THE CORRESPONDENCE BETWEEN MICROSTRUCTURAL FEATURES AND ACOUSTIC NONLINEARITY
ABSTRACT
Ensuring the safety of civil infrastructure is crucial for public safety. Non-destructive evaluation (NDE) techniques provide effective ways to detect interior materials damage without impacting its serviceability. Conventional acoustic/ultrasonic methods, although provide effective ways for detecting and characterizing defects in materials, can only detect defects with sizes on the order of millimeters. Nonlinear acoustic/ultrasonic methods, on the other hand, enable detection of incipient micro-scale damage on the order of micrometers (e.g. micro-cracks, closed cracks and debonded interfaces). Although nonlinear acoustic/ultrasonic methods have shown potential in early damage detection, there are still several factors that prevent them from being widely used in practice. Most importantly, the lack of quantitative relations between the damage and the nonlinear acoustic response impede the interpretation of the results. The aim of this dissertation is to address this limitation.
The first part of the dissertation is focused on the relation between the nonlinear acoustic responses of materials undergoing progressive distributed damage. I proposed a new resonance nonlinear acoustic technique, i.e., single impact-based nonlinear resonant acoustic spectroscopy (SINRAS), in characterizing materials with distributed microscopic damage. I developed a novel signal processing approach to obtain the hysteresis nonlinearity parameter from the changes in the instantaneous resonance frequency in the ring-down vibration signal after a strong impact on test samples. This technique also gives linear resonance frequency ( 0) and the decaying rate of the ring-down signal, thus provides a comprehensive assessment of the state of damage. I demonstrate the use of SINRAS for monitoring the growth of damage in concrete samples under two different types of mechanism: freeze-thaw (FT) cycles and alkali silica reaction (ASR). The results suggest that SINRAS, while being much simpler and faster to conduct, gives comparable results to those from multi-impact based nonlinear resonant acoustic spectroscopy (MINRAS). The extracted
nonlinearity parameters not only clearly differentiate damaged and undamaged samples, but also successfully monitor the microstructural evolution of the samples during the damage process. However, the quantitative relationship between distributed microcracks and nonlinearity is rather empirical due to the complexity of the physical mechanisms involved. Given that distributed cracks are clusters of individual cracks, understanding the nonlinear signatures of a single crack is a necessary first step. Furthermore, the heterogeneity and multi-phase nature of cementitious materials make it challenging to develop basic understanding of the origins of nonlinearity and the processes involved. Therefore, the second part of the dissertation is focused on investigating the physical micro-mechanisms that give rise to contact acoustic nonlinearity (CAN) at a single interface in an otherwise homogenous medium. The experimental focus is on fatigue cracks in several aluminum samples. The cracks’ microscopic features such as crack path roughness and aperture distribution are extracted from scanning electron microscopy (SEM) images. A new configuration of dynamic acousto-elastic testing (DAET) with a surface wave probe is used to measure near-surface CAN. This allows a direct study of CAN for contact interfaces of known micro-geometries. I use a data-driven approach to construct a quantitative relationship between nonlinear acoustic parameters (from DAET) and the microstructural features of contact interfaces (from SEM images). In addition, a physics-based analytical model based on the extracted aperture profile is built to predict CAN in terms of the variation of ultrasound transmission coefficients and phase delay during dynamic perturbation. The predicted nonlinearity is in qualitative agreement with our DAET data. However, the model tends to underestimate the nonlinearity due to some of the underlying assumptions. A comprehensive sensitivity analysis illuminates the influence of some of these assumptions.
Understanding how microscopic features at contact interfaces affect the contact acoustic nonlinearity is a necessary step towards quantitative diagnostics with nonlinear acoustic techniques.
In addition to the field of NDE, the results of this study have important implementations in a wide spectrum of scientific fields including structural health monitoring (SHM), biomedical diagnostics, and earth sciences.
