), \(\theta_1+\theta_2\). Consequently, if these three angles are determined, the values of the three differences of the principal stresses can be obtained. If one of the three principal stresses is equal to zero (a two-dimensional state of stress), then the other two are determined from equation (45).

Hiltscher measured \(\Omega\), \(\theta_1\), and \(\theta_2\), using a convergent beam of polarized light; these angles can also be measured with the universal rotating stage used in petrography.

29. Membrane method

Let us consider the small strain \(\varepsilon_x\) of a linear segment \(PA\) (Fig. 41) of length \(\delta x\), parallel to the \(x\)-axis. Let the displacement of the point \(P\) to \(P'\) be denoted by \(u\); the displacement of the point \(A\) to \(A'\) will be equal to \(u+\left(\dfrac{\partial u}{\partial x}\right)\delta x\). Consequently, the strain of \(PA\) is equal to

\[ \varepsilon_x=\frac{P'A'-PA}{PA} = \frac{\left[(\delta x-u)+u+\left(\frac{\partial u}{\partial x}\right)\delta x\right]-\delta x}{\delta x} = \frac{\partial u}{\partial x}. \tag{46} \]

STUDY OF STRESSES BY THE PHOTOELASTICITY METHOD

Similarly, the deformation \(\varepsilon_y\), parallel to the \(y\)-axis, expressed through the displacement \(v\) in the \(y\) direction, will be equal to:

\[ \varepsilon_y=\frac{dv}{dy}. \tag{47} \]

To obtain the corresponding expression for the shear strain \(\gamma_{xy}\) in the \(x,y\) plane, let us consider the change in the angle between adjacent sides \(PA\) and \(PB\) of an element of this plane (Fig. 42). \(P\) is displaced to \(P'\), \(A\) to \(A'\), and \(B\) to \(B'\). If the distance along the perpendicular from \(P'\) to \(PA\) is \(v\), then, to a first approximation, the distance along the perpendicular from \(A'\) to \(PA\) is \(v+\left(\dfrac{\partial v}{\partial x}\right)\delta x\). Consequently, the sine of the angle between \(P'A'\) and \(PA\) is equal to

Fig. 41. Deformations and displacements along the \(x\)-axis

Fig. 42. Shear strain in the \(x,y\) plane

\[ \frac{\left(v+\left(\dfrac{\partial v}{\partial x}\right)\delta x\right)-v}{\delta x} = \frac{\partial v}{\partial x} \]

and since the angle is small, \(\dfrac{\partial v}{\partial x}\) may be taken as equal to the angle between \(P'A'\) and \(PA\). Similarly, \(\dfrac{\partial u}{\partial y}\) is the angle between \(P'B'\) and \(PB\). The quantity \(\gamma_{xy}\) is equal to the total change in the angle between the two sides, i.e.

\[ \gamma_{xy}=\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}. \tag{48} \]

It is obvious that the three displacement components \(\varepsilon_x\), \(\varepsilon_y\), and \(\gamma_{xy}\) are not independent quantities, since all of them can be expressed through the two displacement components \(u\) and \(v\). To find the relation between them, we differentiate equation (46) twice with respect to \(x\), equation (47) twice with respect to \(y\), and equation (48) once with respect to \(x\) and once with respect to \(y\). Adding, we obtain

\[ \frac{\partial^2 \varepsilon_x}{\partial y^2} + \frac{\partial^2 \varepsilon_y}{\partial x^2} = \frac{\partial^2 \gamma_{xy}}{\partial x\,\partial y}. \tag{49} \]

This expression is known as the compatibility equation; it expresses the law governing the possible changes of the displacement components along the plate, or, in other words, gives the condition of continuity of the material.

Let us now consider the laws relating the components of stress under the condition that the forces acting on an element of the \(x,y\) plane are in equilibrium. Such an element, with the forces acting on it, is shown in Fig. 43.

Summing the forces in the directions of the \(x,y\) axes, we find the following equilibrium equations:

\[ \left. \begin{aligned} \frac{\partial \sigma_x}{\partial x}+\frac{\partial \tau_{xy}}{\partial y}&=0,\\ \frac{\partial \sigma_y}{\partial y}+\frac{\partial \tau_{xy}}{\partial x}&=0 \end{aligned} \right\}. \tag{50} \]

Fig. 43. Stresses acting on a rectangular element in the \(x,y\) plane

Differentiating the first of equations (50) with respect to \(x\), the second with respect to \(y\), and adding them, we obtain

\[ \frac{\partial^2 \sigma_y}{\partial y^2} + \frac{\partial^2 \sigma_x}{\partial x^2} + 2\frac{\partial^2 \tau_{xy}}{\partial x\,\partial y} =0. \tag{51} \]

