Stress Wave Propagation in a Rayleigh–Love Rod with Sudden Cross-Sectional Area Variations Impacted by a Striker Rod

Abstract

This paper presents an in-depth study of the stress wave behavior propagating in a Rayleigh–Love rod with sudden cross-sectional area variations. The analytical solutions of stress waves are derived for the reflection and transmission propagation behavior at the interface of the cross-sectional area change in the rod, considering inertia and Poisson’s effects on the rod material. Examples solved using the finite element method are provided to verify the correctness of the analytical results. Based on the forward analysis of Rayleigh–Love wave propagation in a rod impacted by a striker rod, an impact-echo-type nondestructive testing (NDT) method is proposed to conduct defect assessment in rod-type structural components with sudden cross-sectional area changes within a cover medium. This proposed NDT method can identify the location, extension, and cross-sectional area drop ratios of an irregular zone in the rod to be inspected.

1. Introduction

Stress wave propagation, particularly concerning the nondestructive testing (NDT) of structures, is an important field in engineering. The NDT method is crucial for assessing the safety, reliability, and efficiency of structures [1,2,3,4,5] because it allows for the evaluation of structures and component integrity without causing damage. This approach is particularly vital for detecting defects or corrosive damage in engineering structures [6,7,8,9,10,11,12]. Corrosive damage, often manifested as reductions in the cross-sectional area of rod-type elements, is a significant concern in structural engineering. This type of deterioration typically results from environmental exposures that accelerate the corrosion process, leading to the weakening of the material and potential structural failure. Detecting and quantifying such damage is critical for maintaining structural integrity and safety. The detection of the damage condition in rod-type structural elements embedded inside a medium is an important topic in the NDT field. Among the various theories introduced to understand the longitudinal wave propagation for the impact-echo NDT method, Rayleigh–Love rod theory [13] provides a very good model of the observed real behavior. This theory combines the effects of lateral inertia and Poisson’s ratio on the rod material, which is often neglected in the traditional model of D’Alembert [14]. Incorporating lateral inertia and Poisson’s ratio is crucial for analyzing longitudinal stress wave propagation in rods because it considerably affects wave behavior, as evidenced by the research provided by Yang et al. [15,16].

The primary goal of this study was to develop an NDT method aimed at identifying the positions of cross-sectional area changes based on the forward analysis of stress wave propagation in a rod. Exhibiting abrupt changes in the cross-sectional area of a rod often indicates damage or defects in the rod-type structural member, like corrosion and bulge. To distinguish damage or defects from the effects of wave propagation modifications, reliance on changes in the amplitude of the wave is crucial; specifically, the amplitude values of reflected waves are indicative. The larger the reflected wave, the greater the damage or defect, as it indicates a discontinuity in the rod’s cross-sectional area. This change in the waveform reveals the condition of the rod and helps identify the reasons for the discontinuities affecting the waves. The theoretical foundation of this research is grounded in Rayleigh–Love rod theory [13], which provides a comprehensive understanding of stress wave propagation in rods, particularly in the context of sudden cross-sectional changes. This analytical approach enables a detailed analysis of the complex dynamics of wave propagation, including the critical effects of lateral inertia and Poisson’s ratio.

As shown in Figure 1, the model for this research was developed for using the impact-echo method to determine the changing cross-sections A₂ and A₃ and length L₁ and l₂ based on stress wave propagation signals, where σ_I is the incident stress. σ_R1 and σ_T1 are the reflected and transmitted stresses due to σ_I at L₁. σ_R2 and σ_T2 are the reflected and transmitted stresses due to σ_T1 at L₂. σ_R3 and σ_T3 are the reflected and transmitted stresses due to σ_R2 at L₁. The stress propagation process begins with the impact of the striker on the semi-infinite rod, generating an incident stress (σ_I) that travels along the semi-infinite rod. This incident stress (σ_I) propagates to location L₁, where a change in cross-sectional area causes reflected stress (σ_R1) and transmitted stress (σ_T1). After the stress wave passes through the cross-section change at L₁, the stress wave continues to move to the changing cross-section at L₂. At this location, stress superposition occurs as the transmitted stress (σ_T1) moves toward the cross-section at L₂ and the reflected stress (σ_R1) travels back to the start of the rod. The transmitted stress (σ_T1) will generate reflected stress (σ_R2) and transmitted stress (σ_T2) due to the change in cross-sectional area. The reflected wave (σ_R2) will propagate back to position L₁, generating a transmitted wave (σ_R3) and a reflected wave (σ_R3). A₁, A₂, and A₃ are the cross-sectional areas of segments 1, 2, and 3 of the semi-infinite rod, respectively. E₁, ρ₁, and υ₁ are the Young’s modulus, mass density, and Poisson’s ratio of the rod, respectively. The striker rod is moved with an impact velocity of 2v₀. Then, at the observed position, an incident wave is generated, and a reflection wave travels back due to a change in the cross-sectional area. A signal is measured at a sensing site for back-calculation to determine the position and severity of a cross-sectional area change. The analysis delves into the reflection and transmission in rods with double sudden changes in cross-sectional area. Such modifications lead to alterations in the waveform during propagation. These abrupt changes are often indicative of major defects, including cracks, necking, expansion, and corrosion within the rod. This aspect is crucial because the shape and intensity of the stress waves are significantly affected at the discontinuity interface within a structure [17,18]. Previous studies [17,18] have documented the substantial influence of boundaries on wave behavior, particularly noting how sudden changes in cross-sectional area can modify wave propagation. However, these studies did not fully consider the specific effects of lateral inertia and Poisson’s ratio on stress wave propagation. Results obtained from considering lateral inertia and Poisson’s ratio show that the magnitude of stress increases, ranging from 23.84% to 30.94%, compared to results obtained without considering lateral inertia and Poisson’s ratio. This higher stress value could cause inaccurate structure analysis results if not considered. In this study, we introduce a method to identify changes in the cross-sectional area and their locations along the rod, considering the effects of lateral inertia and Poisson’s ratio. The stress values obtained are consistently higher than those calculated without considering the effects of lateral inertia and Poisson’s ratio. This is an important consideration, as exceeding allowable stress values can lead to structural damage.

