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

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Green’s Function

**P**is the force vector and ω is the predominant angular frequency $\left(\omega =2\pi \nu \right)$ of AE. Dirac’s delta function $\delta \left(\mathit{x}-{\mathit{x}}_{0}\right)$ provides a method for solving spatial problems dealing with a PS. Green’s function $g\left(\mathit{x};{\mathit{x}}_{0}\right)$, defined as

_{ν1}, we obtain Green’s function:

**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 $\xi =r-{r}_{0}$, $\vartheta =\theta -{\theta}_{0}$, and $\eta =z-{z}_{0}$, where ${\xi}_{0}=0$, ${\vartheta}_{0}=0$, and ${\eta}_{0}=0$ (hereafter, referred to as the PS-oriented $\left(\xi ,\vartheta ,\eta \right)$ coordinate system), Equations (13) and (14) can be rewritten as

## 3. Displacement Fields Generated by a Point Source

**u**in an elastic and homogeneous medium subject to a local body force

**f**is given by [31]

**u**in cylindrical coordinates proposed by Morse and Freshbach involves three potential functions $\Phi $ for the P wave, ${\rm X}{\widehat{\mathit{e}}}_{z}$ for the SH wave, and $\Psi {\widehat{\mathit{e}}}_{z}$ for the SV wave. The representation of

**u**for cylindrical geometries is given by

**P**due to an intrinsic point defect, the three potential functions $\Phi $, ${\rm X}$, and $\Psi $ are correlated with the force vector

**P**(referred to as CFIPs). CFIPs in the PS-oriented $\left({\xi}_{i},{\xi}_{j},\eta \right)$ Cartesian coordinates are defined as

**P**s acting in the radial $\left({P}_{j}\right)$ and the axial $\left({P}_{z}\right)$ directions are introduced to solve for $\mathsf{\Phi},{\rm X}{\widehat{\mathit{e}}}_{z}$, and $\Psi {\widehat{\mathit{e}}}_{z}$. By substituting Equation (27) into Equation (21) with coordination conversion, we obtain the CFIP for the P wave

_{j}as

_{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 ${\xi}_{i}$ and ${\xi}_{j}$, satisfying $\sqrt{{\left({\xi}_{i}+{x}_{0i}\right)}^{2}+{\left({\xi}_{i}+{x}_{0i}\right)}^{2}}=a$ for the outer circumference and $\sqrt{{\left({\xi}_{i}+{x}_{0i}\right)}^{2}+{\left({\xi}_{i}+{x}_{0i}\right)}^{2}}=b$ for the inner circumference in Equation (17) for the shell

_{j}and f = z for P

_{z}.

**P**in Equation (1). The CF is an impulsive force acting at the PS-oriented origin ${\xi}_{i}=0,{\xi}_{j}=0$, and $\eta =0$ at t = 0. The arrival time τ of the signal at position $\left({\xi}_{i},{\xi}_{j},\eta \right)$ must be considered. The arrival times of the P and S waves propagating with velocities ${c}_{P}$ and ${c}_{S}$ are given as

^{10}N s

^{−1}and b = 1.0 × 10

^{−5}s

^{−1}.

_{f}can be obtained as

## 4. Simulations

^{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 ${k}_{\eta}$ from Equation (86), resulting in two solutions of ${A}_{mz}$.

_{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.

_{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, $\mathrm{cos}\left(m\vartheta \right)$, and $\mathrm{sin}\left(m\vartheta \right)$, 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 ${x}_{0i}=0,{x}_{0j}=0.45\mathrm{m}$ and ${z}_{0}=1$ m, and ${x}_{0i}=0.35\mathrm{m},{x}_{0j}=0.35\mathrm{m}$ and ${z}_{0}=1$ m. For the first PS, the distance from the PS to the point at ${\xi}_{i}=0$, ${\xi}_{j}=0.5\mathrm{m}$, and $\eta =0$ on the outer surface is the shortest with $\varsigma =0.05\mathrm{m}$.

_{z}excitation of 1 N produces a P wave with a maximum amplitude of tens of centimeters.