The strains are related to the stress by Hooke’s law [equation (5)]. In the case of a plane state of stress we have

\[ \left. \begin{aligned} E\varepsilon_x&=\sigma_x-\nu\sigma_y,\\ E\varepsilon_y&=\sigma_y-\nu\sigma_x,\\ E\gamma_{xy}&=2(1+\nu)\tau_{xy} \end{aligned} \right\}. \tag{52} \]

Substituting equation (52) into (49) and using (51), we obtain

\[ \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) (\sigma_x+\sigma_y)=0. \tag{53} \]

Further, from equations (17) and (18)

\[ (\sigma_x+\sigma_y)=(p+q) \]

we have Laplace’s equation

\[ \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) (p+q)=0; \tag{54} \]

the latter equation governs the change in the sum of the principal stresses. Den Hartog\(^{50}\), Biot\(^{31}\), and Binzeno and Koch\(^{36}\) drew attention to the fact that equation (54) coincides with the equation of a membrane subjected to forces applied only to its contour. This analogy is used in the following manner: in a plate, an opening is cut out of exactly the same shape as the contour of the model; at the edges of the opening, normals to the plane are erected at every point, proportional in magnitude to \(p+q\). A membrane is stretched over this opening, and in such a way that its ordinates at the edges of the opening coincide with the lengths of the normals proportional to the values of \(p+q\). Then the membrane will assume such a shape that its ordinates at all points of the opening will also be proportional to the magnitude \(p+q\) at each point.

This method is easily implemented and is used for all regions near the boundary of the model for which \(p+q\) can be determined in advance. The method described was successfully used by Weibel\({}^{177,178}\) (Figs. 44 and 45), using a soap film as the membrane. Bio and Smith\({}^{35}\), MacGivern and Supper\({}^{121,122}\) used thin-sheet rubber.

Fig. 44. Models used in the membrane analogy (E. E. Weibel)

Fig. 45. Apparatus of the membrane analogy (University of Michigan)

30. Electrical method

If the values of \(p-q\) on the contour are known, then equation (54) can also be solved in another way.

It is known, for example, that the distribution of potential in a two-dimensional electric field obeys the same equation. If an opening of the same shape as the model is cut in a metal plate and an electric potential proportional to \(p+q\) is applied at each point of the contour, then the potential at points inside the opening will be proportional to the value of \(p+q\) at those points.

The method was described by Binzeno and Koch\({}^{36}\), Bio\({}^{31}\), Malavard\({}^{100}\), and Meyer and Tank\({}^{110}\).

31. Computational method

Liebmann\({}^{97}\) showed that Laplace’s equation can be solved for given values on the contour of the model by the method of successive approximations. In this method a grid, consisting of a system of orthogonal straight lines, is superimposed on the region under investigation; approximate values of \(p+q\) are assigned at each point of intersection of the lines. Liebmann points out that the true value of the function at any point will be equal to the average of the four values at neighboring intersection points. In this way, step by step, the values of \(p+q\) can be computed at all points of the region.

Shortley and Weller\({}^{142}\) modified the method, seeking to achieve more rapid convergence of the computation.

V. Special Applications of the Method of Photoelasticity

32. Goodier’s Hydrodynamic Analogy

Goodier[^73] drew attention to the complete analogy between the patterns of photoelasticity and the slow motion of a viscous fluid in a two-dimensional treatment. Isochromatics correspond to lines of constant rotational velocity; lines perpendicular to the isochromatics correspond to lines of constant hydrostatic pressure; isoclines correspond to lines of constant magnitude of shear deformation in the flow; finally, isostatics correspond to lines indicating the orientation of elements having the greatest shear deformation.

This analogy provides a basis for a method of solving hydrodynamic problems by the methods of photoelasticity.

33. Gravitational and Thermal Analogy

Biot[^32–^34] showed how the photoelastic method can be used to investigate the problem of stresses arising in a body owing to the action of gravity and to changes in temperature, without the actual reproduction of body forces and heating of the model.

The stresses caused by the self-weight of a heavy body can be calculated from the stresses of a small model of the same shape if a normal pressure, varying linearly with thickness, is applied to the contour of this model.

The stresses caused by a steady flow of heat in a hollow cylinder can be found by investigating the stresses of a plane model having the shape of the cylinder’s cross-section. This model must be cut along a radius, and the resulting slit must be enlarged or deformed by an amount that can be calculated in advance. Weibull[^179] applied this method to the case of various cross-sections (see Fig. 46, insert sheet III). The stresses in a thick cylinder under unsteady flow can be determined with the aid of a plane model of the same cross-section, loaded along its contour.

