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Research Papers

# A Helmholtz Potential Approach to the Analysis of Guided Wave Generation During Acoustic Emission EventsOPEN ACCESS

[+] Author and Article Information

Department of Mechanical Engineering,
University of South Carolina,
300 Main Street, Room A237,
Columbia, SC 29208
e-mail: haiderm@email.sc.edu

Victor Giurgiutiu

Professor
Fellow ASME
Department of Mechanical Engineering,
University of South Carolina,
300 Main Street, Room A222,
Columbia, SC 29208
e-mail: victorg@sc.edu

1Corresponding author.

Manuscript received May 4, 2017; final manuscript received September 28, 2017; published online October 27, 2017. Assoc. Editor: Paul Fromme.

ASME J Nondestructive Evaluation 1(2), 021002 (Oct 27, 2017) (11 pages) Paper No: NDE-17-1005; doi: 10.1115/1.4038116 History: Received May 04, 2017; Revised September 28, 2017

## Abstract

This paper addresses the predictive simulation of acoustic emission (AE) guided waves that appear due to sudden energy release during incremental crack propagation. The Helmholtz decomposition approach is applied to the inhomogeneous elastodynamic Navier–Lame equations for both the displacement field and body forces. For the displacement field, we use the usual decomposition in terms of unknown scalar and vector potentials, $Φ$ and $H$. For the body forces, we hypothesize that they can also be expressed in terms of excitation scalar and vector potentials, $A*$ and $B*$. It is shown that these excitation potentials can be traced to the energy released during an incremental crack propagation. Thus, the inhomogeneous Navier–Lame equation has been transformed into a system of inhomogeneous wave equations in terms of known excitation potentials $A*$ and $B*$ and unknown potentials $Φ$ and $H$. The solution is readily obtained through direct and inverse Fourier transforms and application of the residue theorem. A numerical study of the one-dimensional (1D) AE guided wave propagation in a 6 mm thick 304-stainless steel plate is conducted. A Gaussian pulse is used to model the growth of the excitation potentials during the AE event; as a result, the actual excitation potential follows the error function variation in the time domain. The numerical studies show that the peak amplitude of A0 signal is higher than the peak amplitude of S0 signal, and the peak amplitude of bulk wave is not significant compared to S0 and A0 peak amplitudes. In addition, the effects of the source depth, higher propagating modes, and propagating distance on guided waves are also investigated.

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## Introduction

Acoustic emission (AE) has been used widely in structural health monitoring and nondestructive testing for the detection of crack propagation and to prevent ultimate failure [15]. AE signals are therefore needed to study extensively in order to characterize the crack in an aging structure. This paper presents a theoretical formulation of AE guided wave due to AE events such as crack extension in a structure. Lamb [6] derived Rayleigh–Lamb wave equations from Navier–Lame elastodynamic equations in an elastic plate. The presence of body force makes the homogeneous Navier–Lame elastodynamic equations into inhomogeneous equations. Helmholtz decomposition principle was used to decompose displacement to unknown scalar and vector potentials, and force vectors to known excitation scalar and vector potentials [7]. There are two types of potentials acting in a plate for straight-crested Lamb waves: pressure potential and shear potential. The assumption of the straight-crested guided wave makes the problem z-invariant (plane strain assumption). Excitation potentials for body forces can be generalized as localized AE event. Excitation potentials can be traced to energy released from the tip of the crack during crack propagation. These potentials satisfy the inhomogeneous wave equations. In this research, inhomogeneous wave equations for unknown potentials were solved due to known generalized excitation potentials in a form, suitable for numerical calculation. The theoretical formulation shows that elastic waves generated in a plate using excitation potentials followed the Rayleigh–Lamb equations.

In numerical studies, the one-dimensional (1D) (straight-crested) AE guided wave propagation was modeled in order to simulate the out-of-plane displacement that would be recorded by an AE sensor placed on the plate surface at some distance away from the source. A real AE source releases energy at certain time rate as a pulse over a finite time period. If the time rate of released energy is known, then this time rate of released energy can be decomposed to time rate of pressure and shear excitation potentials. The time profile of excitation potential is calculated from the time rate of excitation potential. Out-of-plane displacement was calculated numerically on the top surface of the plate. Parameter studies were performed to evaluate: (a) the effect of the pressure and shear potentials, (b) the effect of the thicknesswise location of the excitation potential sources varying from midplane to the top surface (source depth effect), (c) the effect of higher propagating modes, and (d) the effect of propagating distance away from the source.

###### State of the Art.

The basic concepts and equations of elastodynamics have been explained by several authors [6,817]. Lamb [18] presented propagation of vibrations over the surface of semi-infinite elastic solids. The vibrations were assumed due to an arbitrary application of force at a point. Rice [19] discussed a theory of elastic wave emission from damage (e.g., slip and microcracking). A general representation of the displacement field of an AE event was presented in terms of damage process in the source region. The solution is obtained in an unbounded medium.

Miklowitz [20] and Weaver and Pao [21] presented the response of transient loads of an infinite elastic plate, using double integral transforms. Based on the generalized theory of AE, the elastodynamic solution due to internal crack or fault can be analyzed by suitable Green function solution [22]. This Green function solution was obtained for either infinite media or half-space media. Later, Ohtsu and Ono [23] characterized the source of AE on the basis of that generalized theory. AE waveforms due to instantaneous formation of a dislocation were presented in that paper. In an inverse study, deconvolution analysis was done to characterize the AE source. Johnson [24] provided a complete solution to the three-dimensional (3D) Lamb’s problem, the problem of determining the elastic disturbance resulting from a point source in a half-space. Roth [25] extended the theory of dislocations to model deformations on the surface of a layered half-space to calculate deformation inside the medium. However, a theoretical representation of wave propagation in a plate due to body force is not presented.

Wave propagation in an elastic plate is well known from Lamb [6] classical work. Achenbach [8], Giurgiutiu [9], Graff [10], and Viktorov [11] considered Lamb waves, which exist in an elastic plate with traction free boundaries. Achenbach [8] presented Lamb wave in an isotropic elastic layer generated by a time harmonic internal/surface point load or line load. Displacements are obtained directly as summations over symmetric and antisymmetric modes of wave propagation. Elastodynamic reciprocity is used in order to obtain the coefficients of the wave mode expansion. Bai et al. [26] presented three-dimensional steady-state Green functions for Lamb wave in a layered isotropic plate. The elastodynamic response of a layered isotropic plate to a source point load having an arbitrary direction was studied in that paper. A semi-analytical finite element method was used to formulate the governing equations. Using this method, in-plane displacements were accommodated by means of an analytical double integral Fourier transform, while the antiplane displacement approximated by using finite elements. They have used same modal summation technique of eigenvectors as described by Liu and Achenbach [27] and Achenbach [8].

Jacobs et al. [28] presented an analytical methodology by incorporating a time-dependent acoustic emission signal as a source model to represent an actual crack propagation and arrest event. A review paper on the signal analysis used in AE for materials research field published by Ono [29]. This paper reviewed recent progress in methods of signal analysis used in acoustic emission such as deformation, fracture, phase transformation, coating, film, friction, wear, corrosion, and stress corrosion for materials research. A novel experimental mechanics technique using scanning electron microscopy in conjunction with AE monitoring is discussed by Wisner et al. [30]. An acoustic emission-based technique may be used to predict the residual fatigue life ceramic-matrix-composite at high temperature [31]. Transducer characterization may be necessary for high-temperature AE applications [32]. Cuadra et al. [33] proposed a 3D computational model to quantify the energy associated with AE source by energy balance and energy flux approach. The time profile of available energy as AE source was obtained for the first increment of crack. Khalifa et al. [34] proposed a formulation for modeling the AE sources and for the propagation of guided or Rayleigh waves.

