Analytical and Experimental Study of Fatigue-Crack-Growth AE Signals in Thin Sheet Metals

The acoustic emission (AE) method is a very popular and well-developed method for passive structural health monitoring of metallic and composite structures. AE method has been efficiently used for damage source detection and damage characterization in a large variety of structures over the years, such as thin sheet metals. Piezoelectric wafer active sensors (PWASs) are lightweight and inexpensive transducers, which recently drew the attention of the AE research community for AE sensing. The focus of this paper is on understanding the fatigue crack growth AE signals in thin sheet metals recorded using PWAS sensors on the basis of the Lamb wave theory and using this understanding for predictive modeling of AE signals. After a brief introduction, the paper discusses the principles of sensing acoustic signals by using PWAS. The derivation of a closed-form expression for PWAS response due to a stress wave is presented. The transformations happening to the AE signal according to the instrumentations we used for the fatigue crack AE experiment is also discussed. It is followed by a summary of the in situ AE experiments performed for recording fatigue crack growth AE and the results. Then, we present an analytical model of fatigue crack growth AE and a comparison with experimental results. The fatigue crack growth AE source was modeled analytically using the dipole moment concept. By using the source modeling concept, the analytical predictive modeling and simulation of the AE were performed using normal mode expansion (NME). The simulation results showed good agreement with experimental results. A strong presence of nondispersive S0 Lamb wave mode due to the fatigue crack growth event was observed in the simulation and experiment. Finally, the analytical method was verified using the finite element method. The paper ends with a summary and conclusions; suggestions for further work are also presented.


Introduction
Engineering structures are prone to various types of failures during their operation. The failure mechanism during their operation depends on the mechanism of the external loading, working conditions, the microstructural changes inside the structure, etc. Various kinds of damages in metallic structures include fatigue, friction, static failure, etc. It is important to detect the damage initiation in its early stage to prevent catastrophic failure. Structural health monitoring (SHM) is an emerging technology with multiple applications in the evaluation of critical structures. Acoustic emission (AE) is a passive SHM method having a lot of potential for early-stage detection of damages and prevention of the formation of the complete failure of the structure. In the AE method, there are several approaches for damage detection, of which physics-based approaches are receiving a lot of AE instrumentation. During the transmission from the source to the digitization at the data acquisition (DAQ), the wave passes through a few pieces of equipment. The acoustic signals are modified when they pass through the instrumentation according to their specifications. These transformations happening to the acoustic signals are discussed in this section.

AE Signal Flow Diagram
The AE signal flow diagram from the initiation at the source to the digitization at the DAQ is presented in Figure 1. The sudden dynamic disturbance at the AE source location causes the generation of AE waves. In thin metallic plates, the AE waves travel as Lamb waves. The AE waves propagate in the plate and then reach the PWAS transducer, resulting in a strain field. The strain field at the transducer causes it to generate a response in the form of voltage. The voltage sensed by the transducer is then magnified when it reaches the pre-amplifier. Lastly, the amplified signal is recorded by the DAQ as a digitized signal.

AE Signal Flow Diagram
The AE signal flow diagram from the initiation at the source to the digitization at the DAQ is presented in Figure 1. The sudden dynamic disturbance at the AE source location causes the generation of AE waves. In thin metallic plates, the AE waves travel as Lamb waves. The AE waves propagate in the plate and then reach the PWAS transducer, resulting in a strain field. The strain field at the transducer causes it to generate a response in the form of voltage. The voltage sensed by the transducer is then magnified when it reaches the pre-amplifier. Lastly, the amplified signal is recorded by the DAQ as a digitized signal. The block diagram of the signal transformations happening during the signal acquisition process is explained in Figure 2. When the AE signal passes through each element of the AE signal acquisition system, a transformation according to the frequency response of the element happens. Due to an AE source such as a fatigue crack growth in thin plates, the generation of AE waves happens according to the structural transfer function of the source on the plate. In thin-walled plates, the AE waves travel as Lamb waves. When the AE waves reach the PWAS sensor, the Lamb waves are modified by the PWAS transfer function or PWAS tuning, which depends on the Lamb wave mode. A signal amplifier was used between the DAQ and the PWAS. The signal amplifier used in the present experiment was a MISTRAS 2/4/6 pre-amplifier, which is a bandpass filter from 30 to 700 kHz. The DAQ system used has a bandwidth of 0-2 MHz. When the signal passes through each element, a modification happens according to the transfer function/bandwidth of the element.

PWAS Transfer Function
In Section 2.1 we have discussed the transformations happening in the AE signal during each stage in the signal flow path. The signal amplifier and DAQ have a specific bandwidth to transfer the The block diagram of the signal transformations happening during the signal acquisition process is explained in Figure 2. When the AE signal passes through each element of the AE signal acquisition system, a transformation according to the frequency response of the element happens. Due to an AE source such as a fatigue crack growth in thin plates, the generation of AE waves happens according to the structural transfer function of the source on the plate. In thin-walled plates, the AE waves travel as Lamb waves. When the AE waves reach the PWAS sensor, the Lamb waves are modified by the PWAS transfer function or PWAS tuning, which depends on the Lamb wave mode. A signal amplifier was used between the DAQ and the PWAS. The signal amplifier used in the present experiment was a MISTRAS 2/4/6 pre-amplifier, which is a bandpass filter from 30 to 700 kHz. The DAQ system used has a bandwidth of 0-2 MHz. When the signal passes through each element, a modification happens according to the transfer function/bandwidth of the element.

AE Signal Flow Diagram
The AE signal flow diagram from the initiation at the source to the digitization at the DAQ is presented in Figure 1. The sudden dynamic disturbance at the AE source location causes the generation of AE waves. In thin metallic plates, the AE waves travel as Lamb waves. The AE waves propagate in the plate and then reach the PWAS transducer, resulting in a strain field. The strain field at the transducer causes it to generate a response in the form of voltage. The voltage sensed by the transducer is then magnified when it reaches the pre-amplifier. Lastly, the amplified signal is recorded by the DAQ as a digitized signal. The block diagram of the signal transformations happening during the signal acquisition process is explained in Figure 2. When the AE signal passes through each element of the AE signal acquisition system, a transformation according to the frequency response of the element happens. Due to an AE source such as a fatigue crack growth in thin plates, the generation of AE waves happens according to the structural transfer function of the source on the plate. In thin-walled plates, the AE waves travel as Lamb waves. When the AE waves reach the PWAS sensor, the Lamb waves are modified by the PWAS transfer function or PWAS tuning, which depends on the Lamb wave mode. A signal amplifier was used between the DAQ and the PWAS. The signal amplifier used in the present experiment was a MISTRAS 2/4/6 pre-amplifier, which is a bandpass filter from 30 to 700 kHz. The DAQ system used has a bandwidth of 0-2 MHz. When the signal passes through each element, a modification happens according to the transfer function/bandwidth of the element.