TABLE OF CONTENTS
LIST OF FIGURES ………………………………………………………………………………………………….. ix
LIST OF TABLES ……………………………………………………………………………………………………. xvii
LIST OF ACRONYMS …………………………………………………………………………………………….. xviii
ACKNOWLEDGEMENTS ……………………………………………………………………………………….. xix
Chapter 1 Introduction ……………………………………………………………………………………………… 1
1.1 Background …………………………………………………………………………………………………. 2
1.1.1 Nonlinear Acoustic Techniques ……………………………………………………………. 2
1.1.2 Acoustic Test Methods for Cementitious Materials …………………………………. 10
1.1.3 Theoretical Background on Nonlinear Crack–Wave Interactions ………………. 13
1.2 Thesis Objective and Outline …………………………………………………………………………. 15
Chapter 2 Impact-based nonlinear acoustic testing for characterizing distributed damage
in concrete ………………………………………………………………………………………………………… 18
2.1 Introduction …………………………………………………………………………………………………. 20
2.2 Theoretical background …………………………………………………………………………………. 25
2.2.1 Classical and Non-classical Nonlinearity ……………………………………………….. 25
2.2.2 Post-perturbation Slow Recovery ………………………………………………………….. 26
2.3 Materials and Test Methods …………………………………………………………………………… 28
2.3.1 Description of Test Samples ………………………………………………………………… 29
2.3.2 Resonance Frequency Measurements ……………………………………………………. 30
2.3.3 Impact Nonlinear Resonant Acoustic Spectroscopy (INRAS) …………………… 30
2.3.4 Impact Dynamic Acousto-Elastic Testing (IDAET) ………………………………… 31
2.4 Data Analysis Approaches …………………………………………………………………………….. 34
2.4.1 Maximum Strain Calculation ……………………………………………………………….. 34
2.4.2 Analysis of INRAS Data ……………………………………………………………………… 34
2.4.3 Analysis of IDAET Data ……………………………………………………………………… 37
2.5 Results and Discussion ………………………………………………………………………………….. 41
2.5.1 Multi-impact INRAS …………………………………………………………………………… 412.5.2 Single-impact INRAS …………………………………………………………………………. 44
2.5.3 Multi- vs. Single-impact INRAS …………………………………………………………… 48 2.5.4 Evolution of recovery rate ν with damage progress in single-impact
INRAS ……………………………………………………………………………………………….. 49
2.5.5 IDAET ………………………………………………………………………………………………. 50
2.6 Conclusions …………………………………………………………………………………………………. 53
Chapter 3 Single-impact nonlinear resonance acoustic spectroscopy for monitoring the
progressive alkali-silica reaction in concrete …………………………………………………………. 55
3.1 Introduction and Background …………………………………………………………………………. 57
3.2 Materials and Methods ………………………………………………………………………………….. 60
3.2.1 Sample composition and preparation …………………………………………………….. 61
3.2.2 Test methods ……………………………………………………………………………………… 62
3.3 Results ………………………………………………………………………………………………………… 70
3.3.1 Comparison between conventional NRAS and MINRAS for a mortar bar …. 70
3.3.2 Linear expansion ………………………………………………………………………………… 72
3.3.3 MINRAS on concrete prisms ……………………………………………………………….. 73
3.3.4 SINRAS vs. MINRAS …………………………………………………………………………. 77
3.3.5 Single-INRAS for monitoring the progress of ASR…………………………………. 79
3.3.6 Influence of temperature on and ………………………………………. 81
3.4 Discussion …………………………………………………………………………………………………… 83
3.5 Summary and Conclusions …………………………………………………………………………….. 85
Acknowledgements ……………………………………………………………………………………………. 86
Chapter 4 Dynamic Acousto-Elastic Response of Single Fatigue Cracks with Different
Microstructural Features: An Experimental Investigation ……………………………………….. 87
4.1 Introduction …………………………………………………………………………………………………. 89
4.2 Materials and Methods ………………………………………………………………………………….. 91
4.2.1 Test Samples ……………………………………………………………………………………… 91
4.2.2 Dynamic Acousto-Elastic Testing (DAET) with a Surface Wave Probe …….. 92
4.2.3 Scanning Electron Microscopy (SEM) and Image Analysis ……………………… 101
4.3 Results ………………………………………………………………………………………………………… 104
4.3.1 Linear wave velocity and transmission loss ……………………………………………. 1044.3.2 A Comparison of DAET Results for the Two Samples ……………………………. 106
4.3.3 Extracted Crack Features …………………………………………………………………….. 112
4.4 Discussion …………………………………………………………………………………………………… 114
4.5 Conclusion ………………………………………………………………………………………………….. 119
Acknowledgements ……………………………………………………………………………………………. 120
Chapter 5 A Data-driven Approach to Construct a Quantitative Relationship between Microstructural Features of Fatigue Cracks and Contact Acoustic Nonlinearity ………… 121
5.1 Introduction …………………………………………………………………………………………………. 1235.2 Materials and Experimental Methods ……………………………………………………………… 125
5.2.1 Sample Preparation …………………………………………………………………………….. 125