There are numerous publications based on finite element analysis for detecting AE signals. For example, Hamsted et al. [35] reported wave propagation due to buried monopole and dipole sources with finite element technique. Hill et al. [36] compared waveforms captured by AE transducer for a step force on the plate surface by finite element modeling. Hamstad [37] presented frequencies and amplitudes of AE signals on a plate as a function of source rise time. In that paper, an exponential increase in peak amplitude with source rise time was reported. Sause et al. [38] presented a finite element approach for modeling of acoustic emission sources and signal propagation in hybrid multilayered plates.

###### Scope of the Paper.

There are numerous publications to characterize the source from a crack growth or damage. Various deformation sources in solids, such as single forces, step force, point force, dipoles, and moments, can be represented as AE sources [23,3941]. A moment tensor analysis to AE has been studied to elucidate crack types and orientations of AE sources [42]. AE source characteristics are unknown, and the detected AE signals depend on the types of AE source, propagation media, and the sensor response. Therefore, extracting AE source feature from a recorded AE waveform is always challenging. In this paper, we proposed a new technique to predict the AE waveform using excitation potentials. Excitation potentials can be traced to the energy released during an incremental crack propagation. The main contribution of this paper is to simulate AE elastic waves due to energy released during crack propagation by Helmholtz potential approach, for the first time to the authors’ best knowledge. The time profile of available energy as AE source from a crack can be obtained analytically or from a 3D computational model [33].

Our analytical model is developed based on integral transform using Helmholtz potentials. The solution is obtained through direct and inverse Fourier transforms and application of the residue theorem. The resulting solution is a series expansion containing the superposition of all the Lamb waves modes and bulk waves existing for the particular frequency-thickness combination under consideration. It is worth mentioning that there are other methods that can be used to solve wave equation of a structure, for example, Green’s functions for forced loading, semi-analytical method, normal mode expansion method, etc. Green’s functions for forced loading can be solved through an integral formulation relying on the elastodynamic reciprocity principle or integral transform. However, the Green function for forced loading problem requires the knowledge of the point load source that generated the AE event. Extracting information about the point load source that generated an actual AE event recorded in practice is quite challenging. Our approach bypassed this difficulty because it only needs an estimation of the energy released from the tip of the crack during a crack propagation increment. Our proposed method novelty lies on calculating AE waveform using energy released from the tip of the crack. Semi-analytical method requires extensive computational effort and has convergence issue at high frequency. Integral transform provides an exact solution, and easy to solve numerically.

###### Description of Excitation Potentials as Strain Energy Released From a Crack.

When a crack grows into a solid, a region of material adjacent to the free surfaces is unloaded, and it releases strain energy. The strain energy is the energy that must be supplied to a crack tip for it to grow, and it must be balanced by the amount of energy dissipated due to the formation of new surfaces and other dissipative processes such as plasticity.

Fracture mechanics allows calculating strain energy released during propagation of a crack. Figure 1(a) shows a plate with an existing crack length of 2a, and Fig. 1(b) shows the typical energy release rate with crack half-length [43]. For small increment of crack length da (Fig. 1(a)), the incremental strain energy can be calculated as shown in Fig. 1(c). This incremental strain energy can be released at different time rate at different time depending on crack growth and types of material. A real AE source releases energy during a finite time period. If time rate of energy released is known (Fig. 1(c)), integration of that with respect to time gives the time profile of energy (Fig. 1(d)) released from a crack. Total released energy from a crack can be decomposed to shear excitation potential and pressure excitation potential. Sections 2 and 3 presents a theoretical formulation for Lamb wave solution using time profile of excitation potentials: a Helmholtz potential technique.

## Formation of Pressure and Shear Potentials

Navier–Lame equations in vector form for Cartesian coordinates are given as Display Formula

(1)$(λ+μ)∇(∇⋅u)+μ∇2u=ρu¨$

where $u=uxî+uyĵ+uzk̂$, with $î$, $ĵ$, $k̂$ being unit vectors in the $x$, $y$, $z$ directions, respectively, $λ,μ$ are Lamé constants, and $ρ$ is the density.

If the body force is present, then the Navier–Lame equations are to be written as follows: Display Formula

(2)$(λ+μ)∇(∇⋅u)+μ∇2u+ρf=ρu¨$

Helmholtz decomposition states that [7] any vector can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal vector field, where an irrotational vector field has a scalar potential and a solenoidal vector field has a vector potential. Potentials are useful and convenient in several wave theory derivations [9,10].

Assume that the displacement $u$ can be expressed in terms of two potential functions, a scalar potential $Φ$ and a vector potential $H$Display Formula

(3)$u=grad Φ+curl H=∇Φ+∇×H$

where Display Formula

(4)$H=Hxi+Hyj+Hzk$

Equation (3) is known as the Helmholtz equation and is complemented by the uniqueness condition, i.e., Display Formula

(5)$∇⋅H=0$

Introducing additional scalar and vector potentials $A∗$ and $B∗$ for body force $f$ [10,14] Display Formula

(6)$f=grad A∗+curl B∗=∇A∗+∇×B∗$

Here Display Formula

(7)$B¯∗=Bx∗i+By∗j+Bz∗k$

The uniqueness condition is Display Formula

(8)$∇⋅B∗=0$

Using Eqs. (3) and (6) into Eq. (2), wave equations for potentials are to be found [44] Display Formula

(9)$cP2∇2Φ+A∗=Φ¨$
Display Formula
(10)$cS2∇2H+B∗=H¨$

Here, $cP2=(λ+2μ)/ρ$ and $cS2=μ/ρ$.

The expanded form of Eqs. (9) and (10) is given in Ref. [44]. Equations (9) and (10) are the wave equations for the scalar potential and the vector potential, respectively. Equation (9) indicates that the scalar potential, $Φ$, propagates with the pressure wave speed, $cP$, due to excitation potential $A∗$, whereas Eq. (10) indicates that the vector potential, $H$, propagates with the shear wave speed, $cs$, due to excitation potential $B∗$.

The unit of the excitation potentials is $μJ/kg$. Inhomogeneous wave equations for potentials are well known in electrodynamics and electromagnetic fields. Wolski [45], Jackson [46], and Uman et al. [47] treated inhomogeneous wave equation for scalar and vector potential with the presence of current and charge density. The source can be treated as localized event, and Green function can be used as the solution of inhomogeneous wave equations.

The derivation Lamb wave equation will be provided by solving wave Eqs. (9) and (10) for the potentials. In this derivation, we will consider generation of straight-crested Lamb wave due to time harmonic excitation potentials. The assumption of straight-crested waves makes the problem z-invariant. For P + SV waves (Lamb wave) [44], the relevant potentials are $Φ,Hz,A, and Bz$. Equations (9) and (10) are condensed into two equations, i.e., Display Formula

(11)$∇2Φ+1cP2A∗=1cP2 Φ¨; ∇2Hz+1cS2Bz∗=1cS2H¨z$

The P waves and SV waves give rise to the Lamb waves, which consist of a pattern of standing waves in the thickness direction (Lamb wave modes) behaving like traveling waves in the x direction in a plate of thickness h = 2d (Fig. 2). Excitation potentials are considered as z-invariant at $x=0 and y=0$, i.e., source depth $d1=d$. For time harmonic source, the equations for excitation potentials are Display Formula

(12)$A*=cP2Aδ(x)δ(y)e−iωt$
Display Formula
(13)$Bz*=cs2Bzδ(x)δ(y)e−iωt$

Here, $A$ and $Bz$ are the source amplitude.