PWAS Transfer Function
In Section 2.1 we have discussed the transformations happening in the AE signal during each stage in the signal flow path. The signal amplifier and DAQ have a specific bandwidth to transfer the

PWAS Transfer Function
In Section 2.1 we have discussed the transformations happening in the AE signal during each stage in the signal flow path. The signal amplifier and DAQ have a specific bandwidth to transfer the signal. The signal reception by PWAS depends on the tuning of the PWAS transducer on the metallic plate. The tuning of the Lamb waves by the PWAS depends on the Lamb wave mode. A closed-form expression for PWAS sensing of Lamb waves is derived in this section. The PWAS senses the in-plane strain caused by the AE waveform and converts it into an equivalent voltage. A schematic of the sensing of Lamb waves by a PWAS sensor is presented in Figure 3. plate. The tuning of the Lamb waves by the PWAS depends on the Lamb wave mode. A closed-form expression for PWAS sensing of Lamb waves is derived in this section. The PWAS senses the in-plane strain caused by the AE waveform and converts it into an equivalent voltage. A schematic of the sensing of Lamb waves by a PWAS sensor is presented in Figure 3. The interaction between electrical and mechanical variables can be described by linear relations for linear piezoelectric materials. The tensorial form of constitutive relation between electrical and mechanical variables is established as follows: where Ei is the electric field, gikl is the piezoelectric voltage coefficient, σikl is the stress on the PWAS, β T ik is the impermittivity coefficient, Dk is the electric displacement, Sij is the strain on the PWAS, and s E ijkl is the compliance. Considering the one-dimensional (1D) PWAS sensing assumption, we get Eliminating stress between equations, we get Upon rearranging, we get 2 31 1 3 11 31 11 33 3 ( ) where 3 e C V The interaction between electrical and mechanical variables can be described by linear relations for linear piezoelectric materials. The tensorial form of constitutive relation between electrical and mechanical variables is established as follows: where E i is the electric field, g ikl is the piezoelectric voltage coefficient, σ ikl is the stress on the PWAS, β T ik is the impermittivity coefficient, D k is the electric displacement, S ij is the strain on the PWAS, and s E ijkl is the compliance. Considering the one-dimensional (1D) PWAS sensing assumption, we get Eliminating stress between equations, we get Sensors 2020, 20, 5835 6 of 25 Upon rearranging, we get where where C e is the external capacitance, V 3 is the voltage, A is the area of the PWAS, and t a is the thickness of the PWAS. For infinitesimal area dA and voltage dV, substituting Equations (7) and (8) into Equation (6), we get Upon rearranging, we get where dA β T 33 t a = C c , the capacitance of PWAS.
Substituting C c into Equation (10), we get Upon rearranging, we get The external capacitance is usually very small compared to the capacitance of the PWAS, i.e., C e << C c . Considering a very small value of C e , we get This voltage is due to the strain at an infinitesimal area dA. The total voltage sensed by the PWAS can be obtained by integrating the voltage over the area S c . Thus, we get the total voltage sensed by PWAS as follows: Sensors 2020, 20, 5835 7 of 25

Specimen Preparation
For capturing AE during fatigue crack growth in thin metallic plates, an experimental specimen was designed. A commonly used aircraft material, aluminum 2024-T3, was chosen for preparing the test specimens. A shear metal cutting machine was used for preparing coupons of 103 mm width, 305 mm length, and 1 mm thickness from a large plate of aluminum 2024-T3. Such a wide specimen was designed for allowing a long crack formation in the specimen. First, a 1 mm hole was drilled at the geometric center of the specimen. Then, fatigue loading in an MTS machine was applied to the specimen for generating a pre-crack of 4 mm tip-to-tip length. The maximum and minimum loads applied for the cyclic fatigue loading were 13.85 kN and 1.38 kN. Then, the specimen was taken out of the MTS machine, and the PWAS sensor was installed at a distance of 25 mm from the pre-crack. Then, a nonreflective clay boundary (NRB) was implemented on the specimen to prevent the reflection of the AE signals from the boundaries. PWAS sensors were bonded using M-Bond AE-15 adhesive, which is resilient to the disbonding of PWAS during extended durations of cyclic loading. M-Bond AE-15 is an epoxy system with two components: a resin and a curing agent. The resin and curing agent were mixed in a specific proportion and stirred for 5 min. This mixture was used to bond the PWAS to the specimen. The PWAS was bonded to the specimen using the mixture and cured for 3 h at 60 • C as recommended by the epoxy manufacturer. The capacitance of the PWAS was measured before bonding and after bonding. The capacitance values made sure that the PWAS, as well as the bonding, was defect-free. The NRB was applied to the specimen to prevent AE signal reflections from the plate boundaries and, thus, to receive reflection-free, clean AE signals. After the AE sensor and NRB implementation on the specimen (Figure 4), the cyclic fatigue loading was continued. The crack was grown an additional 5.4 mm (until the crack length reached 9.4 mm tip to tip), simultaneously capturing the AE signals. The wide geometry of the specimen was desired for this work so that the acoustic waves generated would travel a longer distance to the edges. This hypothesis, in turn, means the signals die out due to geometric spreading and material damping before reaching the sensor after the reflection from the boundaries. The properties of the Al 2024-T3 specimen were as follows: modulus of elasticity 73 GPa, density 2767 kg/m 3 , and Poisson's ratio 0.33. test specimens. A shear metal cutting machine was used for preparing coupons of 103 mm width, 305 mm length, and 1 mm thickness from a large plate of aluminum 2024-T3. Such a wide specimen was designed for allowing a long crack formation in the specimen. First, a 1 mm hole was drilled at the geometric center of the specimen. Then, fatigue loading in an MTS machine was applied to the specimen for generating a pre-crack of 4 mm tip-to-tip length. The maximum and minimum loads applied for the cyclic fatigue loading were 13.85 kN and 1.38 kN. Then, the specimen was taken out of the MTS machine, and the PWAS sensor was installed at a distance of 25 mm from the pre-crack. Then, a nonreflective clay boundary (NRB) was implemented on the specimen to prevent the reflection of the AE signals from the boundaries. PWAS sensors were bonded using M-Bond AE-15 adhesive, which is resilient to the disbonding of PWAS during extended durations of cyclic loading. M-Bond AE-15 is an epoxy system with two components: a resin and a curing agent. The resin and curing agent were mixed in a specific proportion and stirred for 5 min. This mixture was used to bond the PWAS to the specimen. The PWAS was bonded to the specimen using the mixture and cured for 3 h at 60 °C as recommended by the epoxy manufacturer. The capacitance of the PWAS was measured before bonding and after bonding. The capacitance values made sure that the PWAS, as well as the bonding, was defect-free. The NRB was applied to the specimen to prevent AE signal reflections from the plate boundaries and, thus, to receive reflection-free, clean AE signals. After the AE sensor and NRB implementation on the specimen (Figure 4), the cyclic fatigue loading was continued. The crack was grown an additional 5.4 mm (until the crack length reached 9.4 mm tip to tip), simultaneously capturing the AE signals. The wide geometry of the specimen was desired for this work so that the acoustic waves generated would travel a longer distance to the edges. This hypothesis, in turn, means the signals die out due to geometric spreading and material damping before reaching the sensor after the reflection from the boundaries. The properties of the Al 2024-T3 specimen were as follows: modulus of elasticity 73 GPa, density 2767 kg/m 3 , and Poisson's ratio 0.33.