5.2.2 Dynamic Acousto-Elastic Testing (DAET) with a Rayleigh Wave Probe …… 126
5.2.3 Scanning Electron Microscopy (SEM) …………………………………………………… 1295.2.4 Dynamic Acousto-Elastic Testing (DAET) with a Rayleigh Wave Probe …… 129
5.3 Data Analysis ………………………………………………………………………………………………. 130
5.3.1 Image Processing ………………………………………………………………………………… 130
5.3.2 Statistical Analysis ……………………………………………………………………………… 135
5.4 Results and Discussion ………………………………………………………………………………….. 138
5.4.1 PCA on Predictor Variables …………………………………………………………………. 138
5.4.2 Stepwise MLR ……………………………………………………………………………………. 140
5.4.3 Stepwise PCR …………………………………………………………………………………….. 141
5.4.4 LASSO ……………………………………………………………………………………………… 144
5.4.5 Discussion: Comparison of Regression Results and Physical Interpretation .. 146
5.5 Summary and Conclusions …………………………………………………………………………….. 150
Acknowledgements ……………………………………………………………………………………………. 151
Chapter 6 An Integrated Analytical and Experimental Study of Contact Acoustic
Nonlinearity at Rough Interfaces of Fatigue Cracks ……………………………………………….. 152
6.1 Introduction …………………………………………………………………………………………………. 154
6.2 Theoretical Basis ………………………………………………………………………………………….. 157
6.2.1 Acoustic Wave Interaction with Imperfect Contact Faces ………………………… 157
6.2.2 Elastic Contact Model …………………………………………………………………………. 159
6.3 Methods ………………………………………………………………………………………………………. 162
6.3.1 Determination of the Model Parameters from the Crack Interface …………….. 162 6.3.2 Measuring the Contact Acoustic Nonlinearity (CAN) using Dynamic
Acousto-elastic Testing (DAET) ……………………………………………………………. 169
6.3.3 Estimation of Relative Displacement at Crack Interface ………………………….. 171
6.4 Comparison between Analytical and Experimental Results ……………………………….. 173
6.5 Discussion …………………………………………………………………………………………………… 1776.6 Conclusions …………………………………………………………………………………………………. 186Acknowledgements ……………………………………………………………………………………………. 187
Supplementary Materials ……………………………………………………………………………………. 188
Chapter 7 Conclusions and Recommendations …………………………………………………………….. 189
7.1 Conclusions …………………………………………………………………………………………………. 189
7.1.1 Rapid Assessment of Microscopic Distributed Damage in Concrete ………….. 189
7.1.2 Contact Acoustic Nonlinearity at Fatigue Cracks ……………………………………. 190
7.2 Limitations and Future Work …………………………………………………………………………. 192
7.2.1 Rapid Assessment of Microscopic Distributed Damage in Concrete ………….. 1927.2.2 Contact Acoustic Nonlinearity at Fatigue Cracks ……………………………………. 193
Bibliography ……………………………………………………………………………………………………………. 194
Chapter 1
Introduction
Maintaining the serviceability and sustainability of civil infrastructure (e.g., buildings, towers, bridges, dams, tunnels, nuclear power plants, and offshore structures) is critical to public safety and societal well-being. Non-destructive evaluation (NDE) techniques provide effective ways to detect interior materials damage without impacting its serviceability. Acoustic methods are among the most widely used NDE techniques because acoustic response is directly related to the mechanical properties of materials. Traditional linear acoustic/ultrasound methods rely on the changes in acoustic impedance at defects locations that cause wave reflections and scattering. Although these techniques are suitable for detecting macroscale defects such as open cracks and voids, early microscale defects are still challenging to identify. On the other hand, nonlinear acoustic/ultrasound methods rely on identifying the distortion of stress waves due to the microstructural changes of materials. The size of microstructural features can be orders of magnitude smaller than the wavelength. Therefore, these techniques have shown great potentials for damage detection at early stages. However, the damage diagnostics is based on the empirical observations of increased nonlinearity with the accumulation of microscopic damage. The qualitative nature of the diagnostics is an impediment to a wider adoption of nonlinear acoustic techniques in practice. This dissertation is an effort towards quantifying the relationship between microstructural features of damage and nonlinearity parameters. This work focuses on: 1) the relationship between nonlinearity parameters and progressive volumetric micro-damage in concrete, and 2) investigating the correspondence between microstructural features of contact interface and contact acoustic nonlinearity (CAN) signatures.