Biot and Smith[^35] investigated, by this method, thermal and shrinkage stresses in dams.

34. Three-Dimensional Photoelasticity

In the preceding chapters the method of photoelasticity for a two-dimensional problem was described. The methods mentioned cannot be applied without modification to the study of a three-dimensional distribution of stresses, since in this case the magnitude and direction of all three stress components vary along the path of light propagation. Only an effect “integrated” over the entire length of the ray is observed; it is necessary, however, to investigate the optical properties in each element along the ray path. Up to now no

Fig. 46. Application of the thermal analogy by Bio (E. E. Weibel)

Fig. 48. Stress concentration around a hole in a stretched strip (M. M. Frocht)

were proposed: analytical expressions relating the total effect to the state of stress inside the body.

Favre^55 proposed observing the optical effect of a stressed small cube made of a photoelastic material and embedded, at the points of greatest interest, in a transparent model made of a material for which the optical stress coefficient is zero; however, the experimental difficulties of such a test are apparently insurmountable.

Ménage^108 proposed a method suitable for a limited class of three-dimensional problems. Successful application of the method would lead to the determination of stresses at boundaries caused by forces applied normally to the plane of a plate. The experiment is performed as follows: a thin layer of transparent material is applied to the polished surface of a metal; this layer becomes birefringent under deformation. Bending of the plate in its own plane produces a plane state of stress; the double refraction caused by the stress is measured in light reflected from the polished metallic surface. We are not aware, however, of any successful results from the application of this method.

Timoshenko and Goodier^164 proposed another method for analyzing stresses in a surface layer. They glued a thin layer of polaroid to the surface of a transparent model. A layer of photoelastic material is glued onto it. The light passing through the model passes through the polaroid in such a way that the elliptical polarization disappears. Thus, only a plane-polarized light wave passes through the layer of birefringent material. Observation of the emerging light beam gives an idea of the two-dimensional state of stress in the surface layer.

In 1851 Maxwell^104 carried out an experiment that became the basis of a promising method of three-dimensional photoelasticity. Maxwell made a hollow cylinder of heated gelatin and rotated the inner surface of the cylinder relative to the outer one (through a small angle about the axis of the cylinder). On cooling, double refraction arose in the gelatin, indicating an elastic distribution of stresses.

Maxwell could not explain this phenomenon, and the experiment was forgotten until 1935. Solakian^145, repeating Maxwell’s experiment, heated a thick cylindrical rod of marble, applying a twisting couple to the axis of the rod; the material cooled under the action of the couple. After this, the cylinder was cut into circular-section plates, which were examined in polarized light. The resulting fringe pattern did not, however, correspond to the state of stress predicted by Saint-Venant’s torsion theory, as was pointed out by Hetényi^77.

Oppel^128 carried out a similar experiment with a specimen of trolon. A metal ball was pressed against the heated specimen; then the specimen was cooled under load. Oppel notes that the pattern appearing when plates cut from the cooled specimen are examined corresponds to the state of elastic stresses.

Recent experiments by Hetényi (see § 12) and the studies of Kuske[^93] explain the phenomenon described. At high temperatures the elastic insoluble skeleton resists the applied forces. When the material cools under load, the soluble part freezes around the deformed skeleton and holds the deformation when the load is removed. As a result, the material is in a state of elastic deformation that took place at high temperature. This state is not noticeably disturbed by careful cutting of the model, since equilibrium between the soluble and insoluble parts exists in regions of space of molecular order. The difficulty in carrying out the investigation lies in the fact that, whereas Young’s modulus decreases with increasing temperature in the ratio \(640:1\), the relative optical stress coefficient increases only in the ratio \(26:1\). The deformation required to obtain a larger value of the double refraction is so great that it significantly changes the shape of the model.

The interpretation of the double refraction obtained in a cut plate is much more complicated than the corresponding decoding in the two-dimensional case. In the general case the principal plane of optical symmetry does not coincide with the plane of the cut, so that the relative phase difference will not be a measure of the difference between the principal refractive indices and, consequently, will not be proportional to the difference of the principal stresses. The three differences of the principal stresses can be determined, as Gilcher[^81] showed, by additional measurements, for each point of the cut layer, of the angle between the optical axes \(2\Omega\) and of the angles \(\theta_1\) and \(\theta_2\) between the normal to the cut and the optical axis (see §§ 8 and 28).

35. Special cases of the three-dimensional problem

There are several cases[^114] for which it is not necessary to measure the three angles \(\Omega\), \(\theta_1\), and \(\theta_2\).