Since the potentials are the time harmonic excitation, the response potentials $Φ,Hz$ have the harmonic response, i.e., Display Formula

(14)$Φ(x,y,t)=Φ(x,y)e−iωtHz(x,y,t)=Hz(x,y)e−iωt$

Then, Eq. (11) becomes Display Formula

(15)$∂2Φ∂x2+∂2Φ∂y2+ω2cP2Φ=−Aδ(x)δ(y)$
Display Formula
(16)$∂2Hz∂x2+∂2Hz∂y2+ω2cS2Hz=−Bzδ(x)δ(y)$

Equations (15) and (16) must be solved subject to zero-stress boundary conditions at the free top and bottom surfaces of the plate, i.e., Display Formula

(17)$σyy|y=±d=0,σxy|y=±d=0$

## Solution in Terms of Displacement

Taking Fourier transform of Eqs. (15) and (16), in x direction, $Φ(x,y)→Φ¯(ξ,y)$ and $H(x,y)→H¯(ξ,y)$Display Formula

(18)$−ξ2Φ¯(ξ,y)+Φ¯″(ξ,y)+ω2cp2Φ¯(ξ,y)=−Aδ(y)$
Display Formula
(19)$−ξ2H¯z(ξ,y)+H¯z″(ξ,y)+ω2cS2H¯z(ξ,y)=−Bzδ(y)$

Here, $Aδ(y)=∫−∞∞Aδ(y)[δ(x)e−iξxdx]$ and $Bzδ(y)=∫−∞∞Bzδ(y)[δ(x)e−iξxdx]$.

Upon rearranging Display Formula

(20)$Φ¯″(ξ,y)+ηp2Φ¯(ξ,y)=−Aδ(y)$
Display Formula
(21)$H¯z″(ξ,y)+ηs2H¯z(ξ,y)=−Bzδ(y)$

Here Display Formula

(22)$(ω2cp2−ξ2)=ηp2; (ω2cs2−ξ2)=ηs2$

Equations (20) and (21) are the second-order ordinary differential equation in $y$ direction. The total solution of Eqs. (20) and (21) consists of representation of two solutions [48]:

1. (a)the complementary solution for the homogeneous equation
2. (b)a particular solution of that satisfies source effect

Hence [44] Display Formula

(23)$Φ¯(ξ,y)=C1 sin ηpy+C2 cos ηpy−A2ηpsin ηp|y|$
Display Formula
(24)$H¯z(ξ,y)=i(D1 sin ηsy+D2 cos ηsy−Bz2ηssin ηs|y|)$

The coefficients $C1, C2, D1, and D2$ of the complimentary solutions (23) and (24) are to be determined by using the Fourier transformed boundary conditions, i.e., Display Formula

(25)$σ¯yy|y=±d=0, σ¯xy|y=±d=0$

After applying the boundary conditions, we get [44] Display Formula

(26)$(ξ2−ηs2)(C1 sin ηpd+C2 cos ηpd−A2ηpsin ηpd)+2ξ(D1ηs cos ηsd−D2ηs sin ηsd−ηsBz2ηscos ηsd)=0$
Display Formula
(27)$(ξ2−ηs2)(−C1 sin ηpd+C2 cos ηpd−A2ηpsin ηpd)+2ξ(D1ηs cos ηsd+D2ηs sin ηsd−ηsBz2ηscos ηsd)=0$
Display Formula
(28)$2ξ(C1ηp cos ηpd−C2ηp sin ηpd−ηpA2ηpcos ηpd)+(ξ2−ηs2)(D1 sin ηsd+D2 cos ηsd−Bz2ηssin ηsd)=0$
Display Formula
(29)$2ξ(C1ηp cos ηpd+C2ηp sin ηpd−ηpA2ηpcos ηpd)+(ξ2−ηs2)(−D1 sin ηsd+D2 cos ηsd−Bz2ηssin ηsd)=0$

Equations (26)(29) are a set of four equations with four unknowns. The equations can be separated into a couple of two equations with two unknowns, one for symmetric motion and one for antisymmetric motion.

###### Symmetric Lamb Wave Solution.

Addition of Eqs. (26) and (27), and subtraction of Eq. (28) from Eq. (29), yields Display Formula

(30)$[C2D1]=1Ds[PS(ξ2−ηs2)sin ηsdPS2ξηp sin ηpd]$

Here Display Formula

(31)$Ds=|(ξ2−ηs2)cos ηpd2ξηs cos ηsd−2ξηp sin ηp(ξ2−ηs2)sin ηsd|=(ξ2−ηs2)2 cos ηpd sin ηsd+4ξ2ηsηp cos ηsd sin ηpd$
Display Formula
(32)$PS=((ξ2−ηs2)A2ηpsin ηpd+ξBz cos ηsd)$

$PS$ is the source term for the symmetric solution which contains source potentials $A$ and $BZ$.

By equating to zero the $Ds(ξ)$ term of Eq. (31), one gets the symmetric Rayleigh–Lamb equation.

###### Antisymmetric Lamb Wave Solution.

Subtraction of Eq. (26) from Eq. (27), and addition of Eqs. (28) and (29), yields Display Formula

(33)$[C1D2]=1DA[PA2ξηs sin ηsdPA(ξ2−ηs2)sin ηpd]$

Here Display Formula

(34)$DA=|(ξ2−ηs2)sin ηpd−2ξηs sin ηsd2ξηp cos ηp(ξ2−ηs2)cos ηsd|=(ξ2−ηs2)2 sin ηpd cos ηsd+4ξ2ηsηp sin ηsd cos ηpd$
Display Formula
(35)$PA=ξA cos ηpd+(ξ2−ηs2)Bz2ηssin ηsd$

$PA$ is the source term for the antisymmetric solution which contains source potentials $A$ and $BZ$. By equating to zero the $DA(ξ)$ term of Eq. (34), one gets the antisymmetric Rayleigh–Lamb equation.

###### Displacement Solution.

The Fourier transform of out-of-plane displacement, $uy$, is [44] Display Formula

(36)$u¯y=∂Φ¯∂y−iξH¯z$

Substitution of Eqs. (23) and (24) into Eq. (36) yields the expressions for out-of-plane displacement in the wavenumber domain in terms of coefficients $C1, C2, D1, and D2$ which are functions of the wavenumber ξ. Evaluation of out-of-plane displacement at the plate top surface, $u¯y|y=d$, yields Display Formula

(37)$u¯y=(PSNSDS+PANADA)−A2cos ηpd−ξBz2ηssin ηsd$

where Display Formula

(38)$Ns=2ξ2ηp sin ηpd sin ηsd−(ξ2−ηs2)ηp sin ηpd sin ηsdNA=2ξηsηp sin ηsd cos ηpd+ξ(ξ2−ηs2)sin ηpd cos ηsd$

The complete solution of displacement is the superposition of the symmetric, antisymmetric, and bulk wave solution. Displacement solution in the physical domain is obtained by taking inverse Fourier transform of Eq. (37) [44,4953]; hence, we get Display Formula

(39)$uy=uyL+uyA+uyB$

where Display Formula

(40)$uyL=i(∑j=0js[PS(ξjS)NS(ξjS)Ds′(ξjS)]ei(ξjSx−ωt)+∑j=0jA[PA(ξjA)NA(ξjA)DA′(ξjA)]ei(ξjAx−ωt))$
Display Formula
(41)$uyA=−A4(ωcp)x−12(d2+x2)−12d J1(ωcp[d2+x2]12)e−iωt$
Display Formula
(42)$uyB=−Bz2ωcsx(x2+d2)−12J0[ωcs(x2+d2)12]e−iωt−iBz2(ωcs)x(x2+d2)−12Y1[(ωcs)(x2+d2)12]e−iωt$

$uyL$ is the out-of-plane displacement containing Lamb wave mode, and $uyA,uyB$ are the out-of-plane displacements of bulk wave for excitation potentials $A$ and $Bz$, respectively.