This chapter provide background information on NDE using nonlinear acoustic methods.
1.1 Background
1.1.1 Nonlinear Acoustic Techniques
Typical nonlinear acoustic techniques include higher harmonic generation, nonlinear resonant ultrasound spectroscopy (NRUS), vibro-acoustic modulation (VAM), and dynamic acousto-elastic testing (DAET). This section will give an overview of each aforementioned method and its application.
1.1.1.1 Higher Harmonic Generation
The frequency of propagating monochromatic acoustic waves is constant when they propagate in a linear elastic medium. Conversely, such a wave in a nonlinear elastic medium is distorted leading to higher harmonics generation. The amplitudes of higher harmonics give a quantitative description of nonlinearity. Considering the constitutive equation for quadratic nonlinearity:
= (1 + ) (1.1)
where is strain, is stress, is Young’s modulus, and is the quadratic nonlinearity parameter. Considering a single-frequency ultrasonic wave (longitudinal) propagating along a onedimensional rod, the equation of motion can be written in the following form:
2
2 = (1.2)
where is the density of the medium, is particle displacement, and is the propagating direction. The resulting particle displacement has a solution of the following form:
= 1 cos( − ) + 2 cos(2 − 2 ) + 3 cos(3 − 3 ) + ⋯ (1.3) where is the wavenumber, is the angular excitation frequency. The displacement amplitude corresponding to the second ( 2) harmonic is related to [1] :
2
= 8 22 (1.4)
1
When only cubic nonlinearity ( = (1 + 2)) is considered, only odd harmonic components (3 , 5 , …) are generated, where is the cubic nonlinearity parameter. The value of is proportional to the ratio of the third harmonic amplitude to the fundamental harmonic
3 amplitude: [1,2] . Both quadratic and cubic nonlinearities are so-called classical
1
nonlinearities due to atomic anharmonicity.
When considering a material with purely non-classical hysteretic nonlinearity, the onedimensional constitutive relation between the stress and strain can be derived from a phenomenological description of hysteretic material (Preisach-Mayergoyz space, or PM space) [3– 5] :
= (1 + (∆ + ( ̇) ) (1.5) where is the non-classical hysteretic nonlinearity parameter, ∆ is the maximum strain
amplitude, ̇ is the strain rate, and ( ̇) = 1 if ̇ > 0, ( ̇) = −1 if ̇ < 0. Similar to the
cubic nonlinearity, only odd harmonics are generated in this case. However, it is important to note that unlike , is proportional to the ratio of the third harmonic and the square of the fundamental
3 harmonic amplitudes: [2] . Figure 1-1 compares the monochromatic wave propagation in
1
linear and various nonlinear media [2] .
Figure 1-1: Schematic overview of harmonic amplitude dependence of the monochromatic wave propagating in linear and various nonlinear media [2] .
Higher harmonic generation method is the most commonly used nonlinear ultrasonic technique for NDE. It has been used for characterizing microstructural changes in metals [6–10] and detecting damage in concrete [11,12] . Besides the conventional bulk wave test configuration, higher harmonic generation has also been used with guided wave testing. For example, nonlinear Rayleigh surface wave has been used for testing metals [13–16] and concrete [17,18] ; and nonlinear Lamb wave has been utilized for characterizing microstructural changes of metals and metallic alloys [19,20] .
The aforementioned classical and non-classical hysteresis nonlinearities are often attributed to the distributed microstructural heterogeneities in materials. A different type of nonlinearity – contact acoustic nonlinearity (CAN) – has been observed in solids with localized discontinuities such as cracks, imperfect bonding, and delamination. The mechanism behind this type of nonlinearity is the opening and closing motion of the unbonded contact interface [21] . Solodov et al. [21] modeled the contact interface as a spring with a bi-linear stiffness with a higher value in compression than in tension. When a monochromatic wave is propagating through such an interface, the state of the spring is oscillating between “intact” stiffness and “decreased” stiffness. This model predicts a rectified wave with its harmonic amplitudes modulated by the sinc envelope function.
CAN provides an effective detection tool for “invisible” flaws such as closed fatigue cracks, which often go undetected using traditional methods. Methods relying on CAN have shown satisfactory results in the assessment of fatigue cracks in metals[22] , fatigue damage of adhesive joints [23] , and delamination in composite materials [24] .