In the most important of them the stress distribution has a plane of symmetry. This occurs when the model and the system of forces are geometrically symmetric with respect to some plane. The plane of symmetry is in this case a plane of principal stresses; thus the relative phase difference for a layer cut along this surface is proportional to one of the three differences of the principal stresses. The plane in which the optical axes lie will be parallel or perpendicular to the plane of symmetry. In the latter case the bisector of the angle between the optical axes is either parallel or perpendicular to the plane of the cut. Consequently,
\[ \theta_1=\theta_2=\frac{\pi}{2}, \]
or
\[ \theta_1=\theta_2=\frac{\pi}{2}-\Omega, \]
or
\[ \theta_1=\theta_2=\Omega, \]
depending on which plane of symmetry coincides with the plane of the cut. If \(\theta_1\) and \(\Omega\) are measured in addition to \(\Delta\), then, according to (45), the three differences of the principal stresses can be calculated.

In a prismatic or cylindrical bar subjected to torsion according to Saint-Venant,^1) one of the principal stresses at a point of the bar will be equal to zero, while the other two are equal to each other and opposite in sign. The direction of the zero principal stress is perpendicular to the generator of the cylinder; the other two directions of the principal stresses make an angle of \(45^\circ\) with the generator. This state of stress corresponds to a biaxial crystal with the angle between the optical axes \(\dfrac{\pi}{2}\), since the optical axes are parallel and perpendicular to the generator. A cut of the bar normal to the generator gives a phase difference equal to zero for a normally incident ray; however, a cut made at another angle to the generator gives phase differences from which the principal stresses can be calculated. For the calculation it is necessary to know the angles \(\theta_1\) and \(\theta_2\) which the normal to the wave makes with the optical axis. One of these angles will be constant; it is equal to the angle between the axis of the bar and the normal to the cut; the other will vary from point to point of the section. For each point of the cut it is necessary to find only this angle \(\theta_2\) and the phase difference \(\Delta\). Equation (45) makes it possible to calculate the principal stresses directly if the following are known: the wavelength \(\lambda\), the thickness of the cut plate \(d\), and the optical coefficient \(C\). Hetényi^78 noted that a cut at an angle of \(45^\circ\) gives the greatest number of fringes; it should be noted, however, that the fringes do not indicate the loci of equal differences of the principal stresses. It is necessary to know the angle \(\theta_2\) for every point in the cut at an angle of \(45^\circ\). The angle \(\theta_2\) may be determined in the following way: let \(O\) be a point on the cut at \(45^\circ\); for this point \(\theta_2\) is sought. The normal to the cut at the point \(O\) and a line parallel to the generator of the bar at the same point \(O\) determine a plane, which intersects the plane of the cut along the line \(OA\). Let \(OB\) be the axis of polarization in the cut at the point \(O\) (it is determined from the system of isoclinics in the usual way), and let the angle \(AOB=\alpha\). It can be shown^114 that

\[ \sin \theta_2 = (1+\cos 2\alpha)^{1/2}. \]

Similar formulas are also suitable for sections cut at another angle.

36. Determination of \(\sigma_3\), \(\sigma_2\), and \(\sigma_1\) in the three-dimensional problem

Hiltscher^81 indicated that, by measuring \(\Delta\), \(\Omega\), \(\theta_1\), and \(\theta_2\) for a sufficiently large number of different sections, it is possible to obtain only the directions of the three principal stresses and their differences for all points, except for points on the free contour. To determine the principal stresses in the general case it is necessary to measure \(n_a\), \(n_b\), and \(n_c\), or to use other non-optical measurements.

There exist three standard methods for measuring the principal refractive indices, but apparently none of them is

^1) Timoshenko, Theory of Elasticity, New York, p. 228, 1934.

satisfactory for modern methods of investigating photoelasticity in three dimensions.

In Stokes’ method1 the section is placed on a ruled grating; the latter is observed through a microscope. Owing to the double refraction of the plate, each family of parallel lines will be seen in focus at two positions of the microscope objective. The three refractive indices can be calculated from the relative positions of the focus. However, the photoelastic double refraction is so small that this method is of little use.

The prism method2 requires cutting a small prism out of the model for each point at which the stresses must be determined.

The methods of total reflection3 are very accurate, but they make it possible to measure the optical properties in the surface layer of the specimen. However, the properties at the surface do not give the optical characteristics of the interior points of the specimen.

Coker[^93] noted that, by heating the sliced plate and measuring the change through the thickness, one can obtain sufficient data (together with those previously obtained) for calculating \(\sigma_3\), \(\sigma_2\), and \(\sigma_1\).