2. Analysis of Stress Wave Propagation in a Rayleigh–Love Rod with Sudden Cross-Sectional Area Variations

2.1. Stress Wave Propagation in a Rod

The behavior of stress waves in a material is typically described by the wave equation. It captures the relationship between the displacement of particles within the material and the rate at which these displacements change in the spatial–temporal domains. This dynamic phenomenon is often attributed to the transmission of energy and momentum through the medium. The traditional wave equation for one-dimensional wave propagation in a rod is given by [14] as

$$\frac{{\partial }^{2}u}{\partial t^2}=c_0^2\frac{{\partial }^{2}u}{\partial x^2},$$

(1)

where u (x, t) represents the axial (longitudinal) displacement of a point located at position x in the rod at time t, and $c_0$ is the wave velocity without considering Poisson’s ratio effect. Herein, wave velocity is related to the mass density ρ and Young’s modulus E of the rod by using the equation $c_0=\sqrt{E/\rho }$. But considering the effect of Poisson’s ratio, wave velocity (c) used in the stress wave propagation analysis is calculated as shown in Appendix A.

The wave equation for one-dimensional wave propagation in a rod, which takes into account the effects of lateral inertia and Poisson’s ratio as provided by Rayleigh–Love [13], is expressed as follows:

$$\frac{{\partial }^{2}u}{\partial t^2}=c_0^2\frac{{\partial }^{2}u}{\partial x^2}+{\upsilon }^2{\kappa }^2\frac{{\partial }^{4}u}{\partial x^2\partial t^2},$$

(2)

where u is the axial displacement in the axial direction of stress wave propagation, υ is Poisson’s ratio, and κ is the radius of gyration of the cross-section. The influence of κ leads to oscillations with larger amplitudes compared to scenarios without consideration. The method for determining κ, related to the wave propagation equation, is detailed in Appendix B. In this theory, the stress equation can be expressed as

$$\sigma =E\frac{\partial u}{\partial x}+\rho {\upsilon }^2{\kappa }^2\frac{{\partial }^{3}u}{\partial x\partial t^2}.$$

(3)

Various methods, such as the finite difference method [19,20,21,22,23,24], the finite element method (FEM) [25,26], or Laplace transform [27,28,29], can be employed to solve Equation (2). The Laplace transform, a mathematical tool that transitions functions from the time domain to the s domain, offers an elegant method for simplifying and solving differential equations [30]. When it is applied in solving wave propagation, this transform converts the original differential equation problem into an algebraic equation, making managing and solving easy. The solution is then converted back to the time domain using the inverse Laplace transform, which provides an accurate solution to the original wave propagation problem. Therefore, the Rayleigh–Love rod model, which considers the effects of lateral inertia, can be effectively solved with the Laplace transform. This analytical approach contributes to a detailed interpretation of the wave propagation dynamics and the consequential stress distribution, thereby providing substantial insights into the dynamic response of materials under stress wave propagation.

2.2. Analysis of Rayleigh–Love Rod Impacted by a Striker Rod

Based on the results of the Laplace transform method derived from Yang et al. [16], the exact solution for the transmitted and reflected stress waves in a Rayleigh–Love rod with a sudden cross-sectional variation is derived first. Subsequently, a backward analysis method can be developed to determine the position of the changing cross-section and the ratio of the reduced area to the original area, as illustrated in Figure 2.

In Figure 2, σ_I, σ_R, and σ_T represent the incident, reflected, and transmitted stresses, respectively. D₁ is the diameter of the striker rod and rod segment 1. D₂ is the diameter of rod segment 2. A₁ is the cross-sectional area of the striker rod and rod segment 1. A₂ is the cross-sectional area of rod segment 2. E₁, ρ₁, and υ₁ are the Young’s modulus, mass density, and Poisson’s ratio of the rod, respectively.

Yang et al. [16] focused on building an impact model based on split Hopkinson tests. The process begins with a semi-infinite rod, known as the incident bar, being struck longitudinally by a striker bar. The traditional one-dimensional theory suggests that the resulting impact generates a rectangular pulse if both rods are composed of the same material and have identical cross-sectional areas. However, the distortions of the pulses are expected on theoretical grounds and observed during experiments. Yang et al. [16] derived the analytical solutions for the longitudinal impact problems in dispersive rods. Their discussion focused on a situation where a semi-infinite Rayleigh–Love rod was impacted by a striker rod of length L with the same material and cross-sectional area as the Rayleigh–Love rod, as shown in Figure 3, where σ_I is the incident stress, and D₁ and A₁ are the diameter and the cross-sectional area of the striker rod and the semi-infinite rod, respectively. E₁, ρ₁, and υ₁ are the Young’s modulus, mass density, and Poisson’s ratio of the two rods, respectively. 2v₀ is the impact velocity.

At the initial time t = 0, a finite striker rod moving coaxially at an impact velocity of 2v₀ hits a stationary semi-infinite rod, with both rods assumed to be unstressed. Consequently, the initial conditions relating to the axial displacement u can be expressed as follows:

$$\begin{array}{ccc}u(x,0)=0,& \frac{\partial u(x,0)}{\partial t}=2v_0,& -L\le x<0\end{array},$$

(4)

$$\begin{array}{ccc}u(x,0)=0,& \frac{\partial u(x,0)}{\partial t}=0,& 0\le x<\infty \end{array}.$$

(5)

The boundary conditions at the free end, where $x=-L$, and at infinity are determined by

$$\begin{array}{cc}\sigma (-L,t)=0,& \left|u(\infty ,t)\right|\end{array}=0.$$

(6)

If the two rods are in tight contact in a compressed state following the collision, then the coupled continuity conditions at the contact surface x = 0 can be established as

$$\begin{array}{cc}u(0^{-},t)=u(0^{+},t),& \sigma (0^{-},t)=\sigma (0^{+},t)\end{array}.$$