Theoretical derivation reveals that bulk wave is present in addition to Lamb wave. The Lamb waves consist of a pattern of standing waves in the thickness direction (Lamb wave modes) behaving like traveling waves in the x direction in a plate. Guided waves can propagate long distances and yield an easy inspection of a wide variety of structures. Whereas, bulk waves generate from a source and directly reach to the AE sensor without interfering the boundaries. Bulk waves are categorized into longitudinal (pressure) and transverse (shear) waves. The displacements $uyA$ and $uyB$ are the longitudinal and transverse bulk waves, respectively.

###### Effect of Source Depth.

If the source is present at different locations other than midplane, i.e., $y=y0$ (Fig. 2), then equations for excitation potentials become Display Formula

(43)$A*=cp2Aδ(x)δ(y−y0)e−iωt$
Display Formula
(44)$Bz*=cs2Bzδ(x)δ(y−y0)e−iωt$

For this new location of excitation potentials, Eqs. (20) and (21) become Display Formula

(45)$Φ¯″(ξ,y)+ηp2Φ¯(ξ,y)=−Aδ(y−y0)$
Display Formula
(46)$H¯z″(ξ,y)+ηs2H¯z(ξ,y)=−Bzδ(y−y0)$

Complementary solution of Eqs. (45) and (46) will be the same as before. Only particular solution will change due to the source in right-hand side.

The corresponding source terms for symmetric and antisymmetric solutions (32) and (35) will be Display Formula

(47)$PS=(ξ2−ηs2)A2ηpsin ηpd1+ξBz cos ηsd1$
Display Formula
(48)$PA=ξA cos ηpd1+(ξ2−ηs2)Bz2ηssin ηsd1$

Here, $d1=d−y0$.

The corresponding displacement Eqs. (40)(42) become Display Formula

(49)$uyL=i(∑j=0js[PS(ξjS)NS(ξjS)Ds′(ξjS)]ei(ξjSx−ωt)+∑j=0jA[PA(ξjA)NA(ξjA)DA′(ξjA)]ei(ξjAx−ωt))$
Display Formula
(50)$uyA=−A4(ωcP)(d2+x2)−12Y1(ωcP[x2+d12]12)e−iωt$
Display Formula
(51)$uyB=−Bz2ωcsx(x2+d12)−12J0[ωcs(x2+d12)12]e−iωt−iBz2(ωcs)x(x2+d12)−12Y1[(ωcs)(x2+d12)12]e−iωt$

###### Acoustic Emission Guided Wave Propagation.

Acoustic emission guided waves will be generated by the AE event. AE guided waves will propagate through the structure according to the structural transfer function. The out-of-plane displacement of the guided waves can be captured by conventional AE transducer installed on the surface of the structure, as shown in Fig. 3.

## Numerical Studies

This section includes guided wave simulation in a plate due to excitation potentials.

###### Time-Dependent Excitation Potentials.

The time-dependent excitation potentials depend on time-dependent energy released from a crack. A real AE source releases energy during a finite time period. At the beginning, the rate of energy released from a crack increases sharply with time and reaches a maximum peak value within very short time, then decreases asymptotically toward the steady-state value, usually zero. Time rate of energy released from a crack can be modeled as a Gaussian pulse. Pressure and shear excitation potentials are assumed to follow the same time rate and cumulative profiles (Fig. 4). Figure 4(a) shows the time rate of excitation (pressure and shear potentials). The corresponding equations are Display Formula

(52)$∂A*∂t=A0t2e−t2τ2; ∂Bz*∂t=BZ0t2e−t2τ2$

Here, $A0$ and $BZ0$ are the scaling factors. The time profile of the potentials is to be evaluated by integrating Eq. (52), i.e., Display Formula

(53)$A∗=A0(πτerf(tτ)−2te−t2τ2); Bz∗=Bz0(πτerf(tτ)−2te−t2τ2)$

The time profile of excitation potential is shown in Fig. 4(b). Both pressure and shear excitation potentials follow the same time profile, as shown in Fig. 4(b). The time profile of AE source may follow cosine bell function [37,38] or error function. In this research, a Gaussian pulse is used to model the growth of the excitation potentials during the AE event; as a result, the actual excitation potential follows the error function variation in the time domain. Cuadra et al. [33] obtained similar time profile of AE energy from a crack by using 3D computation method.

The key characteristics of the excitation potentials are:

• peak time: time required to reach the time rate of potential to maximum value

• rise time: time required to reach the potential to 98% of maximum value or steady-state value

• peak value: maximum value of time rate of potential

• maximum potential: maximum value of time profile of potential

Amplitude of the source $A$ and $Bz$ for unit width is Display Formula

(54)$A=hcp2A0(πτerf(tτ)−2te−t2τ2)Bz=hcs2Bz0(πτerf(tτ)−2te−t2τ2)$

Here, plate thickness $h=2d$.

###### Acoustic Emission Guided Wave Propagation in a 6 mm Plate: Effect of Source Depth.

A test case example is presented in this section to show how AE guided waves propagate over a certain distance using excitation potentials. For numerical analysis, a 304-stainless steel plate with 6 mm thickness was chosen as a case study. The signal was received at a distance 500 mm away from the source. Time profile excitation potentials (Fig. 4(b)) were used to simulate AE waves in the plate. Excitation potential can be calculated from the time rate of potential released during crack propagation. Peak time, rise time, peak value, and maximum potential of excitation potentials are 3 $μs$, 6.6 $μs$, 0.28 $μW/kg$, and 1 $μJ/kg$. Based on this information, a numerical study on AE Lamb wave propagation was conducted. Peak time or rise time is one of the major characteristics of AE source. Several researchers investigate potential effect of rise time on AE signal by varying the source rise time in-between 0.1 $μs$ and 15 $μs$ [3739,53]. In numerical analysis, direct Fourier transforms of the time domain excitation signal were performed to get the frequency contents of the signal [44]. Then, the source term and plate transfer function are multiplied with the signal to get out-of-plane displacement in frequency domain (Eq. (37)). Finally, an inverse Fourier transform was done to get the time domain output signal from frequency response of out-of-plane displacement. In this paper, each excitation potential was considered separately to simulate the Lamb waves. Total released energy from a crack can be decomposed to pressure and shear excitation potentials. This paper presents the individual effect of the unit amplitude of potentials (1 $μJ/kg$) on AE signals. However, distributing the energy between the pressure and shear potentials for real AE source is very important, and we recommend it for future study for specific damage assumptions (e.g., a crack propagating under mode I fracture may have a different mix of shear/pressure potentials than a crack propagating under mode II or mode III fracture modes; similarly for mixed-mode fracture).

The excitation sources were located at the top surface, 1.5 mm from the top surface, and 3 mm from the top surface (midplane). Figures 57 show the out-of-plane displacement (S0 and A0 modes and bulk wave) versus time at 500 mm propagation distance for midplane, top surface, and 1.5 mm depth from top-surface source location. The summary of the results is listed in Table 1. Signals are normalized by their individual peak amplitudes, i.e., amplitude/peak amplitude. From Fig. 5, it is observed that the A0 mode appears only while using pressure potential on the top surface, whereas S0 mode appears only while using shear potential. If the AE source is located at top surface, the effect of the excitation pressure and shear potentials on the S0 and A0 modes seems to be decoupled (Eqs. (32) and (35)): pressure potential does not contribute to S0 and bulk wave amplitude, whereas shear potential does not contribute to the A0 and bulk wave amplitude. Despite the fact that pressure and shear potentials are decoupled from each other, they may exist simultaneously for an AE event.

