Modeling Acoustic Emission Due to an Internal Point Source in Circular Cylindrical Structures

: In one of our previous publications, we developed the ﬁrst mathematical model for acoustic emission from an internal point source in a transversely isotropic cylinder. The point source, as an internal defect, is the most fundamental source generating AE in homogeneous media; it is represented by a spatiotemporal concentrated force and generates three scalar potentials for compressional, and horizontally and vertically polarized shear waves. The mathematical formulas for the displacements were derived by introducing the concentrated force-incorporated potentials into the Navier–Lam é equation. Since the publication of that paper, we detected some errors. In this paper, we correct the errors and extend the analytical modeling to a cylindrical shell structure. For acoustic emission in general circular cylindrical structures, we derived solutions by applying the boundary conditions at inner and outer surfaces of the structures. Under these conditions, we solve the radial, tangential, and axial displacement ﬁelds. Analytical simulations of the acoustic emission were carried out at several point source locations for circular cylindrical geometries. We show that the maximum amplitude of the axial displacement is dependent on the point source position and 2 π -aperiodicity of the cylindrical geometry. Our mathematical formulas are very useful for characterizing AE features generated from an internal defect source in cylindrical geometries.


Introduction
Many conventional techniques applied in nondestructive testing are based on an active mode, in which testing loadings are applied during testing to deliver signals or energy from the outside to the test body. In contrast, the acoustic emission (AE) technique is a passive method that does not require the application of external energy to the test structure, as AE is generated by a material as a result of a sudden release of energy (other than heat) from localized sources within the solid, in turn due to a failure of lattice vibrations in materials. AE sources are usually classified as primary or secondary. Primary AE sources include material degradations related to deformation and fracture development, whereas leak, flow, and the fabrication process are secondary AE sources. Due to this unique characteristic, AE techniques are uniquely applicable to structure health monitoring (SHM) [1][2][3][4].
Among the primary AE sources, crack formation and growth are the most important practical nondestructive testing (NDT) because the detecting and monitoring of these failures can prevent or slow further damage. Thus, in SHM, the point source (PS) is adapted as an AE excitation source; for example in seismic displacements, crack fracture and cleavage, and concentrated vertical step forces [5][6][7][8][9][10]. The displacements generated by a PS were first introduced by Stokes [11]. Later, Achenbach presented mathematical formulas for displacements in spherical geometry by defining it as a concentrated force (CF) loaded at a point [12]. In mathematical formulas, Helmholtz potentials for the displacements were derived in terms of scalar potentials generated by PS excitation in an infinite domain [12][13][14]. Although the AE generated by the PS is important for characterizing real signals observed