(7)

Yang et al. [16] converted the values to dimensionless quantities for easy transformation by setting $\overline{u}=\frac{u}{D_1}$, $\overline{x}=\frac{x}{D_1}$, $\overline{t}=\frac{ct}{D_1}$, $\overline{L}=\frac{L}{D_1}$, $b=\upsilon \overline{\kappa }=\upsilon \frac{\kappa }{D_1}$, ${\overline{v}}_0=\frac{v_0}{c}$, and $\overline{\sigma }=\frac{\sigma }{E}$. The Laplace transform and inverse techniques are employed to solve Equation (2). The result of the displacement is shown as Equation (15a) in Yang et al.’s study [16]. The stress state in a semi-infinite rod is then determined as follows:

$$\overline{\sigma }\left(\overline{x},\overline{t}\right)=\frac{\partial \overline{u}}{\partial \overline{x}}+b^2\frac{{\partial }^3\overline{u}}{\partial \overline{x}\partial {\overline{t}}^2}.$$

(8)

By performing the calculation and converting the dimensionless quantities from Equation (8) into dimensional quantities, the result of stress wave propagation in the semi-infinite rod can be obtained as follows:

$${\sigma }_I\left(x,t\right)=-\frac{4\rho c{v}_0}{\pi }{\displaystyle {\int }_0^{\frac{1}{b}}\sin \frac{L\eta }{D_1\sqrt{1-b^2{\eta }^2}}\sin \frac{\left(x+L\right)\eta }{D_1\sqrt{1-b^2{\eta }^2}}}\sin \left(\frac{tc}{D_1}\eta \right)\left(\frac{\sqrt{1-b^2{\eta }^2}}{\eta }\right)d\eta ,$$

(9)

where $b=\upsilon \overline{\kappa },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{\kappa }=\frac{\kappa }{D_1},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\kappa =\sqrt{\frac{I_1}{A_1}}$, $\eta$ is a variable in the contour integration of the Laplace inverse transform, ρ is the mass density, c is the longitudinal wave velocity of the elastic material as expressed by Equation (A16) in Appendix A, and v₀ is the impact velocity.

In this study, the incident stress (${\sigma }_I$) result of Equation (9) is considered to determine the reflected stress $({\sigma }_R)$ and transmitted stress $({\sigma }_T)$ in a Rayleigh–Love rod with a sudden change in cross-sectional area under the effects of Poisson’s ratio and lateral inertia. Based on the balance of forces at the interface between two segments, the total force applied by segment 1 (from the incident and reflected waves) must be equal to the total force applied by segment 2 (from the transmitted wave), as shown in Figure 4, where v_I, σ_I, and u_I are the incident velocity, incident stress, and incident displacement of segment 1, respectively. v_R, σ_R, and u_R are the reflected velocity, reflected stress, and reflected displacement of segment 1, respectively. v_T, σ_T, and u_T are the transmitted velocity, transmitted stress, and transmitted displacement of segment 2, respectively. A₁ and A₂ are the cross-sectional areas of segment 1 and segment 2 of the semi-infinite rod, respectively. E₁, ρ₁, and υ₁ are the Young’s modulus, mass density, and Poisson’s ratio of the two rods, respectively. x_I, x_R, and x_T are the positions determined for the incident, reflected, and transmitted stresses in the semi-infinite rod, respectively. x_b is the position-sensing stress in the striker rod. Thus, the following equation is obtained:

$$A_1({\sigma }_I+{\sigma }_R)=A_2{\sigma }_T,$$

(10)

where A₁ and A₂ are the cross-sectional areas of segment 1 and segment 2, respectively, and σ_I, σ_R, and σ_T are the incident, reflected, and transmitted stresses, respectively.

Based on the continuity of velocity in Figure 4, the velocity at the interface of the two segments must be the same, which is expressed as

$$v_I-v_R=v_T.$$

(11)

The relationship between stress and velocity in the rod is

$$\sigma =\rho cv,$$

(12)

where v is the particle velocity of the material point under stress σ, and c is the longitudinal wave velocity of the elastic material (see Appendix A) and can be determined as

$$c=\sqrt{\frac{(1-\upsilon )}{(1+\upsilon )(1-2\upsilon )}}\sqrt{\frac{E}{\rho }}, \mathrm{where}-1<\upsilon <0.5.$$

(13)

Based on Equation (12), the velocity for incident, reflected, and transmitted waves can be, respectively, presented as

$$v_I=\frac{{\sigma }_I}{{\rho }_1{c}_1},$$

(14)

$$v_R=\frac{{\sigma }_R}{{\rho }_1{c}_1},$$

(15)

$$v_T=\frac{{\sigma }_T}{{\rho }_2{c}_2}.$$

(16)

Substituting Equations (14)–(16) into Equation (11) can obtain

$$\frac{{\sigma }_I}{{\rho }_1{c}_1}-\frac{{\sigma }_R}{{\rho }_1{c}_1}=\frac{{\sigma }_T}{{\rho }_2{c}_2}.$$

(17)

Solving Equations (10) and (17) can obtain the reflected stress ${\sigma }_R$ and transmitted stress ${\sigma }_T$ as

$${\sigma }_{R1}=\frac{A_2{\rho }_2{c}_2-A_1{\rho }_1{c}_1}{A_1{\rho }_1{c}_1+A_2{\rho }_2{c}_2}{\sigma }_I=R_1\times {\sigma }_I,$$

(18)

$${\sigma }_{T1}=\frac{2A_1{\rho }_2{c}_2}{A_1{\rho }_1{c}_1+A_2{\rho }_2{c}_2}{\sigma }_I=T_1\times {\sigma }_I,$$

(19)

where the reflected ratio at L₁ due to incident stress (σ_I) is

$$R_1=\frac{A_2{\rho }_2{c}_2-A_1{\rho }_1{c}_1}{A_1{\rho }_1{c}_1+A_2{\rho }_2{c}_2},$$