Therefore, both A0 and S0 will exist, only pressure potential does not contribute to S0 and shear potential does not contribute to the A0. Another notable characteristic is that the pressure potential contributes to the high amplitude of the trailing edge (low-frequency component) of A0 wave packet for top-surface source. With increasing source depth, the effect of high amplitude of low frequency decreases (Figs. 6(a) and 7(a)) in A0 signal. Figures 6 and 7 show that the peak A0 amplitude increases and peak S0 amplitude decreases with increasing source depth while using pressure excitation potential only.

Figures 5(b), 6(b), and 7(b) show some qualitative and quantitative changes in spectrum while using shear excitation potentials only. The amplitude of S0 signal decreases and A0 signal increases with increasing source depth while using shear excitation potential only (Figs. 6(b) and 7(b)). Qualitative change in S0 signal with source depth refers to the change in frequency content of the signal. It should be noted that the peak amplitude of S0 and A0 signals using shear excitation potential located at 1.5 mm depth from the top surface is more significant compared to pressure potential located at same location (Fig. 6). However, with increasing source depth, S0 signals amplitude becomes more significant due to pressure excitation potential over shear excitation potential (Figs. 6 and 7). For all AE source location, the shear potential part of the AE source has more contribution to the peak A0 amplitude than pressure potential. The peak amplitude of bulk waves increases with increasing source depth (Figs. 6 and 7). However, the amplitude of bulk wave is much smaller than the peak A0 and S0 amplitudes. Therefore, the peak amplitude of bulk wave may not appear in real AE signal.

###### Effect of Higher-Order Lamb Wave Modes.

Since AE signals have a wideband response, higher-order modes should appear in addition to the fundamental S0 and A0 Lamb wave modes. This section discusses the effect of higher-order modes on AE signal due to shear potential excitation and pressure potential excitation. A real AE sensor does not have ultimate high-frequency response; therefore, the signals were filtered at 10 kHz and 700 kHz frequency. Figure 8 shows the higher-order Lamb wave (S1 and A1) modes at 500 mm distance in 6 mm 304-steel plate. The A1 signal is more dispersive for the shear excitation potential than the pressure excitation potential. The peak amplitude of the A1 mode is higher than the peak amplitude of the S1 mode for both shear and pressure potentials. The peak S1 and A1 amplitudes for pressure potential are higher than the shear potential. By comparing Figs. 7 and 8, it can be inferred that the peak amplitude of fundamental Lamb wave mode (S0 and A0) is higher than the peak S1 and A1 amplitudes. Therefore, the peak amplitude of higher-order mode may not be significant in real AE signal.

###### Effect of Propagation Distance.

The attenuation of the peak amplitude of the signal as a function of propagating distance was also determined. Figures 9(a) and 9(b) show the normalized peak amplitude of out-of-displacement (S0, A0, and bulk wave) against propagation distance from 100 mm to 500 mm for pressure potential and shear potential, respectively. The amplitudes are normalized by their individual peak amplitudes.

Figures 9(a) and 9(b) show attenuation of S0, A0, and bulk wave signal with propagating distance from 100 mm to 500 mm. The effect of the propagating distance was found to produce increasingly larger losses in S0 and A0 peak signal amplitudes as the propagation distance varied between 100 mm and 500 mm. Since the medium was assumed lossless, this attenuation in propagation distance attributes to signal dispersion, which is to be expected due to the wideband characteristic of the AE excitation. However, larger attenuation of peak bulk wave amplitude is observed compared to peak S0 and A0 amplitudes. Therefore, far from the source, bulk wave becomes less significant and may not be captured by the AE transducer. Another important observation is that the peak A0 amplitude attenuates more than peak S0 amplitude while using pressure potential only, whereas the peak S0 amplitude attenuates more than peak A0 amplitude while using shear potential only. The attenuation of the peak amplitude is expected due to dispersion of the signal.

## Summary, Conclusions, and Future Work

###### Summary.

The guided waves generated by an AE event were analyzed through a Helmholtz potential approach. The inhomogeneous elastodynamic Navier–Lame equation was expressed as a system of wave equations in terms of unknown scalar and vector solution potentials, $Φ$ and $H$, and known scalar and vector excitation potentials, $A*$ and $B*$. The excitation potentials $A*$ and $B*$ were traced to the energy released during an incremental crack propagation.

The solution was readily obtained through direct and inverse Fourier transforms and application of the residue theorem. The resulting solution took the form of a series expansion containing the superposition of all the Lamb waves modes and bulk waves existing for the particular frequency-thickness combination under consideration.

A numerical study of the AE guided wave propagation in a 6 mm thick 304-steel plate was conducted in order to predict the out-of-plane displacement that would be recorded by an AE sensor placed on the plate surface at some distance away from the source. Parameter studies were performed to evaluate: (a) the effect of the pressure and shear potentials, (b) the effect of the thicknesswise location of the excitation potential sources varying from midplane to the top surface (source depth effect), (c) the effect of higher propagating mode, and (d) the effect of propagating distance away from the source.

###### Conclusions.

The present work has shown that pressure and shear source potentials can be used to model the guided wave generation and propagation due to an AE event associated with incremental crack growth. The advantage of this approach is to decouple the inhomogeneous Navier–Lame equations.

The numerical studies performed over a range of parameters have shown that:

• (i)The peak amplitude of A0 mode is higher than the peak amplitude of the S0 mode.
• (ii)The amplitude of bulk wave is much smaller than peak A0 and S0 amplitude. Therefore, the peak amplitude of bulk waves may not be significant in real AE signal.
• (iii)For midplane AE source location, the shear potential part of the AE source has more contribution to the peak A0 amplitude than pressure potential, whereas the pressure potential part of the AE source has more contribution to the peak S0 amplitude than shear potential.
• (iv)With an increase in the source depth, the peak A0 and S0 amplitude increases by considering pressure potential only, whereas the peak A0 increases and S0 decreases by using shear potential only.
• (v)If the AE source is located at top surface, the effect of the excitation pressure and shear potentials on the S0 and A0 modes seems to be decoupled: pressure potential does not contribute to S0 and bulk wave amplitude, whereas shear potential does not contribute to the A0 and bulk wave amplitude.
• (vi)For top-surface AE source, the pressure excitation potential has contribution to the high-amplitude low-frequency component of the A0 wave packet. This contribution decreases as the source depth increases.

###### Future Work.

Substantial future work is still needed to verify the hypotheses and substantiate the calculation of the AE source potentials that produce the guided wave excitation. The extensive experimental AE monitoring data existing in the literature should be explored to find actual physical signals that could be compared with numerical predictions in order to extract factual data about the amplitude and time-evolution of the AE source potentials. A frequency analysis of time domain signal should be done to analyze the frequency content of the captured AE signals. Frequency content may help to distinguish different source types and source location. If necessary, additional experiments with wider band AE sensors should be conducted. An inverse algorithm could then be developed to characterize the AE source during crack propagation. The source characterization can provide information about amount of energy released from the crack. Therefore, it may help to generate a qualitative as well quantitative description of the crack propagation phenomenon. A further extensive study on the effect of plate thickness, AE rise time, and AE source depth would be recommended. Straight-crested Lamb wave (1D wave guided propagation) was considered in this paper. However, realistic AE guided wave propagation usually takes place events in two-dimensional geometries (pressure vessels, containment vessels, etc.). Hence, the present theory should be extended to circular-crested Lamb waves (two-dimensional wave guided propagation). The contribution of pressure and shear excitation potentials to the total energy release from a crack needs to be studied through types of materials and crack types.