AE Experimental Set-Up
After installing the sensor and NRB, the fatigue loading was continued to grow the crack and capture AE signals simultaneously. The experimental set-up for capturing the AE signal from a fatigue crack growth event is presented in Figure 5. The test specimen installed with the PWAS transducer was mounted on the MTS machine. The bond quality assurance of PWAS sensors was performed periodically by electromechanical impedance spectroscopy (EMIS). The acoustic pre-amplifier connected to the PWAS sensor is a bandpass filter, with a filtering range between 30 kHz to 700 kHz. The pre-amplifier operates with either a single-ended or differential sensor provided with 20/40/60 dB gain (can be selected using a switch). A 40 dB gain was selected in the present experiment. The pre-amplifier was connected to the MISTRAS AE system. For capturing any high-frequency AE signals, a sampling frequency of 10 MHz was chosen. The timing parameters set for the MISTRAS system were as follows: peak definition time (PDT)= 200 µs, hit definition time (HDT) = 800 µs, and hit lockout time (HLT) = 1000 µs.

AE Experimental Set-Up
After installing the sensor and NRB, the fatigue loading was continued to grow the crack and capture AE signals simultaneously. The experimental set-up for capturing the AE signal from a fatigue crack growth event is presented in Figure 5. The test specimen installed with the PWAS transducer was mounted on the MTS machine. The bond quality assurance of PWAS sensors was performed periodically by electromechanical impedance spectroscopy (EMIS). The acoustic preamplifier connected to the PWAS sensor is a bandpass filter, with a filtering range between 30 kHz to 700 kHz. The pre-amplifier operates with either a single-ended or differential sensor provided with 20/40/60 dB gain (can be selected using a switch). A 40 dB gain was selected in the present experiment. The pre-amplifier was connected to the MISTRAS AE system. For capturing any high-frequency AE signals, a sampling frequency of 10 MHz was chosen. The timing parameters set for the MISTRAS system were as follows: peak definition time (PDT)= 200 µs, hit definition time (HDT) = 800 µs, and hit lockout time (HLT) = 1000 µs.

Experimental Results
In this experiment, the AE signals during fatigue crack growth events recorded using PWAS were identified. Two examples of the AE signals are presented in Figure 6a,b. The frequency spectrum of the signals is also presented in Figure 6c,d. As we observe, the time-domain signal has a sharp rise in the beginning. The frequency spectrum of the signal also has specific peaks and valleys in both signals, which are comparable to each other. A detailed discussion of the results of the experiment is presented in Joseph and Giurgiutiu [58].

Experimental Results
In this experiment, the AE signals during fatigue crack growth events recorded using PWAS were identified. Two examples of the AE signals are presented in Figure 6a,b. The frequency spectrum of the signals is also presented in Figure 6c,d. As we observe, the time-domain signal has a sharp rise in the beginning. The frequency spectrum of the signal also has specific peaks and valleys in both signals, which are comparable to each other. A detailed discussion of the results of the experiment is presented in Joseph and Giurgiutiu [58].
The group velocity dispersion curve of a 1 mm aluminum plate is presented in Figure 7. The S0 and A0 Lamb wave modes travel at different velocities with the S0 Lamb wave mode at a higher velocity than the A0 mode. The group velocity dispersion curve of 1 mm aluminum is superimposed in the time-frequency plot of the AE signal in Figure 6e,f. From the time domain, as well as the time-frequency domain representation superimposed with the group velocity dispersion curve of the AE signal, we observe the strong presence of S0 mode in the experimentally observed AE signal. This observation proves that the AE signals from fatigue crack events travel as Lamb waves. The group velocity dispersion curve of a 1 mm aluminum plate is presented in Figure 7. The S0 and A0 Lamb wave modes travel at different velocities with the S0 Lamb wave mode at a higher velocity than the A0 mode. The group velocity dispersion curve of 1 mm aluminum is superimposed in the time-frequency plot of the AE signal in Figure 6e,f. From the time domain, as well as the timefrequency domain representation superimposed with the group velocity dispersion curve of the AE signal, we observe the strong presence of S0 mode in the experimentally observed AE signal. This observation proves that the AE signals from fatigue crack events travel as Lamb waves.

Predictive Modeling of Fatigue Crack Growth AE
In this section, the analytical predictive modeling of AE signals due to the fatigue crack growth

Predictive Modeling of Fatigue Crack Growth AE
In this section, the analytical predictive modeling of AE signals due to the fatigue crack growth process is presented. The schematic of the AE source modeling approach for a fatigue crack growth event is presented in Figure 8. Figure 8a shows the fatigue specimen with a pre-crack and the fatigue loading applied to the specimen. A schematic of the zoom-in of the fatigue crack tip is presented in Figure 8b. During the fatigue crack growth, the crack propagation was observed to happen in the z-direction (creating a fracture surface in y-z plane). Therefore, the normal to the fracture surface is the x-direction. The fatigue loading applied during the experiment was also in the x-direction, as we observe from Figure 8a. A fracture in which the loading and normal to the fracture plane are in the same direction is called a Mode-I fracture. Thus, a fatigue crack growth event in the present experimental set-up is considered as a Mode-I fracture process from the definitions of fracture mechanics.