(20a)

where the transmitted ratio at L₁ due to incident stress (σ_I) is

$$T_1=\frac{2A_1{\rho }_2{c}_2}{A_1{\rho }_1{c}_1+A_2{\rho }_2{c}_2},$$

(20b)

According to Figure 2, the stress wave propagation σ₁ traveling along segment 1 is determined as

$${\sigma }_1(x,t)={\sigma }_I(x,t)+{\sigma }_R(2L_1-x,t),$$

(21)

where

$${\sigma }_I\left(x,t\right)=-\frac{4\rho c{v}_0}{\pi }{\displaystyle {\int }_0^{\frac{1}{b}}\sin \frac{L\eta }{D_1\sqrt{1-b^2{\eta }^2}}\sin \frac{\left(x+L\right)\eta }{D_1\sqrt{1-b^2{\eta }^2}}}\sin \left(\frac{tc}{D_1}\eta \right)\left(\frac{\sqrt{1-b^2{\eta }^2}}{\eta }\right)d\eta ,$$

(22)

$${\sigma }_R\left(2L_1-x,t\right)=-\frac{4\rho c{v}_0{R}_1}{\pi }{\displaystyle {\int }_0^{\frac{1}{b}}\sin \frac{L\eta }{D_1\sqrt{1-b^2{\eta }^2}}\sin \frac{\left(2L_1-x+L\right)\eta }{D_1\sqrt{1-b^2{\eta }^2}}}\sin \left(\frac{tc}{D_1}\eta \right)\left(\frac{\sqrt{1-b^2{\eta }^2}}{\eta }\right)d\eta .$$

(23)

Here, $x\in \left[0,L_1\right]$, where x is the position determined for the incident and reflected stresses (see Figure 4).

The propagation of the transmitted stress propagation σ₂ traveling along segment 2 is determined as

$${\sigma }_2\left(x,t\right)={\sigma }_T=-\frac{4\rho c{v}_0{T}_1}{\pi }{\displaystyle {\int }_0^{\frac{1}{b}}\sin \frac{L\eta }{D_1\sqrt{1-b^2{\eta }^2}}\sin \frac{\left(x+L\right)\eta }{D_1\sqrt{1-b^2{\eta }^2}}}\sin \left(\frac{tc}{D_1}\eta \right)\left(\frac{\sqrt{1-b^2{\eta }^2}}{\eta }\right)d\eta .$$

(24)

Here, $x\in \left[L_1,\infty \right]$, where x is the position determined for the transmitted stress.

The stress wave propagation in the striker rod is determined as follows:

$$\begin{array}{cc}{\sigma }_S\left(x,t\right)=-\frac{4\rho c{v}_0}{\pi }{\displaystyle {\int }_0^{\frac{1}{b}}\sin \frac{L\eta }{D_1\sqrt{1-b^2{\eta }^2}}\sin \frac{\left(x+L\right)\eta }{D_1\sqrt{1-b^2{\eta }^2}}}\sin \left(\frac{tc}{D_1}\eta \right)\left(\frac{\sqrt{1-b^2{\eta }^2}}{\eta }\right)d\eta ,& -L\le x<0\end{array},$$

(25)

Here, $x\in \left[-L,0\right]$, where $x$ is the position for the determined stress wave in the striker rod.

When Poisson’s ratio is set as zero, the incident stress, reflection stress, and transmitted stress are, respectively, expressed as follows:

$${\sigma }_I\left(x,t\right)=-\frac{4\rho c{v}_0}{\pi }{\displaystyle {\int }_0^{\infty }\sin \frac{L\eta }{D_1}\sin \frac{\left(x+L\right)\eta }{D_1}}\sin \left(\frac{tc}{D_1}\eta \right)\left(\frac{1}{\eta }\right)d\eta =\left\{\begin{array}{ll}-\rho c{v}_0,& \frac{x}{c}<t<\frac{x+2L}{c}\\ -\frac{\rho c{v}_0}{2},& t=\frac{x}{c},t=\frac{x+2L}{c}\\ 0,& \mathrm{otherwise}\end{array}\right.,$$

(26)

$${\sigma }_R\left(2L_1-x,t\right)=-\frac{4\rho c{v}_0{R}_1}{\pi }{\displaystyle {\int }_0^{\infty }\sin \frac{L\eta }{D_1}\sin \frac{\left(2L_1-x+L\right)\eta }{D_1}}\sin \left(\frac{tc}{D_1}\eta \right)\left(\frac{1}{\eta }\right)d\eta =\left\{\begin{array}{l}\begin{array}{cc}-\rho c{v}_0{R}_1,& \frac{2L_1-x}{c}<t<\frac{2\left(L+L_1\right)-x}{c}\end{array}\\ \begin{array}{cc}\begin{array}{l}\frac{-\rho c{v}_0{R}_1}{2},\\ 0,\end{array}& \begin{array}{l}t=\frac{2L_1-x}{c},t=\frac{2\left(L+L_1\right)-x}{c}\\ \mathrm{otherwise}\end{array}\end{array}\end{array}\right.,$$

(27)

$${\sigma }_T\left(x,t\right)=-\frac{4\rho c{v}_0{T}_1}{\pi }{\displaystyle {\int }_0^{\infty }\sin \frac{L\eta }{D_1}\sin \frac{\left(x+L\right)\eta }{D_1}}\sin \left(\frac{tc}{D_1}\eta \right)\left(\frac{1}{\eta }\right)d\eta =\left\{\begin{array}{l}\begin{array}{cc}-\rho c{v}_0{T}_1,& \frac{x}{c}<t<\frac{x+2L}{c}\end{array}\\ \begin{array}{cc}\begin{array}{l}\frac{-\rho c{v}_0{T}_1}{2},\\ 0,\end{array}& \begin{array}{l}t=\frac{x}{c},t=\frac{x+2L}{c}\\ \mathrm{otherwise}\end{array}\end{array}\end{array}\right..$$

(28)

The numerical integration provided by Yang et al. [31] was utilized to obtain the stress wave propagation in a Rayleigh–Love rod to solve Equations (21)–(25). When the stress wave propagation does not consider Poisson’s ratio and inertia effects in Equations (21)–(25), they will degenerate to Equations (26)–(28), and the analytical solution can be obtained.