## Funding Data

• U.S. Department of Energy (DOE), Office of Nuclear Energy (Grant Nos. DE-NE 0000726 and DE-NE 0008400).

## References

Harris, D. O. , and Dunegan, H. L. , 1974, “ Continuous Monitoring of Fatigue-Crack Growth by Acoustic-Emission Techniques,” Exp. Mech., 14(2), pp. 71–81.
Han, B. H. , Yoon, D. J. , Huh, Y. H. , and Lee, Y. S. , 2014, “ Damage Assessment of Wind Turbine Blade Under Static Loading Test Using Acoustic Emission,” J. Intell. Mater. Syst. Struct., 25(5), pp. 621–630.
Tandon, N. , and Choudhury, A. , 1999, “ A Review of Vibration and Acoustic Measurement Methods for the Detection of Defects in Rolling Element Bearings,” Tribol. Int., 32(8), pp. 469–480.
Roberts, T. , and Talebzadeh, M. , 2003, “ Acoustic Emission Monitoring of Fatigue Crack Propagation,” J. Constr. Steel Res., 59(6), pp. 695–712.
Bassim, M. N. , Lawrence, S. S. , and Liu, C. D. , 1994, “ Detection of the Onset of Fatigue Crack Growth in Rail Steels Using Acoustic Emission,” Eng. Fract. Mech., 47(2), pp. 207–214.
Lamb, H. , 1917, “ On Waves in an Elastic Plate,” Proc. R. Soc. London A: Math., Phys. Eng. Sci., 93(648), pp. 114–128.
Helmholtz, H. , 1858, “ Uber Integrale der Hydrodynamischen Gleichungen, Welche den Wirbelbewegungen Entsprechen,” J. Reine Angew. Math., 1858(55), pp. 25–55.
Achenbach, J. D. , 2003, Reciprocity in Elastodynamics, Cambridge University Press, Cambridge, UK.
Giurgiutiu, V. , 2014, Structural Health Monitoring With Piezoelectric Wafer Active Sensors, 2nd ed., Elsevier, Amsterdam, The Netherlands.
Graff, K. F. , 1975, Wave Motion in Elastic Solids, Clarendon Press, Oxford, UK.
Viktorov, I. A. , 1967, Rayleigh and Lamb Waves: Physical Theory and Applications, Plenum Press, New York.
Landau, L. D. , and Lifschitz, E. M. , 1965, Teoriya Uprugosti, Nauka, Moscow, Russia [Theory of Elasticity, 2nd ed., Pergamon Press, Oxford, UK (1970)].
Love, A. E. H. , 1944, A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York.
Aki, K. , and Richards, P. G. , 2002, Quantitative Seismology, Vol. 1, University Science Books, Sausalito, CA.
Vvedenskaya, A. V. , 1956, “ The Determination of Displacement Fields by Means of Dislocation Theory,” Izv. Akad. Nauk SSSR, 3(27), pp. 227–284.
Nabarro, F. R. N. , 1951, “ The Synthesis of Elastic Dislocation Fields,” Philos. Mag., 42(334), p. 313.
Maruyama, T. , 1963, “ On the Force Equivalents of Dynamical Elastic Dislocations With Reference to the Earthquake Mechanism,” Bull. Earthquake Res. Inst., Tokyo Univ., 41, pp. 467–486.
Lamb, H. , 1903, “ On the Propagation of Tremors Over the Surface of an Elastic Solid,” Proc. R. Soc. London, 72(477–486), pp. 128–130.
Rice, J. R. , 1980, “ Elastic Wave Emission From Damage Processes,” J. Nondestr. Eval., 1(4), pp. 215–224.
Miklowitz, J. , 1962, “ Transient Compressional Waves in an Infinite Elastic Plate or Elastic Layer Overlying a Rigid Half-Space,” ASME J. Appl. Mech., 29(1), pp. 53–60.
Weaver, R. L. , and Pao, Y.-H. , 1982, “ Axisymmetric Elastic Waves Excited by a Point Source in a Plate,” ASME J. Appl. Mech., 49(4), pp. 821–836.
Ono, K. , and Ohtsu, M. , 1984, “ A Generalized Theory of Acoustic Emission and Green’s Functions in a Half Space,” J. Acoust. Emiss., 3, pp. 27–40.
Ohtsu, M. , and Ono, K. , 1986, “ The Generalized Theory and Source Representations of Acoustic Emission,” J. Acoust. Emiss., 5(4), pp. 124–133.
Johnson, L. R. , 1974, “ Green’s Function for Lamb’s Problem,” Geophys. J. Int., 37(1), pp. 99–131.
Roth, F. , 1990, “ Subsurface Deformations in a Layered Elastic Half-Space,” Geophys. J. Int., 103(1), pp. 147–155.
Bai, H. , Zhu, J. , Shah, A. H. , and Popplewell, N. , 2004, “ Three-Dimensional Steady State Green Function for a Layered Isotropic Plate,” J. Sound Vib., 269(1), pp. 251–271.
Liu, G. R. , and Achenbach, J. D. , 1995, “ Strip Element Method to Analyze Wave Scattering by Cracks in Anisotropic Laminated Plates,” ASME J. Appl. Mech., 62(3), pp. 607–613.
Jacobs, L. J. , Scott, W. R. , Granata, D. M. , and Ryan, M. J. , 1991, “ Experimental and Analytical Characterization of Acoustic Emission Signals,” J. Nondestr. Eval., 10(2), pp. 63–70.
Ono, K. , 2011, “ Acoustic Emission in Materials Research—A Review,” J. Acoust. Emiss., 29, pp. 284–309.
Wisner, B. , Cabal, M. , Vanniamparambil, P. A. , Hochhalter, J. , Leser, W. P. , and Kontsos, A. , 2015, “ In Situ Microscopic Investigation to Validate Acoustic Emission Monitoring,” Exp. Mech., 55(9), pp. 1705–1715.
Momon, S. , Moevus, M. , Godin, N. , R’Mili, M. , Reynaud, P. , Fantozzi, G. , and Fayolle, G. , 2010, “ Acoustic Emission and Lifetime Prediction During Static Fatigue Tests on Ceramic-Matrix-Composite at High Temperature Under Air,” Composites, Part A, 41(7), pp. 913–918.
Haider, M. F. , Giurgiutiu, V. , Lin, B. , and Yu, L. , 2017, “ Irreversibility Effects in Piezoelectric Wafer Active Sensors After Exposure to High Temperature,” Smart Mater. Struct., 26(9), p. 095019.
Cuadra, J. A. , Baxevanakis, K. P. , Mazzotti, M. , Bartoli, I. , and Kontsos, A. , 2016, “ Energy Dissipation Via Acoustic Emission in Ductile Crack Initiation,” Int. J. Fract., 199(1), pp. 89–104.
Khalifa, W. B. , Jezzine, K. , Grondel, S. , Hello, G. , and Lhémery, A. , 2012, “ Modeling of the Far-Field Acoustic Emission From a Crack Under Stress,” J. Acoust. Emiss., 30, pp. 137–152.
Hamstad, M. A. , O’Gallagher, A. , and Gary, J. , 1999, “ Modeling of Buried Monopole and Dipole Sources of Acoustic Emission With a Finite Element Technique,” J. Acoust. Emiss., 17(3–4), pp. 97–110.
Hill, R. , Forsyth, S. A. , and Macey, P. , 2004, “ Finite Element Modelling of Ultrasound, With Reference to Transducers and AE Waves,” Ultrasonics, 42(1), pp. 253–258. [PubMed]
Hamstad, M. A. , 2010, “ Frequencies and Amplitudes of AE Signals in a Plate as a Function of Source Rise Time,” 29th European Conference on Acoustic Emission Testing, Vienna, Austria, Sept. 8–10, pp. 1–8.
Sause, M. G. , Hamstad, M. A. , and Horn, S. , 2013, “ Finite Element Modeling of Lamb Wave Propagation in Anisotropic Hybrid Materials,” Composites, Part B, 53, pp. 249–257.
Michaels, J. E. , Michaels, T. E. , and Sachse, W. , 1981, “ Applications of Deconvolution to Acoustic Emission Signal Analysis,” Mater. Eval., 39(11), pp. 1032–1036.
Hsu, N. N. , Simmons, J. A. , and Hardy, S. C. , 1978, “ Approach to Acoustic Emission Signal Analysis-Theory and Experiment,” Nondestructive Evaluation, La Jolla, CA, July 17–21, p. 31.
Pao, Y. H. , 1978, “ Theory of Acoustic Emission,” Transactions of the 23rd Conference of Army Mathematicians, Hampton, VA, May 11–13, p. 389.
Ohtsu, M. , 1995, “ Acoustic Emission Theory for Moment Tensor Analysis,” J. Res. Nondestr. Eval., 6(3), pp. 169–184.
Fischer-Cripps, A. C. , 2000, Introduction to Contact Mechanics, Springer, New York.
Haider, M. F. , and Giurgiutiu, V. , 2017, “ Full Derivation of the Helmholtz Potential Approach to the Analysis of Guided Wave Generation during Acoustic Emission Events,” University of South Carolina, Columbia, SC, Report No. USC-ME-LAMSS-2001-101.
Wolski, A. , 2011, “ Theory of Electromagnetic Fields,” CAS - CERN Accelerator School: RF for Accelerators, Ebeltoft, Denmark, June 8–17, Paper No. 15.
Jackson, J. D. , 1999, Classical Electrodynamics, Perseus Books, Reading, MA.
Uman, M. A. , McLain, D. K. , and Krider, E. P. , 1975, “ The Electromagnetic Radiation From a Finite Antenna,” Am. J. Phys., 43(1), pp. 33–38.
Jensen, F. B. , Kuperman, W. A. , Porter, M. B. , and Schmidt, H. , 1994, Computational Ocean Acoustics, American Institute of Physics, Woodbury, NY.
Remmert, R. , 2012, Theory of Complex Functions, Vol. 122, Springer Science & Business Media, New York.
Cohen, H. , 2010, Complex Analysis With Applications in Science and Engineering, Springer Science & Business Media, New York.
Krantz, S. G. , 2007, Complex Variables: A Physical Approach With Applications and MATLAB, CRC Press, Boca Raton, FL.
Watanabe, K. , 2014, Integral Transform Techniques for Green’s Function, Springer, Cham, Switzerland. [PubMed] [PubMed]
Haider, M. F. , Giurgiutiu, V. , Lin, B. , and Yu, Y. , 2017, “ Simulation of Lamb Wave Propagation Using Excitation Potentials,” ASME Paper No. PVP2017-66074.
View article in PDF format.

