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Publication Number: FHWA-HRT-13-085
Date: October 2013

 

Accelerated Determination of ASR Susceptibility During Concrete Prism Testing Through Nonlinear Resonance Acoustic Spectroscopy

CHAPTER 2. Theoretical background For nonlinear acoustics

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).

The equation calculates   (stress) as   (strain) times ? subscript 0 (linear elastic modulus) times the quantity: 1 plus   (coefficient of quadratic anharmonicity) times   plus   (coefficient of cubic anharmonicity) times   squared plus   (measure of the material hysteresis) times the quantity:   (strain amplitude) plus   times the sine of   (strain rate) closed quantity, closed quantity.
Figure 1. Equation. Nonlinear stress–strain relationship

where
σ  =  stress, The stress formula N divided by m squared
Ε0 =  linear elastic modulus,The stress formula N divided by m squared
β  =  coefficient of quadratic anharmonicity
δ   =  coefficient of cubic anharmonicity
ε =  strain part of the nonlinear stress-strain reltionship formula where delta L is divided by L, where L is length , where L is length
α  =  measure of the material hysteresis
Δε   =  strain amplitude
part of the nonlinear stress-strain relationship formula where epsilon dot is the strain rate =  strain rate, part of the nonlinear stress-strain relationship formula where epsilon is divided by seconds
sgn(part of the nonlinear stress-strain relationship formula where epsilon dot is the strain rate)=1  if part of the nonlinear stress-strain relationship formula where epsilon dot is the strain rate > 0 , -1 if part of the nonlinear stress-strain relationship formula where epsilon dot is the strain rate < 0 , and 0 if part of the nonlinear stress-strain relationship formula where epsilon dot is the strain rate = 0

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)

The equation calculates quantity: f subscript 0 (linear resonance frequency) minus f (resonance frequency at increased excitation amplitude) closed quantity, divided by f subscript 0 all set equal to C subscript 1 (coefficient proportional to material hysteresis) times.
Figure 2. Equation. Relationship between frequency shift and strain amplitude

where
f0  =  linear resonance frequency, Hz
f  =  resonance frequency at increased excitation amplitude, Hz
C1  =  coefficient proportional to material hysteresis

At higher amplitudes, there will also be an additional quadratic term for the strain amplitudean additional  quadratic term for the strain amplitude D sub1 time delta epsilon squared ; however, because the experiments are performed at low levels of strain excitation, this higher-order 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.

The equation calculates quantity: f subscript 0 (linear resonance frequency) minus f (resonance frequency at increased excitation amplitude) closed quantity, divided by f subscript 0 all set equal to   (scaled hysteresis parameter) times A (signal amplitude).
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)

The equation calculates quantity:   (damping rate at increased excitation amplitude) minus   subscript 0 (linear damping rate) closed quantity, divided by   subscript 0 all set equal to C subscript three (coefficient proportional to material hysteresis) times.
Figure 4. Equation. Relationship between change in damping and strain amplitude

where
ξ0  =  linear damping rate
ξ  =  damping rate at increased excitation amplitude
C3  =  coefficient proportional to material hysteresis

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:

The equation calculates quantity:   (damping rate at increased excitation amplitude) minus   subscript 0 (linear damping rate) closed quantity, divided by   subscript 0 all set equal to   (nonlinear damping parameter) times A.
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 (ηc) can be measured by integration, as shown in the equation in figure 6.

The equation calculates   subscript C (cumulative nonlinearity) as equal to the integral of   times   times d  over the interval from 0 to t (time).
Figure 6. Equation. Integral to calculate “cumulative” nonlinearity

With experimental data, a Riemann sum can be used to approximate this integral (see figure 7).

The equation calculates   subscript C as approximately equal to one-half the sum of quantity: t subscript i minus t subscript (i minus 1) closed quantity, times quantity   times t subscript i plus   times t subscript (i minus 1) closed quantity, over the interval of i from 2 to N.
Figure 7. Equation. Approximation of integral to calculate”cumulative” nonlinearity using a Riemann sum

 

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