Predictive Modeling of Fatigue Crack Growth AE
In this section, the analytical predictive modeling of AE signals due to the fatigue crack growth process is presented. The schematic of the AE source modeling approach for a fatigue crack growth event is presented in Figure 8. Figure 8a shows the fatigue specimen with a pre-crack and the fatigue loading applied to the specimen. A schematic of the zoom-in of the fatigue crack tip is presented in Figure 8b. During the fatigue crack growth, the crack propagation was observed to happen in the zdirection (creating a fracture surface in y-z plane). Therefore, the normal to the fracture surface is the x-direction. The fatigue loading applied during the experiment was also in the x-direction, as we observe from Figure 8a. A fracture in which the loading and normal to the fracture plane are in the same direction is called a Mode-I fracture. Thus, a fatigue crack growth event in the present experimental set-up is considered as a Mode-I fracture process from the definitions of fracture mechanics.  The edge view of the fatigue fracture process is presented in Figure 8c. Assuming a 1D waveguide, and considering that the Lamb wave is decoupled from the shear horizontal waves, one can represent the effective excitation due to such a Mode-I fracture as an M 11 dipole moment excitation, as we observe in Figure 8c. Therefore, the M 11 dipole moment component excitation represents a Mode-I fracture process in a 1D waveguide and the equivalent of the excitation happening during fatigue crack growth event. Next, for the predictive modeling of the AE, using a closed-form solution of the wavefield due to the dipole moment, excitation needs to be derived. The wavefield due to a concentrated moment excitation is derived from the wavefield due to a concentrated force following the approach mentioned by Aki and Richards [5]. The method is discussed in short below.
Suppose a couple moment is generated due to a force vector Q applied at the position ξ(α, β) and another force vector Q applied at ξ (α, β + ∆X j ) in the opposite direction to that of the force applied at ξ (α, β + ∆X j ), where ∆X j is a small distance in the X j direction as shown in Figure 9. excitation is derived from the wavefield due to a concentrated force following the approach mentioned by Aki and Richards [5]. The method is discussed in short below.
Suppose a couple moment is generated due to a force vector Q applied at the position ( , ) ξ α β and another force vector Q applied at '( , ) j X ξ α β + Δ in the opposite direction to that of the force applied at '( , ) Δ is a small distance in the j X direction as shown in Figure 9. The displacement field at an arbitrary point x due to a point force Q at an arbitrary point ( , ) ξ α β and Q at '( , ) j X ξ α β + Δ , as shown in Figure 9, is denoted as i u ξ and ' i u ξ , respectively. If the material is linearly elastic and the displacement field has a linear relation with force applied, we can write where G is the Green function for a point force. The displacement field at x due to a couple with the moment ij i j M Q X = Δ can be written from Equations (16) and (17) as follows: For an infinitesimally small distance j X Δ , we apply the limits to get Using the first-order differential formula, Equation (20) can be written as Figure 9. Couple force applied for generating the moment. As we limit the separation of the forces to zero, the couple becomes a concentrated moment at ξ(α, β).
The displacement field at an arbitrary point x due to a point force Q at an arbitrary point ξ(α, β) and Q at ξ (α, β + ∆X j ), as shown in Figure 9, is denoted as u ξ i and u ξ i , respectively. If the material is linearly elastic and the displacement field has a linear relation with force applied, we can write where G is the Green function for a point force. The displacement field at x due to a couple with the moment M ij = Q i ∆X j can be written from Equations (16) and (17) as follows: For an infinitesimally small distance ∆X j , we apply the limits to get Using the first-order differential formula, Equation (20) can be written as Thus, the displacement field for a moment tensor component with forces acting in the i direction and separated in the j direction is given as

Wavefield Due to M 11 Dipole Excitation
A through-thickness M 11 dipole moment excitation was considered, as presented in Figure 10 for representing the dipole moment excitation due to the present Mode-I fracture case. Therefore, for predictive modeling of the Mode-I fracture case, a closed-form solution for M 11 dipole moment excitation response at the PWAS is required, which is derived in this section.
Thus, the displacement field for a moment tensor component with forces acting in the i direction and separated in the j direction is given as

Wavefield Due to M11 Dipole Excitation
A through-thickness M11 dipole moment excitation was considered, as presented in Figure 10 for representing the dipole moment excitation due to the present Mode-I fracture case. Therefore, for predictive modeling of the Mode-I fracture case, a closed-form solution for M11 dipole moment excitation response at the PWAS is required, which is derived in this section. First, the velocity field due to a through-thickness force excitation was derived. Then, the velocity field due to a through-thickness dipole M11 dipole moment excitation was derived using the principle in Equation (22). Next, the strain field equation was derived from the velocity field equation. Finally, from the strain field equation, a closed-form solution for PWAS voltage response was derived using Equation (15).

In-Plane Line Force Excitation-Normal Mode Expansion (NME)
The wavefield, due to the dipole moment excitation, is derived from the wavefield due to the force excitation. For obtaining the through-thickness M11 moment tensor excitation field, a throughthickness force excitation, as presented in Figure 11, is considered. The limiting process of the through-thickness couple forces separated by an infinitesimal distance converts the force to a moment and the wavefield due to the force to a wavefield due to a moment, as discussed in Figure 9. The separation of the forces required for obtaining the moment M11 in Figure 10 from the force in Figure  11 is in the x-direction. The normal mode expansion (NME) was used for obtaining the solution for the elastodynamic field due to the through-thickness force excitation in a thin plate, as presented in Figure 11.  First, the velocity field due to a through-thickness force excitation was derived. Then, the velocity field due to a through-thickness dipole M 11 dipole moment excitation was derived using the principle in Equation (22). Next, the strain field equation was derived from the velocity field equation. Finally, from the strain field equation, a closed-form solution for PWAS voltage response was derived using Equation (15).