Green's Function
In Ref. [28], the CF as the internal PS located at x 0 is formulated in terms of an oscillating impulse with natural frequencies of the material in a given geometry, as follows: where P is the force vector and ω is the predominant angular frequency (ω = 2πν) of AE. Dirac's delta function δ(x − x 0 ) provides a method for solving spatial problems dealing with a PS. Green's function g(x; x 0 ), defined as Is the solution for the delta function in elastodynamics, and provides the spatial distribution of the CF at a given time. In cylindrical coordinates, Equation (2) is expressed as ∇ 2 [g r (r; r 0 )g θ (θ; θ 0 )g z (z; z 0 )] = δ(r − r 0 )δ(θ − θ 0 )δ(z − z 0 ) r (3) where ∇ 2 is the Laplacian in cylindrical coordinates, and the CF is located at r 0 , θ 0 , and z 0 . At any point in the cylindrical domain except the CF locating point, Equation (3) becomes zero, ∇ 2 [g r (r; r 0 )g θ (θ; θ 0 )g z (z; z 0 )] = 0.
In cylindrical (r, θ, z) coordinates, Equation (4) can be rewritten as r 2 g r ∂ 2 g r ∂r 2 + r g r ∂g r ∂r Substituting the following relations 1 g z ∂ 2 g z ∂z 2 = κ 2 z , 1 g θ ∂ 2 g θ ∂θ 2 = −ν 2 , Into the above PDE results, we obtain r 2 ∂ 2 g r ∂r 2 + r ∂g r ∂r + κ 2 z r 2 − v 2 g r = 0 Note that we selected exponential, rather than oscillating, solutions in the z-direction. This implies that the radial solutions are appropriate for the particular set of boundary conditions under consideration. In ref. [28], we obtained the axial, tangential, and radial components of Green's function under certain conditions, such as the continuity and discontinuity principles, angular symmetry for the cylindrical domain with length l and a radius of a. The inner radius b is introduced to the shell structure as an additional radial boundary condition.
In Equation (7), J v is a Bessel function of the first kind of v-th order. The Bessel function of the second kind is excluded because there is no singularity in the cylindrical domain. The value of κ z can be obtained by applying the boundary conditions at the outer or inner surfaces of the cylindrical shell to Equation (7): As an alternative, the boundary condition g r (r; r 0 )| r=a = 0 was applied [30]. However, this condition resulted in a discontinuous and asymmetric Green's function. Selecting the first root (r v1 ) of the Bessel function, r v1 = κ z (a − r 0 ) or r v1 = κ z (r 0 − b). These relations give For the outer surface, κ z = r v1 (a−r 0 ) . Introducing a parameter, A ν1 , we obtain Green's function: In ref. [28], the parameter A v1 was determined by integrating the delta function in the cylindrical domain. The integrations of δ(θ − θ 0 )δ(z − zs 0 ) for the cylindrical shell are the same as those for the cylinder, because these two delta functions are independent of r. The remaining problem is to integrate Multiplying both sides of Equation (10) by J p (κ z r) rdr and integrating over (b, a) gives the following: The integrations on the left side of the above equation can be divided into two parts: Applying the normalization and orthogonality principles of Bessel functions, And the following concepts of the delta function, To Equation (11), we obtain the constant For the cylindrical problem, the corrected value of A v1 is given in (A1). Introducing Equation (12) to Equation (9) gives the Green's function for the Kronecker delta function: where Since the CF direction is determined by the P vector, Green's function is assumed to be azimuthally independent (v = 0). In addition, it is very convenient to take the location of the PS as the new origin and introduce relative coordinates to the location of the PS, defined as ξ = r − r 0 , ϑ = θ − θ 0 , and η = z − z 0 , where ξ 0 = 0, ϑ 0 = 0, and η 0 = 0 (hereafter, Appl. Sci. 2022, 12, 12032 5 of 19 referred to as the PS-oriented (ξ, ϑ, η) coordinate system), Equations (13) and (14) can be rewritten as In Equation (15), the value of ξ at a given point is the shortest distance from the PS. In cylindrical geometry, For the shell geometry, the calculation of ξ is somewhat complicated due to its hollow interior. No linear distance exists between the two points across the hollow interior. In Figure 1b, an arc connecting the PS and a given point is introduced. This connection should not intersect with the hollow circle. The arc rises at a constant rate between r 0 and (r p < r 0 ), given as R = r P −r 0 where θ is the angle between the PS and a point P. If the angle, dθ, between r 1 and r 2 is infinitesimal, the arc length, dς, connecting the two points, becomes a line. Applying the cosine rule to the triangle, of the PS as the new origin and introduce relative coordinates to the location of the PS, defined as = − 0 , = − 0 , and = − 0 , where 0 = 0 , 0 = 0 , and 0 = 0 (hereafter, referred to as the PS-oriented ( , , ) coordinate system), Equations (13) and (14) can be rewritten as In Equation (15), the value of at a given point is the shortest distance from the PS. In cylindrical geometry, For the shell geometry, the calculation of is somewhat complicated due to its hollow interior. No linear distance exists between the two points across the hollow interior. In Figure 1b, an arc connecting the PS and a given point is introduced. This connection should not intersect with the hollow circle. The arc rises at a constant rate between 0 and ( 0 < ), given as = − 0 ( 0 = √ 0 2 + 0 2 , = √ 2 + 2 ), where θ is the angle between the PS and a point P. If the angle, , between 1 and 2 is infinitesimal, the arc length, , connecting the two points, becomes a line. Applying the cosine rule to the triangle,  Assuming that cos(dθ) ≈ 1 − (dθ) 2 2 and (dθ) 3 ≈ 0, the above equation becomes (18) Figure 2 shows the Green's function calculated for the cylinder (a = 0.5 m, b = 0) and shell (a = 0.5 m, b = 0.4 m). It can be shown that the calculated functions are continuous and symmetric with respect to the PS location. and ( ) 3 ≈ 0, the above equation becomes (18) Figure 2 shows the Green's function calculated for the cylinder (a = 0.5 m, b = 0) and shell (a = 0.5 m, b = 0.4 m). It can be shown that the calculated functions are continuous and symmetric with respect to the PS location.