1.1.1.2 Nonlinear Resonant Ultrasound Spectroscopy (NRUS)
Nonlinear resonant ultrasound spectroscopy (NRUS) is based on quantifying the resonance frequency shift and attenuation enhancing with increasing driving amplitude. This method relies on the non-classical hysteretic nonlinear effects, which is typically observed in materials with a soft mesoscopic structure in a rigid matrix, such as rocks, concrete, soil, and other materials with distributed micro-cracks. Considering a material with non-classical hysteretic nonlinearity according to Eq. 1.5, the resonant frequency shift and attenuation variation is proportional to the strain amplitude (∆ ):
− 0
= ∆
1 (1.6)
0 = ∆
where 0 and 0 are the resonance frequency and quality factor in linear strain regime, and and
are the resonance frequency and quality factor at each incremental strain level. The parameters
and are proportional to the hysteretic nonlinearity parameter in the constitutive relation (Eq. 1.5). Figure 1-2 shows the schematic NRUS test configuration. Typical NRUS consists of exciting the sample by continuous waves or chirps that sweep over a frequency range covering the resonant frequencies of target modes. The excitation amplitude increases from very small (barely above the noise level) to that inducing strains of ~10-6. The frequency spectra for increasing excitation amplitudes are recorded by a receiver (an accelerometer or a laser Doppler Vibrometer).
Figure 1-2: Schematics of NRUS test setup. The resonant frequency shifts downwards with increasing driving amplitudes.
NRUS has been used to assess the mechanical integrity of different materials with volumetric microscopic cracks such as rocks [25] , concrete [26] , composites [27,28] , and bones
[29–32] . Alternative setups such as impact-based NRUS has been applied to large concrete samples
[33–35] .
1.1.1.3 Vibro-Acoustic Modulation (VAM)
Vibro-acoustic modulation (VAM) [36] or nonlinear wave modulation spectroscopy (NWMS) technique [2] is based on the modulation of a continuous high-frequency () ultrasonic wave by low-frequency ( ) vibration, which can be a resonance mode of the test sample. In the presence of cracks, the powerful pumping vibration would cause the cyclic opening and closing of the cracks. The frequency spectrum of the modulated wave includes sidebands at frequencies ± , = 1,2,3, … in addition to the frequencies of the input high-frequency and low-frequency waves and
(Figure 1-3).
Figure 1-3: Schematics showing the principle of nonlinear vibro-acoustic modulation technique.
Similar to the higher harmonic generation method, the level of nonlinearity is quantified
by the amplitudes of side bands [2] . In the case of classical nonlinearity ( = (1 + +2), a modulation of the two input waves with amplitudes and results in the first-order sidebands
( ± ) with amplitudes proportional to . In addition, it also produces second-order sidebands ( ± 2 ) with amplitudes proportional to 2, with being a constant related
to both and . In case of pure non-classical hysteretic nonlinearity, only second-order sidebands ( ± 2 ) are generated and their amplitudes are proportional to , with being the
hysteretic nonlinear parameter. In practice, the sideband amplitudes are normalized by the amplitude of incident input waves. Klepka et al. [37] used one parameter defined as = ( + + − )/ to detect impact damage in laminated composites, where + and − are the
spectral amplitudes of the first pair of sidebands. Similarly, some studies used a so-called modulation index or modulation intensity (MI), which is defined as the ratio of sidebands amplitudes to the product of the amplitudes of low- and high-frequency input waves, to assess the damage severity caused by fatigue cracking in metallic samples [38,39] as well as delamination and kissing bonds in composites [40] . Alternatively, some researchers simply used the amplitude of sidebands to monitor the growth of damage [41,42] . The ratio of the sideband energy relative to the incident energy is also used [2,28,43,44] especially when the sidebands cannot be clearly distinguished due to either high attenuation or weak nonlinearity.
1.1.1.4 Dynamic Acousto-Elastic Testing (DAET)
Dynamic acousto-elastic testing (DAET) is based on the monitoring of the changes in wave velocity and attenuation (measured by a probe) under dynamic loading (exerted by a pump). In fact, DAET uses a pump and probe schemes and has a similar configuration to that described for the VAM technique. Figure 1-4 shows the schematics of a typical DAET configuration. The test setup consists of a continuous low frequency vibration (pump) and repeated high frequency ultrasonic pulses (probe): The pump typically matches a resonant mode to induce a cyclic strain field, while the probe senses the variation of wave velocity and attenuation at different dynamic strain levels.