In-Plane Line Force Excitation-Normal Mode Expansion (NME)
The wavefield, due to the dipole moment excitation, is derived from the wavefield due to the force excitation. For obtaining the through-thickness M 11 moment tensor excitation field, a through-thickness force excitation, as presented in Figure 11, is considered. The limiting process of the through-thickness couple forces separated by an infinitesimal distance converts the force to a moment and the wavefield due to the force to a wavefield due to a moment, as discussed in Figure 9. The separation of the forces required for obtaining the moment M 11 in Figure 10 from the force in Figure 11 is in the x-direction. The normal mode expansion (NME) was used for obtaining the solution for the elastodynamic field due to the through-thickness force excitation in a thin plate, as presented in Figure 11.
Thus, the displacement field for a moment tensor component with forces acting in the i direction and separated in the j direction is given as

Wavefield Due to M11 Dipole Excitation
A through-thickness M11 dipole moment excitation was considered, as presented in Figure 10 for representing the dipole moment excitation due to the present Mode-I fracture case. Therefore, for predictive modeling of the Mode-I fracture case, a closed-form solution for M11 dipole moment excitation response at the PWAS is required, which is derived in this section. First, the velocity field due to a through-thickness force excitation was derived. Then, the velocity field due to a through-thickness dipole M11 dipole moment excitation was derived using the principle in Equation (22). Next, the strain field equation was derived from the velocity field equation. Finally, from the strain field equation, a closed-form solution for PWAS voltage response was derived using Equation (15).

In-Plane Line Force Excitation-Normal Mode Expansion (NME)
The wavefield, due to the dipole moment excitation, is derived from the wavefield due to the force excitation. For obtaining the through-thickness M11 moment tensor excitation field, a throughthickness force excitation, as presented in Figure 11, is considered. The limiting process of the through-thickness couple forces separated by an infinitesimal distance converts the force to a moment and the wavefield due to the force to a wavefield due to a moment, as discussed in Figure 9. The separation of the forces required for obtaining the moment M11 in Figure 10 from the force in Figure  11 is in the x-direction. The normal mode expansion (NME) was used for obtaining the solution for the elastodynamic field due to the through-thickness force excitation in a thin plate, as presented in Figure 11.  We let the elastodynamic state 1 represent the field due to a body force excitation, F 1 , acting on the control volume. Hence, we represent state 1 as the sum of normal modes.
where a m (x) is the modal participation factor or normal mode expansion coefficient, which corresponds to the excitation. This coefficient needs to be determined. The summation contains all wave modes that significantly contribute to the total wave field. The notations v and T represents the velocity and stresses of the wave fields. Since v 1 and T 1 represent a single-mode component, the same set of amplitude coefficients is used for particle velocity and stress field. The normal mode expansion coefficient a m is a function of x. The wave mode amplitude consolidation for different modes changes with the nature of the excitation source. v m (y) is the velocity modeshape, and T m (y) is the stress modeshape of the normal mode. Elastodynamic state 2 is assumed as the wave field of mode n, expressed as follows: where v n (y) is the velocity modeshape, T n (y) is the stress modeshape, and ξ n is the wavenumber of the Lamb wave mode. For two elastodynamic states of the plate, the complex reciprocity relationship can be given as where~represents the complex conjugate. Expanding the "del" operator, substituting Equations (23) and (24) into Equation (25), and simplifying, we get Upon further simplifying, we get 4P nn ∂ ∂x e iξ n x · a n (x) = e iξ n x +d −d ( v n (y) · F 1 )dy, (27) where P mn is the orthogonality relation in the general form represented as follows: Due to the orthogonality condition of the Lamb wave modes, P mn = 0 [59] for m n, in the case of a propagating Lamb wave mode. Thus, only P nn exists in Equation (27).
Considering the Lamb wave propagating in a 1D plate, the lamb velocity vector is given as In Equation (25), F 1 is the body force excitation. For a through-thickness dynamic force excitation acting as shown in Figure 11, F 1 can be expressed as follows: Simplifying Equation (27) using Equations (29) and (30), assuming the location of fatigue crack source as the origin and waveform measurement location as x, the closed-form expression for normal mode expansion coefficient is given as Substituting the normal mode expansion coefficient into Equation (23), we get the expression for the velocity field due to a through-thickness dynamic force excitation in the x-direction (as shown in Figure 11) as follows: For unit force excitation, the equation becomes Line Moment (M 11 ) Excitation Field Two force excitations separated by an infinitesimal distance and the limiting process of that distance that goes to zero create a dipole moment excitation, as shown in Figure 12a. In the figure, PWAS is located at a distance of r c from the source, and s r is the radius of the PWAS. The limiting process of the separation turns the excitation to a moment excitation. The through-thickness M 11 dipole moment excitation is presented in Figure 12b. In Equations (16)- (22), it was shown that the wavefield due to through-thickness dipole moment excitation can be obtained from the displacement field due to through-thickness force excitation from the limiting process, explained as follows: Sensors 2020, 20, x FOR PEER REVIEW 14 of 25 Simplifying Equation (27) using Equations (29) and (30), assuming the location of fatigue crack source as the origin and waveform measurement location as x, the closed-form expression for normal mode expansion coefficient is given as Substituting the normal mode expansion coefficient into Equation (23), we get the expression for the velocity field due to a through-thickness dynamic force excitation in the x-direction (as shown in Figure 11) as follows: For unit force excitation, the equation becomes Line Moment (M11) Excitation Field Two force excitations separated by an infinitesimal distance and the limiting process of that distance that goes to zero create a dipole moment excitation, as shown in Figure 12a. In the figure, PWAS is located at a distance of rc from the source, and sr is the radius of the PWAS. The limiting process of the separation turns the excitation to a moment excitation. The through-thickness M11 dipole moment excitation is presented in Figure 12b. In Equations (16)- (22), it was shown that the wavefield due to through-thickness dipole moment excitation can be obtained from the displacement field due to through-thickness force excitation from the limiting process, explained as follows: (a) (b) Figure 12. The through-thickness M11 moment excitation. Figure 12. The through-thickness M 11 moment excitation.
Here, the G is the velocity field due to the force excitation presented in Equation (33). Following the limiting process in Equation (34), the velocity field due to M 11 line source is obtained from Equation (33) as follows: The displacement field can be obtained from the velocity field as follows: where ω is the angular frequency measured in radians per second. The in-plane strain is the spatial derivative of the displacement field as follows: The strain field is sensed by the PWAS and converted to the equivalent voltage by the PWAS. Following Equation (15), the total voltage sensed by PWAS can be obtained from the integral as follows: where S c represents the limits of integration over the PWAS. Substituting Equation (37) into Equation (38), we get Rearranging the integral term, we get Suppose the PWAS is located at a distance of r c from the source. Additionally, let s r be the radius of the PWAS. Then, the limit of integration S c is from r c to 2s r . Substituting the limits of integration, we get Performing the integration and substituting the limits, we get Rearranging the terms, we get a closed-form expression for the voltage sensed by the PWAS as follows:

Mode-1 Fracture AE Simulation
The dipole moment source definition for Mode-I fatigue crack growth was used for simulation of fatigue crack growth AE. For the time domain of the dipole source excitation, a commonly used wide-band cosine bell function excitation from the literature [14] was assumed. The mathematical representation for the function is given as follows: The flow chart for predictive simulation of AE signal propagation is presented in Figure 13. First, the Fourier transform of the cosine bell excitation function, E(t), was performed. Then, the structural transfer function G of the Mode-I fracture formation was calculated. A Mode-I fracture has an equivalent AE source excitation of the M 11 dipole moment. Thus, the structural transfer function G is the closed-form solution derived in Equation (43) for a unit excitation. Then, the structural transfer function for a Mode-I fracture was multiplied with the cosine bell excitation function in the frequency domain to obtain the frequency domain of the signal at the receiver. Finally, the inverse Fourier transform of the frequency domain of the signal at the receiver was performed to obtain the AE signal prediction at the receiver PWAS.
The variables in the numerical simulation were taken from the experiment. The dimension of the PWAS used during the experiment was 7 mm. For the simulation, the PWAS dimension, 2 s r , was also taken as 7 mm. The location of the PWAS r c was taken as 25 mm. The material properties corresponding to the aluminum 2024-T3 specimen were considered (70 GPa Young's modulus, 0.33 Poisson's ratio, and 2780 kg/m 3 ). The rise time parameter of the equation was adjusted to obtain the best similarity between the experiment and simulation. The frequency content of the excitation function varies according to the rise time, which is assumed. This, in fact, varies the frequency content of the resulting predictive simulation. The rise time was adjusted to have a good similarity between the experimental observation and simulation. The cosine bell function excitation was adjusted to a rise time of s, as shown in Figure 14, to match the simulation closely with the experimental observation.
The wavenumber plot for S0 and A0 Lamb wave modes for 1 mm aluminum 2024-T3 is presented in Figure 15. By using the expression in Equation (43), the normalized PWAS tuning curve for M 11 moment excitation was plotted for the S0 and A0 Lamb wave mode, as shown in Figure 16. For an M 11 excitation, a strong S0 mode content was observed, and no A0 mode response was excited. equivalent AE source excitation of the M11 dipole moment. Thus, the structural transfer function G is the closed-form solution derived in Equation (43) for a unit excitation. Then, the structural transfer function for a Mode-I fracture was multiplied with the cosine bell excitation function in the frequency domain to obtain the frequency domain of the signal at the receiver. Finally, the inverse Fourier transform of the frequency domain of the signal at the receiver was performed to obtain the AE signal prediction at the receiver PWAS.  The variables in the numerical simulation were taken from the experiment. The dimension of the PWAS used during the experiment was 7 mm. For the simulation, the PWAS dimension, 2 sr, was also taken as 7 mm. The location of the PWAS rc was taken as 25 mm. The material properties corresponding to the aluminum 2024-T3 specimen were considered (70 GPa Young's modulus, 0.33 Poisson's ratio, and 2780 kg/m 3 ). The rise time parameter of the equation was adjusted to obtain the best similarity between the experiment and simulation. The frequency content of the excitation function varies according to the rise time, which is assumed. This, in fact, varies the frequency content of the resulting predictive simulation. The rise time was adjusted to have a good similarity between the experimental observation and simulation. The cosine bell function excitation was adjusted to a rise time of s, as shown in Figure 14, to match the simulation closely with the experimental observation. The wavenumber plot for S0 and A0 Lamb wave modes for 1 mm aluminum 2024-T3 is presented in Figure 15. By using the expression in Equation (43), the normalized PWAS tuning curve for M11 moment excitation was plotted for the S0 and A0 Lamb wave mode, as shown in Figure 16. For an M11 excitation, a strong S0 mode content was observed, and no A0 mode response was excited.  The variables in the numerical simulation were taken from the experiment. The dimension of the PWAS used during the experiment was 7 mm. For the simulation, the PWAS dimension, 2 sr, was also taken as 7 mm. The location of the PWAS rc was taken as 25 mm. The material properties corresponding to the aluminum 2024-T3 specimen were considered (70 GPa Young's modulus, 0.33 Poisson's ratio, and 2780 kg/m 3 ). The rise time parameter of the equation was adjusted to obtain the best similarity between the experiment and simulation. The frequency content of the excitation function varies according to the rise time, which is assumed. This, in fact, varies the frequency content of the resulting predictive simulation. The rise time was adjusted to have a good similarity between the experimental observation and simulation. The cosine bell function excitation was adjusted to a rise time of s, as shown in Figure 14, to match the simulation closely with the experimental observation. The wavenumber plot for S0 and A0 Lamb wave modes for 1 mm aluminum 2024-T3 is presented in Figure 15. By using the expression in Equation (43), the normalized PWAS tuning curve for M11 moment excitation was plotted for the S0 and A0 Lamb wave mode, as shown in Figure 16. For an M11 excitation, a strong S0 mode content was observed, and no A0 mode response was excited.  Using the procedure in Figure 13 and the cosine bell function excitation in Figure 14, the simulation of fatigue crack growth AE signal was performed. The resulting predicted waveform after performing the bandpass filtering corresponding to the pre-amplifier (Section 2.1) is presented in Figure 17a. We observe the nondispersive S0 mode in the simulation. The time-frequency representation of the numerical simulation and experiment was performed using Choi-Williams transform, which is also presented in Figure 18a. A side-by-side comparison of the time-domain simulation signal and the experimental signal and the Choi-Williams transform of the simulation signal and the experimental signal is presented in Figures 17 and 18, respectively. In the experimental observation, a nondispersive S0 wave packet was also observed at the beginning of the signal. However, in the experiment, apart from the initial nondispersive wave packet, the presence of a slow traveling wave packet was also observed. This wave packet is insignificant in example 2 and weakly present in example 1. This later arriving wave packet observed in the experiment is due to the scattering of the AE waves generated at the crack tip due to the crack formation at the crack and reaching the PWAS sensors following the initial wave packet. At present, in the simulation, we consider an ideal two-dimensional (2D) case in which a crack length in the third dimension is not considered. Therefore, the present simulation does not predict the scattered later weak wave packet.
Sensors 2020, 20, x FOR PEER REVIEW 18 of 25 Figure 16. Lamb wave strain response due to M11 excitation on 1 mm thick aluminum specimen received with 7 mm PWAS receiver.
Using the procedure in Figure 13 and the cosine bell function excitation in Figure 14, the simulation of fatigue crack growth AE signal was performed. The resulting predicted waveform after performing the bandpass filtering corresponding to the pre-amplifier (Section 2.1) is presented in Figure 17a. We observe the nondispersive S0 mode in the simulation. The time-frequency representation of the numerical simulation and experiment was performed using Choi-Williams transform, which is also presented in Figure 18a. A side-by-side comparison of the time-domain simulation signal and the experimental signal and the Choi-Williams transform of the simulation signal and the experimental signal is presented in Figures 17 and 18, respectively. In the experimental observation, a nondispersive S0 wave packet was also observed at the beginning of the signal. However, in the experiment, apart from the initial nondispersive wave packet, the presence of a slow traveling wave packet was also observed. This wave packet is insignificant in example 2 and weakly present in example 1. This later arriving wave packet observed in the experiment is due to the scattering of the AE waves generated at the crack tip due to the crack formation at the crack and reaching the PWAS sensors following the initial wave packet. At present, in the simulation, we consider an ideal two-dimensional (2D) case in which a crack length in the third dimension is not considered. Therefore, the present simulation does not predict the scattered later weak wave packet. Using the procedure in Figure 13 and the cosine bell function excitation in Figure 14, the simulation of fatigue crack growth AE signal was performed. The resulting predicted waveform after performing the bandpass filtering corresponding to the pre-amplifier (Section 2.1) is presented in Figure 17a. We observe the nondispersive S0 mode in the simulation. The time-frequency representation of the numerical simulation and experiment was performed using Choi-Williams transform, which is also presented in Figure 18a. A side-by-side comparison of the time-domain simulation signal and the experimental signal and the Choi-Williams transform of the simulation signal and the experimental signal is presented in Figures 17 and 18, respectively. In the experimental observation, a nondispersive S0 wave packet was also observed at the beginning of the signal. However, in the experiment, apart from the initial nondispersive wave packet, the presence of a slow traveling wave packet was also observed. This wave packet is insignificant in example 2 and weakly present in example 1. This later arriving wave packet observed in the experiment is due to the scattering of the AE waves generated at the crack tip due to the crack formation at the crack and reaching the PWAS sensors following the initial wave packet. At present, in the simulation, we consider an ideal two-dimensional (2D) case in which a crack length in the third dimension is not considered. Therefore, the present simulation does not predict the scattered later weak wave packet.