3. Examples

3.1. Example 1: Study of the Stress Wave Propagation in a Rod with A₁ ≤ A₂

3.1.1. Verification of the Analytical Solution from Equations (21)–(25) with the FEM Solution

This section presents the stress wave propagation in the rods consisting of two segments with different cross-sectional areas, where cross-sectional area A₂ is greater than A₁, as shown in Figure 2. First, the accuracy of Equations (21)–(25) will be confirmed by comparing the results derived from these equations and the results obtained from the FEM. The parameters of material properties and geometries of the striker rod and the segment rods used in this study are shown in

Table 1. The parameters of material properties and geometries of the striker rod and the segment rods used in example 1.

Parameters	Values
Diameter of the striker rod	30 mm
Diameter of segment 1	30 mm
Diameters of segment 2	30, 42, 52, 60, 67 mm
Young modulus, E	195 GPa
Poisson’s ratio, υ	0.3
Mass density, ρ	7850 kg/m3
Striker rod length, L	0.15 m
Length of segment 1, L1	1 m
Length of segment 2, L2	1.5 m
Impact velocity, 2v0	11.6 m/s
Wave speed of the semi-infinite rod material, c	5782.69 m/s
Wave speed of the striker rod material, cs	5782.69 m/s

.

The geometrical shape and finite element mesh of the Rayleigh–Love rod model are plotted in Figure 5a,b, and the relevant parameters used in the Rayleigh–Love rod model are shown in Table 1. The elastic material wave velocity c of the rod and striker materials was computed as 5782.69 m/s. In the Abaqus finite element analysis code, the solid element-type C3D8R was employed, with 10,947 nodes and 8448 elements in the model, as shown in Figure 5b. The detailed stress variations are demonstrated in Figure 5c–h at the time from the initial state to 450 μs.

For the cross-sectional area ratio A₂/A₁ = 2, i.e., A₂ = 1.41 × 10⁻³ mm², the comparisons of the stress responses over time at varying positions at the striker rod, segment 1, and segment 2 between both methods are illustrated in detail in Figure 6, Figure 7 and Figure 8, respectively, through the whole wave propagation duration. A high consistency presents positive evidence for a high degree of match between the analytical solutions expressed by Equations (21)–(25) and the FEM simulation results. This verifies the correctness of the numerical algorithm [31] used to solve Equations (21)–(25) in this study.

3.1.2. Investigation of Stress Wave Propagation in a Rod at Various Positions

Figure 9 presents the two types of stress responses, incident and reflected stresses, at x = 0.5 m, i.e., point A in Figure 5b, in segment 1, within the rod based on Equation (21). The shape of the incident stresses remains unchanged in the period of 90–140 μs, as expected. By contrast, the reflected stress in the period of 260–310 μs significantly varies with cross-sectional area A₂. In particular, an increase in cross-sectional area A₂ results in a corresponding increase in the magnitude of the reflected stresses. When the cross-sectional areas are equal, i.e., A₁ = A₂, no reflected stresses are observed, signifying the absence of reflected stresses. Moreover, the incident and reflected stresses exhibit the same compression stress responses.

Figure 9 also shows the stress responses without considering Poisson’s effect. The peak stress values of the incident stress wave are −333.24 and −263.286 MPa, respectively. This finding indicates that the stress under Poisson’s effect increases by 25.57% compared with that without Poisson’s effect.

Table 2. Reflected stresses in case of A₂ ≥ A₁.

A2/A1	${\mathit{\sigma }}_{\mathit{R}}^{\mathit{p}}(\mathit{\upsilon })$ (MPa)	${\mathit{\sigma }}_{\mathit{R}}^{\mathit{p}}(\mathit{\upsilon }=0)$ (MPa)	Diff. (%)
1	0	0	-
2	−108.688	−87.762	23.84
3	−163.362	−131.643	24.09
4	−196.481	−157.972	24.38
5	−218.610	−175.524	24.55

shows that the reflected stress values range from 23.84% to 24.55% for the cross-sectional area ratio, A₂/A₁, varying from two to five. This finding demonstrates that Poisson’s effect has similar stress responses for different cross-sectional area ratios.

Figure 10 illustrates the stress responses at x = 1.5 m, i.e., point B in Figure 5b, in segment 2 within the rod based on Equation (24). The first peak values of this transmitted stress vary with cross-sectional area A₂. The transmitted stresses are equal to the incident stress values when the cross-sectional area A₁ is equal to A₂. In particular, the resulting transmitted stress responses become increasingly intensive as the cross-sectional area A₂ increases.

Table 3. Transmitted stresses in case of A₂ ≥ A₁.

A2/A1	${\mathit{\sigma }}_{\mathit{T}}^{\mathit{p}}(\mathit{\upsilon })$ (MPa)	${\mathit{\sigma }}_{\mathit{T}}^{\mathit{p}}(\mathit{\upsilon }=0)$ (MPa)	Diff. (%)
1	−333.24	−263.286	26.57
2	−222.16	−175.524	26.57
3	−166.62	−131.643	26.57
4	−133.296	−105.314	26.57
5	−111.08	−87.762	26.57

shows that the stress values have similar stress responses, with and without considering Poisson’s effect for different cross-sectional area ratios, A₂/A₁. The waveforms of nonzero Poisson’s ratio oscillate about the waveform of zero Poisson’s value.

Figure 11 illustrates the stress wave travel time history at various positions in a Rayleigh–Love rod. The stress wave propagation in the striker rod is presented at points I₁, I₂, I₃, and P₀ (left). Points P₀ (right) to P₃ represent the presence of the incident and reflected waves. Points P₅ to P₇ describe the transmitted wave in the semi-infinite rod. The dotted lines represent stress in the striker rod, the solid lines represent the incident and reflected waves, and the dashed lines correspond to the transmitted wave.