Displacement Fields Generated by a Point Source
The NL equation for the displacement field u in an elastic and homogeneous medium subject to a local body force f is given by [31] where λ and μ are the Lamé constants, and is the density of the media. The displacement field u in cylindrical coordinates proposed by Morse and Freshbach involves three potential functions for the P wave, ̂ for the SH wave, and ̂ for the SV wave. The representation of u for cylindrical geometries is given by where a is the radius of the cylinder. For the displacement field generated by the force vector P due to an intrinsic point defect, the three potential functions , , and are correlated with the force vector P (referred to as CFIPs). CFIPs in the PS-oriented ( , , ) Cartesian coordinates are defined as

Displacement Fields Generated by a Point Source
The NL equation for the displacement field u in an elastic and homogeneous medium subject to a local body force f is given by [31] where λ and µ are the Lamé constants, and ρ is the density of the media. The displacement field u in cylindrical coordinates proposed by Morse and Freshbach involves three potential functions Φ for the P wave, Xê z for the SH wave, and Ψê z for the SV wave. The representation of u for cylindrical geometries is given by where a is the radius of the cylinder. For the displacement field generated by the force vector P due to an intrinsic point defect, the three potential functions Φ, X, and Ψ are correlated with the force vector P (referred to as CFIPs). CFIPs in the PS-oriented ξ i , ξ j , η Cartesian coordinates are defined as where φ, χ, and ψ are scalar functions for P, SH, and SV waves, respectively. These scalar functions are expressed by Combining Equations (19)- (23) with Equations (1) and (2), and replacing the Lamé constants and ρ by the longitudinal wave speed c P = λ + 2µ/ρ and transverse wave speed c S = µ/ρ leads to where k p = ω c p and k s = ω c s , corresponding to the angular wavenumbers of the P and the S waves, respectively. The solutions of these PDEs in the PS-oriented (ξ, ϑ, η) cylindrical coordinates are as follows: where The corrections for the particular solutions in Ref. [28] are given in Appendix A.
In the same manner as in ref. [28], the force vectors Ps acting in the radial P j and the axial (P z ) directions are introduced to solve for Φ, Xê z , and Ψê z . By substituting Equation (27) into Equation (21) with coordination conversion, we obtain the CFIP for the P wave In Equation (31), where ϕ is the angle between an observation point and the x i axis. Similarly, we obtain the CFIP for the SH wave where The CFIP for the SV wave is given by All three CFIPs for the P, SH, and SV waves were completely determined in the given CF direction. In ref. [28], detailed derivation of the displacement components for the cylindrical geometry are described in terms of Φ, X, and Ψ. For the cylindrical geometries, applying gradient, divergence, and curl operators, Equation (20) results in the displacement components in the (r, θ, z; r 0 , θ 0 , z 0 ) coordinates as where u r , u θ , and u z are the radial, tangential, and axial displacements, respectively. In the (ξ, ϑ, η) coordinates, these displacement components are as follows: Substituting Equations (30), (33), and (36) into Equations (38)-(40) gives the displacement d component due to P j as Notably, the component F 4 d f (f = j or z) is obtained from the particular solutions for P and SH potentials associated with Green's function. In this paper, the components F 4 d f in ref. [28] is corrected.
For the radial component u rj , For the tangential component u θj For the axial component u zj , Similarly, the displacement d due to P z is expressed by For the radial component u rz , For the tangential component u θz , For the axial component u zz , As expressed by Equations (41) and (54), the displacement components involve the coupling constants A m , B m , and C m . These constants can be determined directly by applying a fundamental set of linear elastic boundary problems. The outer and inner surfaces of the cylindrical shell studied in the present paper are stress-free. Thus, the following stress components are zero at ξ i and ξ j , satisfying (ξ i + x 0i ) 2 + (ξ i + x 0i ) 2 = a for the outer circumference and (ξ i + x 0i ) 2 + (ξ i + x 0i ) 2 = b for the inner circumference in Equation (17) for the shell In ref. [28], by using the stress displacement relations, we obtained a system of linear algebraic equations for the TIC, given by where f = j for P j and f = z for P z . For P j , all elements in Equation (64) are nonzero, as given by For P z , For P z , by substituting Equations (77)-(85) into Equation (64), we derive Equation (86) allows us to obtain the value of κ z at a given point on the circumference. The only remaining task to complete the displacement fields is to introduce retardation times into the CF P in Equation (1). The CF is an impulsive force acting at the PS-oriented origin ξ i = 0, ξ j = 0, and η = 0 at t = 0. The arrival time τ of the signal at position ξ i , ξ j , η must be considered. The arrival times of the P and S waves propagating with velocities c P and c S are given as respectively, where ς is given by Equation (18). Introducing P 0 and b parameters, determining the amplitude and duration of the wave, respectively, yields the CF acting in the f direction as In the simulation, P 0 =1.0 × 10 10 N s −1 and b = 1.0 × 10 −5 s −1 . Finally, the displacements generated by P f can be obtained as For practical purposes, the P, SH, and SV waves are introduced as