Verification of the Analytical Method
For verification of the theoretical modeling equations, an FEM numerical analysis of the PWAS response was performed. A 400 mm long and 1 mm thick 2D model was developed using the ANSYS software package (Figure 19a). In the FEM model, the material properties corresponding to the aluminum 2024-T3 specimen were considered (70 GPa Young's modulus, 0.33 Poisson's ratio, and 2780 kg/m 3 ). The element chosen for the finite element modeling was the plane-183 element. The AE waves due to the excitation were received using a receiver PWAS of 7 mm width and 0.5 mm thickness with the material properties of APC-850, located at 25 mm from the AE source. For the modeling of PWAS, plane-82 elements were used. For eliminating the reflections from the boundaries of the plate, nonreflective boundaries (NRBs) were applied at both ends of the model using the spring damper element COMBIN14 in ANSYS [60]. NRBs were applied, as presented in Figure 19a. The COMBIN14 elements were implemented at the top and bottom surfaces and both ends of the plate. The damping coefficients of the elements were varied gradually, starting from zero in a sinusoidal pattern on the plate. At the edges of the plate, the maximum stiffness and damping values were applied. The maximum stiffness and damping values at the edges of the plate were KN = 0, CN = LρCp, KT = 0, and CT = LρCs. Here, K is the stiffness coefficient, and C is the damping coefficient. L is the element length (0.05 mm). N and T are the normal and tangential directions. ρ is the material density. Cp and Cs are the bulk wave speeds of the material, 6100 m/s and 3070 m/s, respectively, for the present study. The damping coefficient values at the plate top and bottom edges were varied from zero to