The interface of the cross-sectional area jumps off the rod, and all three types of waves (incident, reflected, and transmitted waves) are observed at P₄. The incident and reflected waves are detected on the left side of point P₄, whereas the transmitted wave is discernible on the right side. The oscillations of the wave, influenced by Poisson’s effect, are consistently found at each position, leading to stress values surpassing those from traditional wave equations that disregard Poisson’s effect. This finding emphasizes the pertinence of the theoretical approach in aligning closely with the actual wave propagation observed in real tests and verifies its applicability to structural analysis works.

3.2. Example 2: Analysis of the Signal Obtained for the Rod Where A₁ ≥ A₂

This section presents stress wave propagation where cross-sectional area A₂ is less than cross-sectional area A₁, as shown in Figure 12, where σ_I, σ_R, and σ_T represent the incident, reflected, and transmitted stresses, respectively. D₁ is the diameter of the striker rod and rod segment 1. D₂ is the diameter of rod segment 2. A₁ is the cross-sectional areas of the striker rod and rod segment 1. A₂ is the cross-sectional area of rod segment 2. E₁, ρ₁, and υ₁ are the Young’s modulus, mass density, and Poisson’s ratio of the rod, respectively. Except for the diameters of segment 2, the relevant parameters used in the Rayleigh–Love rod model are listed in Table 1. The diameters of segment 2 are set as 30, 21, 17, 15, and 13 mm instead.

Figure 13 depicts the stress responses and incident and reflected stresses, at x = 0.5 m, i.e., point A in Figure 5b, in segment 1, within the rod based on Equation (21) when cross-sectional area A₁ is greater than cross-sectional area A₂. The shape of the incident stresses remains invariant in the 90–140 μs period regardless of the changes in cross-sectional area A₂, as expected. Conversely, the reflected stress in the 260–310 μs period significantly varies with alterations in cross-sectional area A₂. If A₂ is smaller than A₁, this leads to intensive reflected stresses. When the cross-sectional areas are equal, i.e., A₁ = A₂, no reflected stress responses are observed, signifying the absence of reflected stresses. In this scenario, the incident and reflected stresses exhibit opposite stress states, indicating that the reflected wave phase is strongly affected by the cross-sectional area ratio at the interface.

Table 4. Reflected stresses in case of A₂ ≤ A₁.

A2/A1	${\mathit{\sigma }}_{\mathit{R}}^{\mathit{p}}(\mathit{\upsilon })$ (MPa)	${\mathit{\sigma }}_{\mathit{R}}^{\mathit{p}}(\mathit{\upsilon }=0)$ (MPa)	Diff. (%)
1	0	0	-
1/2	114.9129	87.762	30.94
1/3	170.8699	131.643	29.80
1/4	204.1672	157.9716	29.24
1/5	226.3654	175.524	28.97

shows that the stress values have similar stress responses with and without Poisson’s effect for different cross-sectional area ratios.

Figure 14 illustrates the stress responses at x = 1.5 m, i.e., point B in Figure 5b, in segment 2 within the rod based on Equation (24). When cross-sectional area A₁ is greater than cross-sectional area A₂, it leads to more intensive transmitted stresses than the incident stress. When cross-sectional area A₁ is equal to A₂, the transmitted stresses are equal to the incident stress values. In particular, the resulting transmitted stress responses become increasingly intensive as cross-sectional area A₂ increases.

Table 5. Transmitted stresses in case of A₂ ≤ A₁.

A2/A1	${\mathit{\sigma }}_{\mathit{T}}^{\mathit{p}}(\mathit{\upsilon })$ (MPa)	${\mathit{\sigma }}_{\mathit{T}}^{\mathit{p}}(\mathit{\upsilon }=0)$ (MPa)	Diff. (%)
1	−333.240	−263.286	26.57
1/2	−444.320	−351.048	26.57
1/3	−499.860	−394.929	26.57
1/4	−533.184	−421.2576	26.57
1/5	−555.400	−438.81	26.57

shows that the stress values have similar stress responses, A₂/A₁, with and without considering Poisson’s effect for different cross-sectional area ratios.

4. Identification of the Locations of Sudden Cross-Sectional Area Change in the Rod

4.1. Reflection and Transmission of Stress Wave in a Rod with Single Sudden Cross-Sectional Area Variation

4.1.1. Developing a Formula to Determine the Damaged Zone

In some engineering diagnosis problems, the damaged position and the condition of the rod inside a cover medium must be identified. The impact-echo method is usually applied for this purpose. The variations in the reflected and transmitted stresses can be presented in terms of cross-sectional area ratio A₂/A₁ to illustrate the influence of the cross-sectional area change in the rod. The reflected stress ${\sigma }_R$ and transmitted stress ${\sigma }_T$ can be expressed as follows by letting α = A₂/A₁ and substituting it into Equations (18) and (19):

$${\sigma }_R=\frac{\alpha {\rho }_2{c}_2-{\rho }_1{c}_1}{{\rho }_1{c}_1+\alpha {\rho }_2{c}_2}{\sigma }_I,$$

(29)

$${\sigma }_T=\frac{2{\rho }_2{c}_2}{{\rho }_1{c}_1+\alpha {\rho }_2{c}_2}{\sigma }_I,$$

(30)

where α = A₂/A₁ is the cross-sectional area ratio.

Equations (29) and (30) show the relationship between the cross-sectional area ratio and the reflected and transmitted stress. When the cross-sectional area ratio A₂/A₁ approaches 0, it follows that σ_R approaches −σ_I, and σ_T approaches 2σ_I; as A₂/A₁ approaches ∞, it follows that σ_R approaches σ_I, and σ_T approaches 0.