Simulations
In ref. [28], simulations of the displacement fields in TIC were confined to the case for the azimuthal independence (m = 0) of wave propagation. In this paper, we extended the simulation to the case of azimuthally dependent tangential displacements, i.e., 2π-aperiodic solutions. Stainless steel cylindrical structures (a = 0.50 m, l = 2.0 m, ρ = 7.80 × 10 3 kg/m 3 , c P = 5.98 km/s, c s = 3.30 km/s, and ν = 155.4 kHz) were used as the test specimens. First, we determined k η from Equation (86), resulting in two solutions of A mz .
On the outer circumference, k z = r v1 (a−r 0 ) , the first root of the function Was solved at a given PS location as a function of η (= l − z 0 ) and the shortest distance ς o from the PS to a given point (0.5 m, 45 • , η) on the outer circumference of the cylindrical structures. Figure 3 shows the dependence of k η /π on η for the cylinder (b = 0) and shell (b = 0.4 m). As shown in the figure, the k η values are independent of the inner diameter b. When PS is located on a radial axis, the dependence of the η-dependency of k η values is very simple: all values are divided into two groups with even m and odd m. It should be noted that the k η values are almost independent of ϑ.

Simulations
In ref. [28], simulations of the displacement fields in TIC were confined to the case for the azimuthal independence ( = 0) of wave propagation. In this paper, we extended the simulation to the case of azimuthally dependent tangential displacements, i.e., 2πaperiodic solutions.
Was solved at a given PS location as a function of η (= − 0 ) and the shortest distance from the PS to a given point (0.5 m, 45°, η) on the outer circumference of the cylindrical structures. Figure 3 shows the dependence of / on η for the cylinder (b = 0) and shell (b = 0.4 m). As shown in the figure, the values are independent of the inner diameter b. When PS is located on a radial axis, the dependence of the η-dependency of values is very simple: all vales are divided into two groups with even m and odd m. It should be noted that the values are almost independent of . Using these results, we simulated the displacement fields at the outer surfaces of the cylinder (b = 0 m) and cylindrical shell (b = 0.4 m). Figure 4 shows the 2π-periodic (m = 0) displacements and their wave properties at the (0.5 m, 45°, 1 m) position, generated by the PS located at the center of the cylinder ( 0 = 0, 0 = 0, 0 = 1 m). The Pj and Pz excitations produce an axial displacement stronger than the radial and tangential displacements. For the Pj excitation, the displacement results in the P wave are the main wave, with a minor SH wave and very weak SV wave. In Figure 4, the displacement amplitudes generated by Pz excitation differ significantly from those generated by the Pj excitation, in which the Pz excitation produces only the P wave. For the Pj excitation, the maximum values of and at the (0.5 m, 45°, 1 m) position were 6.7 and 17.4 nm, respectively, while for the Pz excitation they were 22.1 and 127.0 nm, respectively. The amplitudes of Using these k η results, we simulated the displacement fields at the outer surfaces of the cylinder (b = 0 m) and cylindrical shell (b = 0.4 m). Figure 4 shows the 2π-periodic (m = 0) displacements and their wave properties at the (0.5 m, 45 • , 1 m) position, generated by the P f PS located at the center of the cylinder x 0i = 0, x 0j = 0, z 0 = 1 m . The P j and P z excitations produce an axial displacement stronger than the radial and tangential displacements. For the P j excitation, the displacement results in the P wave are the main wave, with a minor SH wave and very weak SV wave. In Figure 4, the displacement amplitudes generated by P z excitation differ significantly from those generated by the P j excitation, in which the P z excitation produces only the P wave. For the P j excitation, the maximum values of u rj and u zj at the (0.5 m, 45 • , 1 m) position were 6.7 and 17.4 nm, respectively, while for the P z excitation they were 22.1 and 127.0 nm, respectively. The amplitudes of the displacements due to P z excitation are much stronger than those due to the P j excitation. the displacements due to Pz excitation are much stronger than those due to the Pj excitation. The angular dependence of the displacement was also calculated as a function of m (m = 0, 1, and 2), as shown in Figure 5. For m = 0, when the PS is located at the center of the circular plane, the angular dependences of , , and arise only from  and , defined in Equations (32) and (35), respectively. For Pz excitation, the displacements of and are free from these factors. When the distances from the PS to the circumference are not equivalent, the angular dependences of the radial and axial displacements are highly significant. These effects are due not only to  and , but also to the superposition of the Bessel functions involved in the displacement Equations. When m is nonzero, additional azimuthal factors, m, cos( ), and sin( ), result in complex angular dependency. It should be noted that at a certain angle, some values of , , and in Equation (64) become too small to cause the sudden increase in displacement, hereafter referred to as "computational divergency".  