The Deviatoric Field and Photoelasticity

The Deviatoric Field and Photoelasticity

The isotropic component of the stress or strain field is therefore a scalar poten-tial function. That is to say, the distance to the center of Mohr's Circle is a Laplacian or Poisson field at every point in the structure. This field can be determined analytically or experimentally in two dimensions by plotting a flow net, assuming, as is seldom the case, that either the mean stress or its nor-mal derivative is known around a closed boundary and body forces are con-stant or zero.

The deviatoric component of the stress tensor is more complicated in that vector fields are involved rather than a simple scalar field. In three dimen-sions, the deviatoric tensor involves three magnitudes, S_i, (principal values) and associated angles, _i, (orientation to the reference frame) and can,

therefore, be thought of as three vectors, S_i, describing the size and aspect ratios of the Mohr's Circles.

In the planar care, there is just one deviatoric vector, q , of magnitude _max = ₁₃ = _q in the orientation 2 on Mohr's Circle. Thus the deviatoric field is a vec-tor field expressing the two pieces of information, magnitude and direction, at each point in the structure as they flow from one boundary to another in accor-dance with the field equations. Figure 4.4 illustrates this vector property of the deviatoric stress at a point by showing, for a specific example, how two states of stress (or strain) at a point can be added together by adding the isotropic mean stresses algebraically and the deviatoric vectors by the parallelogram rule.*

Photoelasticity is a beautiful experimental method of discovering and dis-playing this deviatoric vector field with photographs. Like Moiré for dis-placements and interferometry for the isotropic stress, photoelasticity is a full-field method and is much easier, accurate, and more versatile. Two sets

of interference fringes are observed when plane, polarized light is transmit-ted through a stressed sheet (a model) of almost any transparent material in a crossed polariscope. One of these sets is the isochromatic fringe pattern given by the fundamental relationship or stress-optic law:

where the stresses are in the plane of the model, t is the thickness of the model, N are either integer, 0, 1, 2 ... or half-order (1/2, 3/2, 5/2...) depending on whether the polariscope is crossed or aligned, and f is the stress-optic

fringe coefficient. This fringe coefficient, in turn, is a property of the transparent material being used for the model and the wave length, , of the transmitted light used to illuminate it.* Thus in white light, the colors are extinguished in order giving a beautiful spectrum effect, which led Maxwell to name these fringes isochromatics. For accurate experimental analysis, monochromatic light or filters on the camera are used to give clearly defined black lines on the photograph. An example of a diametrically loaded thick ring is shown in Figure 4.5a. Thus in one photograph, a complete contour map of the magni-tude of the in-plane deviatoric stress tensor field, S or _max is determined and easily visualized.

Isoclinic fringes are, as the name implies, contour lines of equal orienta-tion of the principal stresses to the Cartesian axes of the crossed polarizing plates (usually horizontal and vertical). Isoclinics are black in white light

FIGURE 4.5

and can therefore be differentiated from the colored isochromatics.* From the isoclinics, stress-trajectories or paths of the principal stresses or maxi-mum shears can easily be constructed graphically. Observed isoclinics and the principal stress trajectories for a diametrically loaded thick ring are shown in Figure 4.5b.

Not only does photoelasticity directly determine the complete deviatoric vector field, it provides boundary conditions for the isotropic field. Since on a boundary one principal stress is known (zero if unloaded), the other can be computed from the isochromatic value for ₁ = ₂. The flow-net for the har-monic mean stress and its conjugate can then be drawn or calculated by computer without ambiguity and the entire stress field determined. Even three-dimensional solutions are possible with photoelastic analysis of slices from a stress-frozen model.