37. Dynamic Problems

The photoelastic method is used in solving three dynamic problems: 1) the investigation of a constant state of stress, 2) the investigation of a periodic change in stresses, and 3) the investigation of the instantaneous state of stresses.

Frocht and Uytcomb[^61] investigated stresses in a rotating disk at constant angular velocity. This is a case in which the stresses do not vary with time.

Investigations of periodically varying stresses in rotating gear wheels were carried out by Timos and Bodom[^82]. Stresses in vibrating beams were studied by Rivilli[^136] and Moppett[^119]. In Fig. 47 (see inset sheet IV) a photograph by Moppett is shown of a cantilever beam with vibrations of 60 periods per second. Instantaneous stresses arising under impact were considered by Tuzi[^169], Tuzi and Nizida[^170],[^171], Frocht[^66], and Tuvenin[^163]. Moppett[^119] pointed out the difficulties encountered in investigations of instantaneous states of stress.

38. Fields of Application of the Photoelastic Method

The photoelastic method has been used in solving a wide class of problems in the fields of mechanical engineering, steel and concrete structures, mining, shipbuilding, aeronautics, materials testing, physical research, the glass industry, soil and foundation mechanics, railway

Fig. 47. Photograph of the bands of a cantilever beam vibrating at 60 oscillations per second (V. M. Murray)

Fig. 49. Stress concentration at circular notches in a strip under tension (M. M. Frocht)

Fig. 50. Stress concentration in the fillets of a cantilever beam (E. E. Weibel)

Fig. 51. Curved bar under pure bending
(E. E. Weibel)

Fig. 52. Chain link (V. M. Murray)

(a)              (b)

Fig. 53. Stresses in gear wheels
(R. I. Dolan)

Fig. 54. Stresses in the body of a plate connected by a riveted seam.

Fig. 56. Stresses in the wheel of a railway car (R. I. Dolan): I—hub, II—wheel rim.

Fig. 55. Study of stress in bridge rollers (I. G. A. Bratt): (a) roller loaded centrally, (b) enlarged view of the region near the concentrated load, (c) roller loaded near the edge.

...engineering and applied physics, and many other fields. The literature through which one may become acquainted with applications of the photoelasticity method in various fields is listed below:

Arch bridges ^23, 107, 127	Retaining walls ^42
Bridge trusses ^22	Plates with holes ^149
Car wheels ^47, 141	Rotating disks ^71
Stresses in concrete ^98, 148	Rubber industry ^161, 162
Dams ^33, 41, 160	Screw threads ^74, 86
Fatigue tests ^131, 132	Shrinkage stresses ^35
Flat bars ^164	Soil mechanics ^84
Fluid flows ^73, 133, 139	Stress concentration ^67, 68, 137, 176, 176, 177
Gear wheels ^37, 82	Machine parts ^35, 42, 158, 159, 168
Glass products ^85, 154, 155	Thermal stresses ^31, 32, 133, 179
Various structures ^87	Torsion ^77, 78
Wedges and keys ^147	Tunnels ^143
Testing of materials ^93, 101, 120, 156	Welded joints ^46, 48, 144
Reinforced concrete ^30, 102

Some typical photographs are given on the inserts. Figs. 48, 49, and 50 show the characteristic pattern of stress distribution near holes and fillets. Fig. 51 shows a bar in pure bending. Fig. 52 shows bands of a loaded link of a chain. Stresses in gear wheels are shown in

Fig. 57. Polariscope-centrifuge apparatus (R. B. Bugg. ^42)
1—lens, 2—analyzer, 3—polarizer, 4—rotating box, 5—photographic plate, 6—mercury tube, 7—synchronous switch, 8—tachometer, 9—voltmeter, 10—motor control, 11—stroboscopic light control.

Fig. 53. Stresses in the body of a plate connected by a riveted seam are shown in Fig. 54. Figs. 55(a) and (c) illustrate studies of bridge rollers. Fig. 55(a) shows a roller loaded along its axis; Fig. 55(b) shows, enlarged, the region near the application of a concentrated load. Fig. 55(c) shows a roller loaded near the edge.

A photograph obtained in studying the stresses in the wheel of a railway car under various loading conditions is shown in Fig. 56.

An interesting device for photoelastic investigation of stresses arising as a result of the action of gravity,

given by Bucky, Solakian, and Baldin^42. The model rotates in a centrifuge, as a result of which forces similar to gravitational forces act on the points of the model; the pattern of fringes is observed with a stroboscope (Fig. 57).

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1. Walker, Analytical Theory of Light, p. 225, 1904. ↩↩↩

2. Ibid., p. 230. ↩

3. Ibid., p. 242. ↩