Photoelastic techniques are based on the bi-refringent characteristics exhibited by transparent plastic materials of specific tailored compounds of plexiglas, polycarbonate, and epoxy resins.  These plastics, under load, develop an isochromatic fringe pattern that can be directly related to the maximum shear stresses in the geometry being analyzed.  The photoelastic materials can be selected to match with the expected elongation of the substrate material.  In Table 7.4.1, the photoelastic test materials are bracketed into three levels by expected elongation range.  The maximum measurable strain for a particular photoelastic coating depends upon its stress-strain curve and the linearity of photoelastic behavior. 

Table 7.4.1.  Coating Selection for Elongation Levels

                Chart courtesy of Vishay Measurements Group, Inc.

The bi-refringent sensitivity is another important factor to consider when choosing a photoelastic coating [Vishay Measurements Group, Inc., 2001].  The overall sensitivity of the strain measurement system depends on:

·        The sensitivity of the coating is expressed by the fringe value, f.  The fringe value represents the difference in principle strains, or the maximum shear strain, required to produce one fringe.  The lower this parameter, the more sensitive the coating,

·        The sensitivity of the polariscope system for examining the photoelastic pattern and determining the fringe order, N.

The primary difference between the approach used for two- and three-dimensional work is that two-dimensional models can be directly analyzed under load whereas the three-dimensional model must be reduced to a two-dimensional model before the crack tip fringe information can be recovered.  To obtain the fringe results from the three-dimensional model, the isochromatic fringe pattern must first be frozen in place while the model is under load; the stress freezing is accomplished through a thermal treatment that takes the material above a critical temperature for a hold-time period which is followed by a slow cooling.  Subsequent to the stress freezing operation, the three-dimensional model is sliced up to obtain a two-dimensional slice that contains the crack segment of interest.  This two-dimensional slice is then interrogated with normal photoelastic equipment (polariscope) to recover the imbedded fringe information.

A new development for building 3-D structural models is by using stereolithography (SLA).  [TECH, Inc. 2001]  SLA is a rapid prototyping process by which a product is created using an ultra-violet (UV) curable liquid resin polymer and advanced laser technology.  Using a CAD package such as Pro/Engineer, SolidWorks, or other solid modeling software, a 3-D solid model is exported from the CAD package as an .stl file. The .stl file is then converted into thin layers. The sliced model, in layers, is then sent to the SLA machine. The SLA machine uses its laser to cure the shape of the 3-D CAD model on a platform in the vat of resin from the bottom up, one layer at a time. As each layer is cured, the platform is lowered the thickness of one layer so that when the part is completely built, it is entirely submerged in the vat.  Stereolithography is capable of creating the most complex geometries quickly and precisely.

Figure 7.4.1.  Stereolithography process diagram (Courtesy of TECH, Inc.)

The analysis of crack tip fringe information is the same for both the two- and three-dimensional models.  For Mode 1 loading, the stress-intensity factor (K) is obtained using:

where s_o is an unknown pseudo-boundary stress, r is the distance directly above the crack tip on an axis perpendicular to the crack path, and t_max is the maximum shear stress obtained from the stress-optic law

with n the photoelastic fringe order, f the material fringe value, and B the thickness of the two-dimensional model or slice.  The shear stress (t_max) is typically analyzed using a truncated Taylor series that describes the behavior in the crack tip region, i.e.

where Smith [1975] suggests N is chosen to be the lowest possible number that results in Equation 7.4.6 providing a good fit to the shear stress data.  Figures 7.4.2 and 7.4.3 illustrate the two basic steps used in determining the stress-intensity factor from photoelastic experiments [Smith, 1975].  For both three-dimensional surface crack models considered, the thin two-dimensional slice that was analyzed for the crack-tip fringe pattern was taken through the point p.  The slice was perpendicular to the crack plane and oriented so that the slice was through the thickness; thus the slice had the appearance of a single edge cracked geometry.

Figure 7.4.2 describes the shear stress distribution (points) and the corresponding least-squares derived truncated Taylor series expansion (curve) for the two surface crack geometries considered.  Figure 7.4.3 illustrates how Equation 7.4.6 and 7.4.4 are combined to extrapolate the photoelastic data to the crack tip.  Figure 7.4.3 portrays the stress-intensity factor based on photoelastic data (K_AP) as the ratio of the photoelastic result to the preexisting theoretical result.  Note that the photoelastic result is calculated from Equation 7.4.4 where the pseudo boundary stress (s_o) is taken as zero.  This stress is accounted for through the N=0 term of Equation 7.4.6.  The curves in Figure 7.4.3 are based on the truncated Taylor series solutions obtained from the data in Figure 7.4.2.  In both cases shown, the extrapolations lead to reasonable estimates of the theoretical results and are somewhat typical of what one might expect from photoelastic estimates of the stress-intensity factor.

Figure 7.4.2.  Typical Maximum Shear Stress Data Modeled with a Truncated Taylor Series Equation [Smith, 1975]

Figure 7.4.3.  Extrapolation of Equation 7.4.4 Based on the Truncated Taylor Series Equation Results Presented in Figure 7.4.2 [Smith, 1975]