Assuming that the variation of wave velocity (∆ ) with dynamic strain relative to the wave velocity at rest ( 0) is small, we can estimate the relative variation of elastic modulus as follows:
∆ ∝ ( 0+∆ )2− 02 ≈ 2 ∆ = + 2 + ⋯ + (1.7)
0 02 0
where and are the quadratic and cubic nonlinear parameters, respectively, and describes the hysteresis nonlinearity ( from NRUS measurement). Similarly, the variations of attenuation (∆ ) during low frequency pumping can also be derived from the changes in the amplitude ( ) of the probing ultrasonic signals:
∆ + 2 + ⋯ + (1.8)
where the subscript denotes the nonlinear parameters for attenuation properties. The right panel of Figure 1-4 shows the typical DAET results.
Figure 1-4: The schematic DAET setup showing the test configuration and typical results. The HF wave velocity and attenuation are “modulated” by the LF pump.
In addition to the classical and hysteresis nonlinearity parameters, it was found that the socalled conditioning and slow dynamics phenomena can also be captured by DAET [45–48] . When the sample is excited by an elastic wave of sufficiently high amplitude, the modulus decrease almost immediately, and continue to decrease for some period of time if the excitation persists, until the material reaches a new non-equilibrium steady state (Conditioning). When the excitation terminates, the modulus and quality factor do not return to the initial equilibrium state immediately, but recover slowly over a period of hours or even days (Slow Dynamics). Some phenomenological models have been proposed to explain conditioning and slow dynamics and show qualitative agreements with experiment results [49–51] . However, the physical mechanisms behind these phenomena are still not well understood, although it is believed that the gradual healing of the bond system (inter-grain contacts, dynamic/static friction, etc.) is responsible for the slow recovery of elastic modulus [46] .
In the laboratory setting, DAET has been successfully used for the microstructure or damage assessment in rocks [25,52–54] , concrete [55–60] , trabecular bone [61,62] , and metals
[63,64] . Several studies have applied this technique to large-scale structures and in-situ conditions. For example, Moradi-Marani et al. [65] proposed the application of DAET on large concrete specimens subjected to simulated traffic load as pump in a laboratory environment. Renaud et al. [66] developed a DAET approach for in-situ conditions to measure the nonlinear elastic properties
of soil.
1.1.2 Acoustic Test Methods for Cementitious Materials
This section will give an overview of acoustic test methods, including both linear and nonlinear, for damage assessment of cementitious materials.
1.1.2.2 Linear Acoustic Techniques
Most common linear acoustic techniques include ultrasonic pulse-echo, pulse velocity, pulse attenuation, and resonant frequencies methods.
The ultrasonic pulse-echo method (ASTM E114) detects flaws in materials by identifying the echo of pulsed stress waves at flawed locations [67] . The elastic wave propagation theory states that a part of the incident stress wave will be reflected at an interface between two media with different acoustic impedances. When the material contains damage, such as cracks or voids, the incident stress waves will be reflected at the interface of flaws. Although this technique is easy to implement, the limitation lies in the detectable size of flaws, which is dependent on the wavelength of incident wave. One solution is to increase the frequency to have a smaller size resolution. However, by increasing the frequency beyond a certain threshold, the attenuation will be significantly increased, making it difficult to detect the echo due to a low signal-noise ratio.
Ultrasonic pulse velocity method (ASTM C597) measures the velocity of an ultrasonic pulse passing through a concrete structure [68] . This test is conducted by sending an ultrasonic pulse through the test material and measuring the time-of-flight, the time it takes for the pulse to travel through the material. Slower velocity suggests lower elastic modulus, which may indicate concrete with cracks or voids. However, this method was found to have a low sensitivity to early damage in concrete materials [69] .
Ultrasonic pulse attenuation method (ASTM C1332) is based on measuring the energy loss of acoustic wave propagating in materials [70] . The attenuation of a damaged material is expected to be higher than that of an undamaged intact one due to the scattering of incident wave by damageinduced heterogeneities such as cracks and voids. Generally speaking, the pulse-amplitude method is more sensitive to the early damage than ultrasonic pulse velocity method. However, the testing results are highly dependent on the coupling condition between the transducers and samples.
The ASTM C215 prescribes the test setup and protocol for measuring the fundamental transverse, longitudinal, and torsional resonant frequencies of concrete prisms and cylinders for the purpose of calculating dynamic modulus of elasticity and Poisson’s ratio [71] . The fundamental resonant frequencies of test specimen can be determined by either forced or impact resonance method. Lower frequencies indicates lower modulus, an indication of low quality. Similar to the ultrasonic pulse velocity method, this method is not sensitive to early damage.
