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Publication Number: FHWAHRT13085 Date: October 2013 
Publication Number: FHWAHRT13085 Date: October 2013 
It is well known that cracks within a material decrease its resonance frequency by decreasing the overall stiffness of the structure. In addition to this linear change in frequency, researchers have demonstrated that cracks are also responsible for nonlinear effects, including the strain amplitude dependent resonance frequency shift.^{(16,17)} Microcracks inside a material form a network that acts as a nonlinear bond system. The nonlinear behavior of this bond system can be attributed to Hertzian contact of crack faces and/or opening and closing of cracks in response to wave motion. Using the phenomenological model for hysteresis and classical nonlinear constitutive relationships, researchers have shown that the nonlinear stress–strain relationship can be shown as the equation in figure 1.^{(9,13)}.
Figure 1. Equation. Nonlinear stress–strain relationship
Assuming that effects of hysteresis are dominant in microcracked materials, the relationship between frequency shift and strain amplitude shown in figure 2 is valid for low levels of strain excitation.^{(13)}
Figure 2. Equation. Relationship between frequency shift and strain amplitude
At higher amplitudes, there will also be an additional quadratic term for the strain amplitude ; however, because the experiments are performed at low levels of strain excitation, this higherorder term can be ignored. In these experiments, the amplitude of the signal, A, which is proportional to the strain amplitude, Δε, is measured instead of the strain amplitude. As a result, the absolute hysteresis parameter, α, is not measured. Instead, a scaled hysteresis parameter (η) proportional to α is used as a measure of the material's nonlinearity. The relationship used in this investigation is given by the equation shown in figure 3.
Figure 3. Equation. Relationship between the frequency shift, the scaled hysteresis parameter, and the signal amplitude
Chapter 4 explains in detail the extraction of the parameter η from recorded data. An additional effect observed for hysteretic materials is the increase in damping for the sample. The equation in figure 4 shows that a linear relationship exists between the change in damping and the strain amplitude.^{(13)}
Figure 4. Equation. Relationship between change in damping and strain amplitude
Because the signal amplitude is proportional to strain amplitude, the relationship between the change in damping and the signal amplitude is the equation shown in figure 5:
Figure 5. Equation. Relationship between the change in damping and the signal amplitude
where Ω is termed the nonlinear damping parameter.
Because the nonlinearity is attributed to nonlinear interaction of cracks, relatively large and open cracks will not contribute to nonlinearity. Under this assumption, the nonlinearity parameter can be thought of as an “instantaneous” measure of nonlinearity. Because the measurements for tracking nonlinearity in CPT samples are taken at rather long intervals of time, the “cumulative” nonlinearity (η_{c}) can be measured by integration, as shown in the equation in figure 6.
Figure 6. Equation. Integral to calculate “cumulative” nonlinearity
With experimental data, a Riemann sum can be used to approximate this integral (see figure 7).
Figure 7. Equation. Approximation of integral to calculate”cumulative” nonlinearity using a Riemann sum