_{zz}generated by two PSs located at ${x}_{0i}=0,{x}_{0j}=0.45\mathrm{m}$ and ${z}_{0}=1$ m, and ${x}_{0i}={x}_{0j}=0.35\mathrm{m}$ 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.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

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**Figure 1.**(

**a**) Geometry of a thick cylindrical shell, (

**b**) its radial cross-section involving the point source (PS) and a given point P, in which the red dotted line represents an arc connecting PS and P, and (

**c**) two forms of the PS vector along the ${x}_{j}$ and z directions used in analytical modeling.

**Figure 2.**Calculated Green’s function for: (a) a cylinder and (b) a shell, in which the PS locates at $\left({x}_{0i}=0,{x}_{0j}=0\right)$ and $\left({x}_{0i}=0,{x}_{0j}=0.45\right)$, respectively. The plots are conducted by applying [th, r] = meshgrid(linspace(0, 2*pi, 2000), linspace(b, a, 2000)) and [X, Y] = pol2cart(th, r) in MATLAB

^{®}(MathWorks, Natick, MA, USA). The peak position in [X, Y] corresponds to the (x

_{0,i},x

_{0j}) Cartesian coordinates.

**Figure 3.**Spectra of ${k}_{\eta}$ values calculated as a function of η (l–z

_{0}): (

**a**) ${x}_{0i}=0,{x}_{0j}=0.45,{z}_{0}=1\mathrm{m}$ and (

**b**) ${x}_{0i}=0.35,{x}_{0j}=0.35,{z}_{0}=1\mathrm{m}$.

**Figure 4.**The 2π-periodic (m = 0) displacements at the (0.5 m, 45°, 1 m) position generated by the PS of 1.0 N located at the center of the cylinder (P

_{j}: upper, P

_{z}: lower): (

**a**,

**c**) radial (black line), axial (red line), and tangential (green line) components; and (

**b**,

**d**) P (black line), SH (red line), and SV (green line) waves.

**Figure 5.**Angular dependences of the maximum displacements on the circumference, with $r=0.5\mathrm{m}$ and $z=1\mathrm{m}$ for the cylinder: radial (black line), axial (red line), and tangential (green line) displacements generated by (

**a**) P

_{j}and (

**b**) P

_{z}of 1.0 N, located at ${x}_{0i}=0$, ${x}_{0j}=0$, and ${z}_{0}=1$ m.

**Figure 6.**Angular dependences of the maximum displacements of ${u}_{zz}$ on the circumference at $z=1\mathrm{m}$ for the cylinder (a = 0.5 m, b = 0 m), generated by the PS located at (

**a**) (0 m, 0.45 m, 1 m), and (

**b**) (0.35 m, 0.35 m, 1 m).

**Figure 7.**Angular dependences of the maximum displacements of ${u}_{zz}$ on the circumference at $z=1\mathrm{m}$ for the shell $\left(a=0.5\mathrm{m},b=0.4\mathrm{m}\right)$, generated by the PS located at: (

**a**) (0 m, 0.45 m, 1 m), and (

**b**) (0.35 m, 0.35 m, 1 m).

**Figure 8.**The 2π-periodic displacements at the (0.5 m, 45°, 1 m) position generated by the PS of 1.0 N located at (0 m, 0.45 m, 1 m) on the cylindrical shell (P

_{j}: upper, P

_{z}: lower): (

**a**,

**c**) radial (black line), axial (red line), and tangential (green line) components; and (

**b**,

**d**) P (black line), SH (red line), and SV (green line) waves.

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**MDPI and ACS Style**

Kim, K.B.; Kim, B.K.; Kang, J.-G. Modeling Acoustic Emission Due to an Internal Point Source in Circular Cylindrical Structures. *Appl. Sci.* **2022**, *12*, 12032.
https://doi.org/10.3390/app122312032

**AMA Style**

Kim KB, Kim BK, Kang J-G. Modeling Acoustic Emission Due to an Internal Point Source in Circular Cylindrical Structures. *Applied Sciences*. 2022; 12(23):12032.
https://doi.org/10.3390/app122312032

**Chicago/Turabian Style**

Kim, Kwang Bok, Bong Ki Kim, and Jun-Gill Kang. 2022. "Modeling Acoustic Emission Due to an Internal Point Source in Circular Cylindrical Structures" *Applied Sciences* 12, no. 23: 12032.
https://doi.org/10.3390/app122312032