1.1.2.3 Nonlinear Acoustic Techniques
A brief introduction to common nonlinear acoustic/ultrasonic methods is given in Section 1.1.1. In this section, the focus is on the application of these techniques to the evaluation of cementitious materials. As previously discussed, using linear acoustic techniques for assessing damage in cementitious materials could be difficult due to their low sensitivity to microcracking and high attenuation. Nonlinear acoustic techniques, however, show considerable promise in micro-damage detection due to their higher sensitivity, when compared to linear methods [27] . Unlike metallic materials, where the nonlinearity originates from atomistic disorder such as dislocations, the source of nonlinearity in concrete has been identified as the weak bonding system which holds the aggregates together [72] . The nonlinearity manifests itself as among others, amplitude-dependent resonance frequencies (NRUS), higher harmonic generation, modulated frequencies (VAM), amplitude-dependent wave velocity and attenuation (DAET), and slow dynamics (DAET). Van den Abeele and De Visscher [73] were the first to apply a nonlinear acoustic technique to concrete. In their study, NRUS was used to assess static load-induced damage in a reinforced concrete beam. Afterwards, NRUS and its variants have been used for characterizing damages due to thermal loading [26] , ASR [34,35,74,75] , sulfate attack and FT cycles [76] and impact force [33] . Antonaci and Shah [11,12] characterized the distributed micro-cracking of concrete under compressive load based on higher harmonic generation. Other than bulk wave test configurations, higher harmonics generation method using Rayleigh surface waves has also been applied for detecting micro-cracks induced by autogenous and drying shrinkage [17] and quantifying microstructural changes due to carbonation [18] . The VAM technique was also shown to be able to quantitatively track the evolution of ASR in cement-based materials [77] . In recent years, DAET has also drawn substantial attention in monitoring the microstructural change in this class of materials [55–60] .
1.1.3 Theoretical Background on Nonlinear Crack–Wave Interactions
Contact acoustic nonlinearity, or CAN is a type of non-classical nonlinearity that occurs specifically at unbonded contact interfaces. This phenomenon originates from the repeated collisions between the two contact interfaces due to the passing of incident stress waves. For a pair of smooth contact interfaces, the waves can propagate through during compression phase, but not during tension phase. The transmitted waves become nearly half-wave rectified, which is an obvious sign of nonlinearity.
The most common way of simulating CAN is by modeling the contact interface as a spring with bi-linear stiffness. Solodov et al. [21] proposed a piece-wise discontinuous stress-strain relationship at the near-surface region of the interface:
σ = 01 − ( )(∆ / 0) (1.9)
where ( ) is the Heaviside unit step function, and ∆ = 0 − ( / ) >0. This model assumes an intact linear stiffness when the material is in compression and a zero? stiffness when the material is in tension. A spectrum containing a number of higher harmonics modulated by a sinc-envelope function is predicted by this model. A similar concept was also applied to describing the nonlinear vibrations of beams with a single crack [78–80] . However, the bi-linear stiffness model is over simplified since it does consider the real physical microstructure of the crack.
Real crack surfaces are usually rough, thus only part of the interface is in contact, and the asperities in contact are deformable. Therefore, when two rough surfaces under compressive loading are in contact, the load is supported by a distribution of asperities. When the load is increased, the supporting asperities are further deformed while some other asperities come newly in contact. This leads to a complex nonlinear relationship between contact pressure and separation distance of two surfaces. Considering this complexity, models based on the theory of contact mechanics are needed. Greenwood and Williamson [81] developed an analytical contact model (also known as the GW model) that considers the statistical distribution of asperity heights to predict the relationship between contact pressure and separation distance of two surfaces. GW model simplifies the rough contact interface as a nominally flat surface and a composite rough surface. The rough surface is composed of asperities with their heights randomly distributed. The peaks of asperities are assumed to be spheres of equal radii. The contact between each asperity and the flat plane follows the Hertzian contact theory [81] . The GW model has been used for studying the interaction of linear ultrasonic waves (transmission and reflection) with rough surfaces in contact [82,83] .