The angular dependence of the displacement was also calculated as a function of m (m = 0, 1, and 2), as shown in Figure 5. For m = 0, when the PS is located at the center of the circular plane, the angular dependences of u rj , u zj , and u tj arise only from Ξ and Σ, defined in Equations (32) and (35), respectively. For P z excitation, the displacements of u rz and u zz are free from these factors. When the distances from the PS to the circumference are not equivalent, the angular dependences of the radial and axial displacements are highly significant. These effects are due not only to Ξ and Σ, but also to the superposition of the Bessel functions involved in the displacement Equations. When m is nonzero, additional azimuthal factors, m, cos(mϑ), and sin(mϑ), result in complex angular dependency. It should be noted that at a certain angle, some a ij values of u rj , u zj , and u rz in Equation (64) become too small to cause the sudden increase in displacement, hereafter referred to as "computational divergency". Figure 6 shows the angular dependence of a relatively strong u zz generated by two PSs located at x 0i = 0, x 0j = 0.45 m and z 0 = 1 m, and x 0i = 0.35 m, x 0j = 0.35 m and z 0 = 1 m. For the first PS, the distance from the PS to the point at ξ i = 0, ξ j = 0.5 m, and η = 0 on the outer surface is the shortest with ς = 0.05 m.  Remarkably, P z excitation of 1 N produces a P wave with a maximum amplitude of tens of centimeters.   Similarly, the displacement fields and their wave properties excited by the PS were also calculated as a function of m (m = 0, 1, and 2) for the shell (a = 0.5 m, b = 0.4 m). Figure 7 shows the angular dependences of the axial component u zz generated by two PSs located at x 0i = 0, x 0j = 0.45 m and z 0 = 1 m, and x 0i = x 0j = 0.35 m and z 0 = 1 m. There is no difference in the spectral features of the corresponding angular dependences between the cylinder and shell geometries. However, the maximum amplitudes of the shell are much smaller than those of the cylinder. Figure 8 shows 2π-periodic (m = 0) displacements at the (0.5 m, 45 • , 1 m) position. For the shell, the most striking feature of the displacement fields is the tangential displacement u tj due to P j excitation, the amplitude of which is comparable to that of the axial displacement u zj , as shown in Figure 8a.
Similarly, the displacement fields and their wave properties excited by the PS w also calculated as a function of m (m = 0, 1, and 2) for the shell (a = 0.5 m, b = 0.4 m). Fig  7 shows the angular dependences of the axial component generated by two PS cated at 0 = 0, 0 = 0.45 m and 0 = 1 m, and 0 = 0 = 0.35 m and 0 = 1 There is no difference in the spectral features of the corresponding angular dependen between the cylinder and shell geometries. However, the maximum amplitudes of shell are much smaller than those of the cylinder. Figure 8 shows 2π-periodic (m = 0) placements at the (0.5 m, 45°, 1 m) position. For the shell, the most striking feature of displacement fields is the tangential displacement due to Pj excitation, the amplit of which is comparable to that of the axial displacement , as shown in Figure 8a.

Conclusions
In this paper, we provided a mathematical model for AE generated by an intrinsic PS in cylindrical geometries, including a cylinder and shell. As an internal crack, the PS produces CFIPs for P, SH, and SV waves. Introducing CFIPs into the NL equation provides solutions for the radial, tangential, and axial displacements, involving azimuthal functions in cylindrical geometries. The main advantage of our model is that it provides an exact solution for the AE features from PS generation, propagation, and reception in cylindrical geometries. In conjunction with experimental data, this mathematical model can be used for NDT of cylindrical structures.

Conclusions
In this paper, we provided a mathematical model for AE generated by an intrinsic PS in cylindrical geometries, including a cylinder and shell. As an internal crack, the PS produces CFIPs for P, SH, and SV waves. Introducing CFIPs into the NL equation provides solutions for the radial, tangential, and axial displacements, involving azimuthal functions in cylindrical geometries. The main advantage of our model is that it provides an exact solution for the AE features from PS generation, propagation, and reception in cylindrical geometries. In conjunction with experimental data, this mathematical model can be used for NDT of cylindrical structures.

Conflicts of Interest:
The authors declare no conflict of interest.