## References

Harris, D. O. , and Dunegan, H. L. , 1974, “ Continuous Monitoring of Fatigue-Crack Growth by Acoustic-Emission Techniques,” Exp. Mech., 14(2), pp. 71–81.
Han, B. H. , Yoon, D. J. , Huh, Y. H. , and Lee, Y. S. , 2014, “ Damage Assessment of Wind Turbine Blade Under Static Loading Test Using Acoustic Emission,” J. Intell. Mater. Syst. Struct., 25(5), pp. 621–630.
Tandon, N. , and Choudhury, A. , 1999, “ A Review of Vibration and Acoustic Measurement Methods for the Detection of Defects in Rolling Element Bearings,” Tribol. Int., 32(8), pp. 469–480.
Roberts, T. , and Talebzadeh, M. , 2003, “ Acoustic Emission Monitoring of Fatigue Crack Propagation,” J. Constr. Steel Res., 59(6), pp. 695–712.
Bassim, M. N. , Lawrence, S. S. , and Liu, C. D. , 1994, “ Detection of the Onset of Fatigue Crack Growth in Rail Steels Using Acoustic Emission,” Eng. Fract. Mech., 47(2), pp. 207–214.
Lamb, H. , 1917, “ On Waves in an Elastic Plate,” Proc. R. Soc. London A: Math., Phys. Eng. Sci., 93(648), pp. 114–128.
Helmholtz, H. , 1858, “ Uber Integrale der Hydrodynamischen Gleichungen, Welche den Wirbelbewegungen Entsprechen,” J. Reine Angew. Math., 1858(55), pp. 25–55.
Achenbach, J. D. , 2003, Reciprocity in Elastodynamics, Cambridge University Press, Cambridge, UK.
Giurgiutiu, V. , 2014, Structural Health Monitoring With Piezoelectric Wafer Active Sensors, 2nd ed., Elsevier, Amsterdam, The Netherlands.
Graff, K. F. , 1975, Wave Motion in Elastic Solids, Clarendon Press, Oxford, UK.
Viktorov, I. A. , 1967, Rayleigh and Lamb Waves: Physical Theory and Applications, Plenum Press, New York.
Landau, L. D. , and Lifschitz, E. M. , 1965, Teoriya Uprugosti, Nauka, Moscow, Russia [Theory of Elasticity, 2nd ed., Pergamon Press, Oxford, UK (1970)].
Love, A. E. H. , 1944, A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York.
Aki, K. , and Richards, P. G. , 2002, Quantitative Seismology, Vol. 1, University Science Books, Sausalito, CA.
Vvedenskaya, A. V. , 1956, “ The Determination of Displacement Fields by Means of Dislocation Theory,” Izv. Akad. Nauk SSSR, 3(27), pp. 227–284.
Nabarro, F. R. N. , 1951, “ The Synthesis of Elastic Dislocation Fields,” Philos. Mag., 42(334), p. 313.
Maruyama, T. , 1963, “ On the Force Equivalents of Dynamical Elastic Dislocations With Reference to the Earthquake Mechanism,” Bull. Earthquake Res. Inst., Tokyo Univ., 41, pp. 467–486.
Lamb, H. , 1903, “ On the Propagation of Tremors Over the Surface of an Elastic Solid,” Proc. R. Soc. London, 72(477–486), pp. 128–130.
Rice, J. R. , 1980, “ Elastic Wave Emission From Damage Processes,” J. Nondestr. Eval., 1(4), pp. 215–224.
Miklowitz, J. , 1962, “ Transient Compressional Waves in an Infinite Elastic Plate or Elastic Layer Overlying a Rigid Half-Space,” ASME J. Appl. Mech., 29(1), pp. 53–60.
Weaver, R. L. , and Pao, Y.-H. , 1982, “ Axisymmetric Elastic Waves Excited by a Point Source in a Plate,” ASME J. Appl. Mech., 49(4), pp. 821–836.
Ono, K. , and Ohtsu, M. , 1984, “ A Generalized Theory of Acoustic Emission and Green’s Functions in a Half Space,” J. Acoust. Emiss., 3, pp. 27–40.
Ohtsu, M. , and Ono, K. , 1986, “ The Generalized Theory and Source Representations of Acoustic Emission,” J. Acoust. Emiss., 5(4), pp. 124–133.
Johnson, L. R. , 1974, “ Green’s Function for Lamb’s Problem,” Geophys. J. Int., 37(1), pp. 99–131.
Roth, F. , 1990, “ Subsurface Deformations in a Layered Elastic Half-Space,” Geophys. J. Int., 103(1), pp. 147–155.
Bai, H. , Zhu, J. , Shah, A. H. , and Popplewell, N. , 2004, “ Three-Dimensional Steady State Green Function for a Layered Isotropic Plate,” J. Sound Vib., 269(1), pp. 251–271.
Liu, G. R. , and Achenbach, J. D. , 1995, “ Strip Element Method to Analyze Wave Scattering by Cracks in Anisotropic Laminated Plates,” ASME J. Appl. Mech., 62(3), pp. 607–613.
Jacobs, L. J. , Scott, W. R. , Granata, D. M. , and Ryan, M. J. , 1991, “ Experimental and Analytical Characterization of Acoustic Emission Signals,” J. Nondestr. Eval., 10(2), pp. 63–70.
Ono, K. , 2011, “ Acoustic Emission in Materials Research—A Review,” J. Acoust. Emiss., 29, pp. 284–309.
Wisner, B. , Cabal, M. , Vanniamparambil, P. A. , Hochhalter, J. , Leser, W. P. , and Kontsos, A. , 2015, “ In Situ Microscopic Investigation to Validate Acoustic Emission Monitoring,” Exp. Mech., 55(9), pp. 1705–1715.
Momon, S. , Moevus, M. , Godin, N. , R’Mili, M. , Reynaud, P. , Fantozzi, G. , and Fayolle, G. , 2010, “ Acoustic Emission and Lifetime Prediction During Static Fatigue Tests on Ceramic-Matrix-Composite at High Temperature Under Air,” Composites, Part A, 41(7), pp. 913–918.
Haider, M. F. , Giurgiutiu, V. , Lin, B. , and Yu, L. , 2017, “ Irreversibility Effects in Piezoelectric Wafer Active Sensors After Exposure to High Temperature,” Smart Mater. Struct., 26(9), p. 095019.
Cuadra, J. A. , Baxevanakis, K. P. , Mazzotti, M. , Bartoli, I. , and Kontsos, A. , 2016, “ Energy Dissipation Via Acoustic Emission in Ductile Crack Initiation,” Int. J. Fract., 199(1), pp. 89–104.