Suppose that one knows the first peak stress values, labeled ${\sigma }_I^p$ as the peak incident stress and ${\sigma }_R^p$ as the peak reflected stress. In this case, one can determine the change in the cross-sectional area A₂ from the acquired signal. From Equation (29), the cross-sectional area ratio α can be rewritten as

$$\alpha =\frac{A_2}{A_1}=\frac{{\rho }_1{c}_1\left({\sigma }_I^p+{\sigma }_R^p\right)}{{\rho }_2{c}_2\left({\sigma }_I^p-{\sigma }_R^p\right)}=\frac{{\rho }_1{c}_1\left(1+{\sigma }_R^p/{\sigma }_I^p\right)}{{\rho }_2{c}_2\left(1-{\sigma }_R^p/{\sigma }_I^p\right)}.$$

(31)

Assume that one measures the wave traveling time t, which is the interval of the first peak of the wave traveling from the sensing point and the first peak of the reflected wave received at the same sensing point. In this case, one can determine the length of L_m between the sensing point and the reflected interface as

$$L_{\mathrm{m}}=\frac{1}{2}t{c}_1,$$

(32)

where t is the time duration from the first peak stress to the second peak stress, and c₁ is the longitudinal wave speed of the rod material. The sensing point is located at a distance L_s from the rod end that is smaller than the embedded length L_e, as shown in Figure 15.

We assume that ${\rho }_1={\rho }_2=\rho ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{c}_1=c_2=c$. Figure 15 shows the relation curve (labeled as a red line) between the cross-sectional area ratio A₂/A₁ and the reflected ratio ${\sigma }_R^p/{\sigma }_I^p$ and the relation curve (labeled as a blue straight line) between the length L_m between the sensing point and the reflected interface and the wave traveling time t. ${\sigma }_R^p/{\sigma }_I^p=-1$ implies that A₂ = 0, indicating that the reflected wave is equal in magnitude but opposite in sign to the incident wave. It also represents a free surface at position L₁. ${\sigma }_R^p/{\sigma }_I^p=1$ implies that $A_2\to \infty$, indicating that the reflected wave is equal to the incident wave in magnitude. When ${\sigma }_R^p/{\sigma }_I^p=0$, this scenario implies that A₂/A₁ = 1. If no reflected wave is received, the cross-sectional area in segment 2 remains unchanged, i.e., A₂ = A₁. For $-1<{\sigma }_R^p/{\sigma }_I^p<0$, the cross-sectional area A₂ is smaller than A₁. When $0<{\sigma }_R^p/{\sigma }_I^p<1$, the cross-sectional area A₂ is greater than A₁. The length of L₁ = L_m + L_s can be determined from Figure 15. The application in Figure 15 is demonstrated in the following subsection.

4.1.2. Examples: Determination of Cross-Sectional Area A₂ and Length L₁ Based on Signals Obtained with Changing Cross-Sectional Area Ratios and Impact Velocities

• Case A: Cross-sectional area ratio A₂/A₁ = 5

As shown in Figure 16a, Case A indicates the stress responses at the position of x = L_s = 0.2 m by using Equation (21). The relevant parameters used in the Rayleigh–Love rod model are identical to those used in Example 1, i.e., Table 1. The diameter of segment 2 is chosen as 67 mm. The first peak incident stress is ${\sigma }_I^p=-327.257\hspace{0.17em}\mathrm{MPa}$, and the first peak reflected stress is ${\sigma }_R^p=-220.398\hspace{0.17em}\mathrm{MPa}$, leading to a reflected ratio of ${\sigma }_R^p/{\sigma }_I^p=0.67$. Figure 16 shows the corresponding cross-sectional area ratio $\alpha =A_2/A_1=5$, which implies A₂ = 5A₁. The signal received by the reflected stress at time t is 458.9 μs, indicating that L_m = 1.3 m, from the corresponding curve in Figure 15. Given that L_s = 0.2 m is chosen, L₁ is calculated as L_m + L_s, equal to 1.5 m.

2.

Case B: Cross-sectional area ratio A₂/A₁ = 0.2

As shown in Figure 16b, Case B indicates the stress responses at the position of x = L_s = 0.2 m by using Equation (21). The relevant parameters of materials are identical to those used in Case A except for the diameter of segment 2. The diameter of segment 2 is set as 13 mm. The first peak incident stress is ${\sigma }_I^p=-327.257\hspace{0.17em}\mathrm{MPa}$, and the first peak reflected stress is ${\sigma }_R^p=220.411\hspace{0.17em}\mathrm{MPa}$, indicating the reflected ratio of ${\sigma }_R^p/{\sigma }_I^p=-0.67$. Figure 16 shows the corresponding cross-sectional area ratio $A_2/A_1=0.2$; thus, $A_2=A_1/5$. The received signal of the reflected stress at t = 282.8 μs corresponds to L_m = 0.8 m in Figure 15. Given that L_s = 0.2 m is chosen, L₁ is calculated as L_m + L_s, equal to 1.0 m.

3.

Case C: Influences of different impact velocities and striker lengths on the characteristics of response signals

All parameters of Case C are identical to those of Case B, except for the different impact velocities and striker lengths. As shown in

Table 6. The initial impact kinetic energy $\left(1/2m{v}_0^2\right)$ of the striker in each case in Figure 17.

No.	Cross-Section (m2)	Density (kg/m3)	Impact Length (m)	Initial Velocity (m/s)	Kinetic Energy (J)
1	7.069 × 10−4	7850	0.15	2.9	3.500
2				5.8	14.000
3				11.6	55.999
4			0.075	2.9	1.750
5				5.8	7.000
6				11.6	27.999

, the length of the striker rod varies between 0.15 m and 0.075 m, respectively. Velocities were used for the cases as 2.9 m/s, 5.8 m/s, and 11.6 m/s, respectively. The impact energy of each case that the striker applied on the rod is also listed in Table 6. Stress responses for the different cases of impact velocity and striker length combination are shown in Figure 17. This figure demonstrates that the first peak stress values varied with the impact energy of the striker input to the rod. It is noted that the time interval t between the first peak of the incident and reflected wave signal is fixed as t = 282.3 μs, and the stress ratio obtained between the cases of reflected and incident waves is the same for all cases, as calculated in Equation (33).

$$\alpha =\frac{A_2}{A_1}=\frac{{\rho }_1{c}_1\left({\sigma }_I^p+{\sigma }_R^p\right)}{{\rho }_2{c}_2\left({\sigma }_I^p-{\sigma }_R^p\right)}=\frac{{\rho }_1{c}_1\left(1+{\sigma }_R^p/{\sigma }_I^p\right)}{{\rho }_2{c}_2\left(1-{\sigma }_R^p/{\sigma }_I^p\right)}.$$

(33)

Since all of the cases have the same arrival time interval t and the same first peak stress ratio ${\sigma }_R^p/{\sigma }_I^p$, they all have the same length L_m and cross-sectional area A₂ as Figure 15. This case study shows that the impact-echo test method proposed in this paper provides suitable impact energy to the inspected rod and measures the response signal at a sensing point. The measurements of the impact velocity and the length of the striker are not required. The location and cross-sectional area ratio of the rod with cross-sectional area jump can be identified through the curves in Figure 15.