Recently, nonlinear ultrasonic responses of rough contacting interfaces have attracted much attention. Pecorari [84] used the framework of the GW model to take the interface roughness into account to predict nonlinearity. The nonlinear stiffness is expanded in powers of the variation in the relative interface separation up to the second order term. The level of nonlinearity based on the amplitude of second harmonic is a function of both the first and second order stiffness constants. Although this study takes the surface roughness into consideration by assuming an inverted chisquared distribution of asperity heights, the actual parameters such as the degrees of freedom for the asperity height distribution, number of asperities in contact, and the average asperity radius are not extracted from aperture profile. Instead, these key parameters are estimated by minimizing the error for the first order stiffness constant from ultrasonic measurements and that derived from the GW model. Kim et al. [85,86] studied the nonlinear acoustic properties of interfaces between solids by including the elastoplastic contact behavior of the interface, where the elastoplastic deformation of asperities is based on the finite element simulations by Kogut and Etsion [87] . Again, the key parameters related to the interface roughness are not extracted from the real aperture profile. Biwa et al. [88,89] took a fundamentally different approach. They investigated this problem directly from a macroscopic point of view by introducing a simple phenomenological power-law pressuredependent function to model the first and second order stiffness constants. This model shows a good qualitative agreement with previous experimental results in the form of the second harmonic amplitude measured for contact interfaces of two separate blocks. Although the parameters in the power-law pressure-dependent function are dependent on the surface roughness, their physical meanings are not known.
1.2 Thesis Objective and Outline
The objective of this dissertation is to quantify the relationship between micro-damage and nonlinearity parameters. The thesis answers two research questions: 1) how does the nonlinearity evolve in concrete with volumetric microscopic damage? and 2) how microstructural features of contact interface influence CAN?
To answer the first question, two studies on concrete samples subjected to two types of damage – freeze-thaw (FT) and alkali silica reaction (ASR) – are conducted. In both studies, we demonstrate the use of single impact-based nonlinear resonance acoustic spectroscopy (SINRAS) for monitoring the microstructural changes within concrete samples. We develop a new processing technique to obtain the nonlinearity parameters from the instantaneous resonance frequency variation during the ring-down of the vibrational response to a strong impact. Moreover, we demonstrate how to extract both classical and non-classical nonlinear parameters from the impactversion of DAET (IDAET).
The answer to the second question is investigated using an integrated analytical and experimental approach. We have prepared four aluminum alloy samples (30×40×170mm3), each having a single fatigue crack. The cracks on the samples are of similar length but have dissimilar microstructural features due to the different stress intensity factors used during the fatigue tests. The geometric features of the cracks, including crack width and roughness, are extracted from a series of scanning electron microscopy (SEM) images. These microstructural features are used as independent variables in a regression analysis to quantify their relationship to nonlinear acoustic response. In addition, the aperture profile is used as an input to an analytical model based on the contact mechanics theory (GW model) considering the topography of the contact interface. The calculated stiffness-separation relation at the contact interfaces can be used to predict acoustic nonlinearity by a displacement discontinuity model. Theoretical predictions are validated by dynamic acousto-elastic testing (DAET) with a surface wave probe.
This dissertation is organized as follows:
Chapter 2 (published) and Chapter 3 (revisions under review) investigate the effect of distributed micro-cracking on the materials nonlinearity. In Chapter 2, a set of concrete samples are subjected to an increasing number of FT cycles; while in Chapter 3, two concrete prisms undergo accelerated ASR. We propose a simple impact-based nonlinear testing procedure which has an identical test setup to that prescribed by ASTM C215. A novel data analysis procedure is developed that simultaneously yields wo nonlinearity parameters in addition to the resonance frequency of the sample.
In Chapter 4 (published), we make a qualitative comparison of CAN signatures obtained for two aluminum alloy samples, each with a single fatigue crack. The cracks on the two samples are of similar length but have dissimilar microstructural features. The micro-geometrical features of each crack are obtained using SEM imaging. The near-surface nonlinearity is measured using an unconventional DAET setup. The use of DAET with a Rayleigh wave probe makes it possible to analyze the nonlinearity signatures in relation to the crack microstructural features.
In Chapter 5, we make a quantitative connection between the nonlinearity parameters and microstructural features of crack interfaces through a regression analysis. The physical interpretation of the regression models is provided.
Chapter 6 utilizes an analytical approach towards understanding the relation between crack aperture profiles and CAN. The model parameters are extracted directly from the measured aperture profiles. The predicted nonlinearity is compared to the DAET experimental results.
Chapter 7 summarizes the findings and presents the future direction of this work.