Verification of the Analytical Method
For verification of the theoretical modeling equations, an FEM numerical analysis of the PWAS response was performed. A 400 mm long and 1 mm thick 2D model was developed using the ANSYS software package (Figure 19a). In the FEM model, the material properties corresponding to the aluminum 2024-T3 specimen were considered (70 GPa Young's modulus, 0.33 Poisson's ratio, and 2780 kg/m 3 ). The element chosen for the finite element modeling was the plane-183 element. The AE waves due to the excitation were received using a receiver PWAS of 7 mm width and 0.5 mm thickness with the material properties of APC-850, located at 25 mm from the AE source. For the modeling of PWAS, plane-82 elements were used. For eliminating the reflections from the boundaries of the plate, nonreflective boundaries (NRBs) were applied at both ends of the model using the spring damper element COMBIN14 in ANSYS [60]. NRBs were applied, as presented in Figure 19a. The COMBIN14 elements were implemented at the top and bottom surfaces and both ends of the plate. The damping coefficients of the elements were varied gradually, starting from zero in a sinusoidal pattern on the plate. At the edges of the plate, the maximum stiffness and damping values were applied. The maximum stiffness and damping values at the edges of the plate were K N = 0, C N = LρC p , K T = 0, and C T = LρC s . Here, K is the stiffness coefficient, and C is the damping coefficient. L is the element length (0.05 mm). N and T are the normal and tangential directions. ρ is the material density.  Finite element meshing was performed by selecting a 0.05 mm element size for the length and thickness of the model. The FE model, after the meshing, is presented in Figure 19b. Afterward, the M11 dipole excitation was defined. The M11 dipole excitation was modeled in the FEM by using dipole forces. The modeling details of the dipole force is presented in Figure 19b. A zoom-in of the details is shown in Figure 19c. Equal and opposite nodal forces were applied to define the M11 dipole excitation. A cosine bell function excitation was applied as the time profile of the excitation with 0.5 µs as the rise time of the excitation ( Figure 14). The finite element simulation was performed to solve the acoustic waveforms generated due to the dipole excitation. After performing the simulation, the inplane displacement field at the thickness nodes at 25 mm from the excitation location was extracted. The thickness modeshape of the in-plane displacement at 25 mm from the excitation was plotted and presented in Figure 19d, while the thickness modeshape of the analytical simulation is presented in Figure 19f. Due to the through-thickness dipole excitation, the in-plane displacement varies symmetrically with y-axis. The in-plane displacement modeshape resembles the modeshape of S0 Finite element meshing was performed by selecting a 0.05 mm element size for the length and thickness of the model. The FE model, after the meshing, is presented in Figure 19b. Afterward, the M 11 dipole excitation was defined. The M 11 dipole excitation was modeled in the FEM by using dipole forces. The modeling details of the dipole force is presented in Figure 19b. A zoom-in of the details is shown in Figure 19c. Equal and opposite nodal forces were applied to define the M 11 dipole excitation. A cosine bell function excitation was applied as the time profile of the excitation with 0.5 µs as the rise time of the excitation ( Figure 14). The finite element simulation was performed to solve the acoustic waveforms generated due to the dipole excitation. After performing the simulation, the in-plane displacement field at the thickness nodes at 25 mm from the excitation location was extracted. The thickness modeshape of the in-plane displacement at 25 mm from the excitation was plotted and presented in Figure 19d, while the thickness modeshape of the analytical simulation is presented in Figure 19f. Due to the through-thickness dipole excitation, the in-plane displacement varies symmetrically with y-axis. The in-plane displacement modeshape resembles the modeshape of S0 Lamb wave mode. In the analytical prediction, it was concluded that the through-thickness dipole excitation causes the generation of pure S0 mode. As we observe from Figure 16, a strong S0 mode is generated due to the through-thickness M 11 excitation, and no A0 mode is present. Analytical and FEM modeshapes were observed to be very closely matching, as we observe from the side-by-side comparison in Figure 19d,f. This comparison verifies the validity of the analytical simulation.
Next, the FEM PWAS response simulation was compared with the analytical prediction. The resulting PWAS response after performing the filtering corresponding to the pre-amplifier bandwidth discussed in Section 2.1 is presented in Figure 20. Figure 20a presents the normalized FEM PWAS response simulation result. Figure 20b shows the normalized analytical simulation result. A very good match of the FEM simulation and analytical simulation was observed. Both simulations showed a nondispersive AE wave sensed by the PWAS in the simulation. Afterward, the Choi-Williams transform of the FEM simulation ( Figure 20c) and analytical simulation (Figure 20d) was compared. The FEM simulation and analytical simulation also showed a very good match in the Choi-Williams transform comparison. The comparison of the FEM simulation and theoretical simulation of the PWAS response proved that the theoretical simulation is valid. Lamb wave mode. In the analytical prediction, it was concluded that the through-thickness dipole excitation causes the generation of pure S0 mode. As we observe from Figure 16, a strong S0 mode is generated due to the through-thickness M11 excitation, and no A0 mode is present. Analytical and FEM modeshapes were observed to be very closely matching, as we observe from the side-by-side comparison in Figure 19d,f. This comparison verifies the validity of the analytical simulation. Next, the FEM PWAS response simulation was compared with the analytical prediction. The resulting PWAS response after performing the filtering corresponding to the pre-amplifier bandwidth discussed in Section 2.1 is presented in Figure 20. Figure 20a presents the normalized FEM PWAS response simulation result. Figure 20b shows the normalized analytical simulation result. A very good match of the FEM simulation and analytical simulation was observed. Both simulations showed a nondispersive AE wave sensed by the PWAS in the simulation. Afterward, the Choi-Williams transform of the FEM simulation ( Figure 20c) and analytical simulation (Figure 20d) was compared. The FEM simulation and analytical simulation also showed a very good match in the Choi-Williams transform comparison. The comparison of the FEM simulation and theoretical simulation of the PWAS response proved that the theoretical simulation is valid.

Summary and Conclusions
This paper presented analytical predictive modeling of AE signals sensed using PWAS sensors during a fatigue crack growth event. This paper started with an introduction of the instrumentation used for sensing AE signals using PWAS transducers and a study of the transformations that AE signals undergo while passing through the instrumentation. A closed-form expression for sensing of acoustic waves using the PWAS sensor was derived. Next, the paper discussed a review of an in situ fatigue experiment performed for the detection of AE signals during a fatigue crack growth in a 1 mm aluminum 2024-T3 plate. The fatigue crack growth-related AE signals detected were presented

Summary and Conclusions
This paper presented analytical predictive modeling of AE signals sensed using PWAS sensors during a fatigue crack growth event. This paper started with an introduction of the instrumentation used for sensing AE signals using PWAS transducers and a study of the transformations that AE signals undergo while passing through the instrumentation. A closed-form expression for sensing of acoustic waves using the PWAS sensor was derived. Next, the paper discussed a review of an in situ fatigue experiment performed for the detection of AE signals during a fatigue crack growth in a 1 mm aluminum 2024-T3 plate. The fatigue crack growth-related AE signals detected were presented in this section. It was observed that the fatigue crack growth AE signals recorded has a strong S0 Lamb wave mode content.
After discussing the characteristics of fatigue crack growth AE signals briefly, the paper presented the analytical modeling of guided wave propagation during the fatigue crack growth event. The dipole moment excitation concept was used for modeling the fatigue crack growth AE source. Normal mode expansion of Lamb wave modes was used for obtaining a closed-form analytical expression for predicting the AE signal recorded using PWAS sensors. Later, the paper presented the numerical prediction results using the analytical expression. A 7 mm diameter PWAS was considered as the receiver, and a cosine bell excitation of 0.5 µs was chosen as the rise time of the excitation. The prediction was compared with experimentally recorded results. First, the time-domain AE signal prediction was compared with the time-domain experimental AE signals. Next, the time-frequency representation of the AE signal prediction and experimental AE signals was obtained using Choi-Williams transform, and the results were compared. The simulation result and the experimental results showed a close match with the presence of a strong S0 Lamb wave mode in the AE signal. However, in the experimental signal, after the strong S0 mode, a weak wave packet was also observed. This phenomenon is due to the scattering of the AE waves generated at the crack. While in the analytical prediction, a 1D waveguide was considered without the effect of AE signal scattering at the crack. Finally, the paper presented the verification of the analytical method using the FEM method. The dipole moment AE source modeling and simulation were performed using the FEM method. The modeshape of the waveform and the PWAS response were calculated. The analytically calculated modeshape and PWAS response were compared with the FEM modeshape and PWAS response. The results from analytical prediction and FEM prediction were very closely matching, which verified the analytical prediction.

Future Work
The analytical model presented in this paper is preliminary and a proof-of-concept study. Further research needs to be done to understand better the interaction of AE signals generated at the crack tips with the crack and improve the model. A three-dimensional (3D) FEM analysis in a model with crack can be performed to study the PWAS response due to fatigue crack growth. Numerical correlation analysis for the comparison of the analytical prediction and the FEM prediction result also can be performed.
Author Contributions: In this article, the formal analysis was done by R.J. The detailed methodology was provided by R.J. and V.G. R.J. prepared the original draft; review and editing were done by R.J. and V.G. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.