Khalifa, W. B. , Jezzine, K. , Grondel, S. , Hello, G. , and Lhémery, A. , 2012, “ Modeling of the Far-Field Acoustic Emission From a Crack Under Stress,” J. Acoust. Emiss., 30, pp. 137–152.
Hamstad, M. A. , O’Gallagher, A. , and Gary, J. , 1999, “ Modeling of Buried Monopole and Dipole Sources of Acoustic Emission With a Finite Element Technique,” J. Acoust. Emiss., 17(3–4), pp. 97–110.
Hill, R. , Forsyth, S. A. , and Macey, P. , 2004, “ Finite Element Modelling of Ultrasound, With Reference to Transducers and AE Waves,” Ultrasonics, 42(1), pp. 253–258. [PubMed]
Hamstad, M. A. , 2010, “ Frequencies and Amplitudes of AE Signals in a Plate as a Function of Source Rise Time,” 29th European Conference on Acoustic Emission Testing, Vienna, Austria, Sept. 8–10, pp. 1–8.
Sause, M. G. , Hamstad, M. A. , and Horn, S. , 2013, “ Finite Element Modeling of Lamb Wave Propagation in Anisotropic Hybrid Materials,” Composites, Part B, 53, pp. 249–257.
Michaels, J. E. , Michaels, T. E. , and Sachse, W. , 1981, “ Applications of Deconvolution to Acoustic Emission Signal Analysis,” Mater. Eval., 39(11), pp. 1032–1036.
Hsu, N. N. , Simmons, J. A. , and Hardy, S. C. , 1978, “ Approach to Acoustic Emission Signal Analysis-Theory and Experiment,” Nondestructive Evaluation, La Jolla, CA, July 17–21, p. 31.
Pao, Y. H. , 1978, “ Theory of Acoustic Emission,” Transactions of the 23rd Conference of Army Mathematicians, Hampton, VA, May 11–13, p. 389.
Ohtsu, M. , 1995, “ Acoustic Emission Theory for Moment Tensor Analysis,” J. Res. Nondestr. Eval., 6(3), pp. 169–184.
Fischer-Cripps, A. C. , 2000, Introduction to Contact Mechanics, Springer, New York.
Haider, M. F. , and Giurgiutiu, V. , 2017, “ Full Derivation of the Helmholtz Potential Approach to the Analysis of Guided Wave Generation during Acoustic Emission Events,” University of South Carolina, Columbia, SC, Report No. USC-ME-LAMSS-2001-101.
Wolski, A. , 2011, “ Theory of Electromagnetic Fields,” CAS - CERN Accelerator School: RF for Accelerators, Ebeltoft, Denmark, June 8–17, Paper No. 15.
Jackson, J. D. , 1999, Classical Electrodynamics, Perseus Books, Reading, MA.
Uman, M. A. , McLain, D. K. , and Krider, E. P. , 1975, “ The Electromagnetic Radiation From a Finite Antenna,” Am. J. Phys., 43(1), pp. 33–38.
Jensen, F. B. , Kuperman, W. A. , Porter, M. B. , and Schmidt, H. , 1994, Computational Ocean Acoustics, American Institute of Physics, Woodbury, NY.
Remmert, R. , 2012, Theory of Complex Functions, Vol. 122, Springer Science & Business Media, New York.
Cohen, H. , 2010, Complex Analysis With Applications in Science and Engineering, Springer Science & Business Media, New York.
Krantz, S. G. , 2007, Complex Variables: A Physical Approach With Applications and MATLAB, CRC Press, Boca Raton, FL.
Watanabe, K. , 2014, Integral Transform Techniques for Green’s Function, Springer, Cham, Switzerland. [PubMed] [PubMed]
Haider, M. F. , Giurgiutiu, V. , Lin, B. , and Yu, Y. , 2017, “ Simulation of Lamb Wave Propagation Using Excitation Potentials,” ASME Paper No. PVP2017-66074.

## Figures

Fig. 1

(a)–(d) Plate with existing crack length 2a, energy release rate with crack length, time rate of energy from a crack, and total released energy from a crack

Fig. 2

Plate of thickness 2d in which straight-crested Lamb waves (P + SV) propagate in the x direction due to concentrated potentials

Fig. 3

AE propagation and detection by a sensor installed on a structure

Fig. 4

Excitation profile: (a) time rate plot and (b) cumulative plot

Fig. 5

Lamb waves (S0 and A0 modes) and bulk waves propagation at 500 mm distance in 6 mm 304-steel plate for (a) pressure potential excitation and (b) shear potential excitation (peak time = 3 μs) located on top surface

Fig. 6

Lamb waves (S0 and A0 modes) and bulk waves propagation at 500 mm distance in 6 mm 304-steel plate for (a) pressure potential excitation and (b) shear potential excitation (peak time = 3 μs) located at 1.5 mm depth from top surface

Fig. 7

Lamb waves (S0 and A0 modes) and bulk waves propagation at 500 mm distance in 6 mm 304-steel plate for (a) pressure potential excitation and (b) shear potential excitation (peak time = 3 μs) located at midplane

Fig. 8

Higher-order Lamb waves (S1 and A1) modes at 500 mm distance in 6 mm 304-steel plate for (a) pressure potential excitation and (b) shear potential excitation (peak time = 3 μs) located at midplane

Fig. 9

(a) Effect of pressure excitation potential and (b) effect of shear excitation potential: variation of out-of-plane displacement (S0, A0, and bulk wave) with propagation distance in 6 mm 304-steel plate for source (peak time = 3 μs) located at the midplane

## Tables

Table 1 Comparison of peak amplitudes of Lamb waves (S0 and A0 modes) and bulk waves for different source depths

## Discussions

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