4.2. Reflection and Transmission of Stress Wave in a Rod with Double Sudden Cross-Sectional Area Variations

This section discusses a rod with double cross-sectional area variations. Initially, the cross-section is A₁, which then changes to A₂ and A₃ at positions L₁ and L₂, respectively. Using the model depicted in Figure 18, a formula is developed to determine the signal received at the sensor location. Based on this formula, changes in the cross-sectional area and the positions of these changes can be identified.

A stress wave is generated by a striker, and upon encountering a change in the cross-section, it causes a reflected wave and a transmitted wave, as described in Section 2.2. There are two locations of sudden cross-section change, as illustrated in Figure 18, where σ_I is the incident stresses. σ_R1 and σ_T1 are the reflected and transmitted stresses due to σ_I at x = L₁. σ_R2 and σ_T2 are the reflected and transmitted stresses due to σ_T1 at x = L₂. σ_R3 and σ_T3 are the reflected and transmitted stresses due to σ_R2 at L₁. A₁, A₂, and A₃ are the cross-sectional areas of segments 1, 2, and 3 of the semi-infinite rod, respectively. E₁, ρ₁, and υ₁ are the Young’s modulus, mass density, and Poisson’s ratio of the rod, respectively. Therefore, the stress wave at the sensor location is determined as follows:

$${\sigma }_S(x,t)={\sigma }_I(x,t)+{\sigma }_{R1}(2L_1-x,t)+{\sigma }_{T3}(2L_2-x,t),$$

(34)

where ${\sigma }_I(x,t)$ is the incident stress as shown in Equation (22); ${\sigma }_{R1}(2L_1-x,t)$ is the reflected stresses due to interface at L₁ as shown in Equation (23); and ${\sigma }_{T3}(2L_2-x,t)$ is the reflected stresses obtained at the sensor position due to interface at L₂ and is determined as follows:

$${\sigma }_{T3}(2L_2-x,t)=-\frac{4\rho c{v}_0{R}_{T3}}{\pi }{\displaystyle {\int }_0^{\frac{1}{b}}\sin \frac{L\eta }{D_1\sqrt{1-b^2{\eta }^2}}\sin \frac{\left(2L_2-x+L\right)\eta }{D_1\sqrt{1-b^2{\eta }^2}}}\sin \left(\frac{tc}{D_1}\eta \right)\left(\frac{\sqrt{1-b^2{\eta }^2}}{\eta }\right)d\eta ,$$

(35)

where $R_{T3}$ is the reflected ratio due to interface at L₁ and L₂, and the method for determining the reflection ratio is similar to that described in Section 2.2, calculated as follows:

$$R_{T3}=\left(\frac{2A_1{\rho }_2{c}_2}{A_1{\rho }_1{c}_1+A_2{\rho }_2{c}_2}\right)\left(\frac{A_3{\rho }_3{c}_3-A_2{\rho }_2{c}_2}{A_2{\rho }_2{c}_2+A_3{\rho }_3{c}_3}\right)\left(\frac{2A_2{\rho }_1{c}_1}{A_1{\rho }_1{c}_1+A_2{\rho }_2{c}_2}\right).$$

(36)

Let $\alpha =\frac{A_2}{A_1},\hspace{0.17em}\beta =\frac{A_3}{A_2}$ be the cross-sectional area ratio; substituting them into Equation (36) and combining with Equation (35) obtains β, as follows:

$$\beta =\frac{{\rho }_2{c}_2}{{\rho }_3{c}_3}\left(\frac{1+\gamma }{1-\gamma }\right), \mathrm{where} \gamma =\frac{{\left({\rho }_1{c}_1+\alpha {\rho }_2{c}_2\right)}^2}{4\alpha {\rho }_1{c}_1{\rho }_2{c}_2}\frac{{\sigma }_{T3}}{{\sigma }_I}.$$

(37)

To verify the signal received at the sensor location, determined by Equation (38), a comparison with the FEM is illustrated in Figure 19. The parameters of the material properties and geometries used are listed in Table 1. Additionally, there is a modification where A₁ = A₃ = 2A₂ and the length L of the striker rod is 0.1 m. The results obtained indicate that there is good agreement between the analytical solution and the FEM results.

The general steps of the processed NDT method to identify defect sizes in the rod (see Figure 20) are as follows:

• First, use a striker rod with a length of L to impact the semi-infinite rod with an initial velocity of 2v₀ to generate a stress wave propagating in the rod.
• Place a strain sensor at position x = L_s on the semi-infinite rod to receive the deformation signal generated by the striker rod.
• The first part of the signal received at position x = L_s is the incident stress, with a peak stress value of ${\sigma }_I^p$ and a duration of $T_i$. The wave then oscillates around zero.
• The second part of the signal obtained is the reflected stress ${\sigma }_{R1}^p$ and a time interval from the first stress peak to the second stress peak of t₁. From this signal, A₂ and L₁ can be determined as follows: the first peak incident stress is ${\sigma }_I^p$, and the first peak reflected stress is ${\sigma }_{R1}^p$, leading to a reflected ratio of ${\sigma }_{R1}^p/{\sigma }_I^p$. Figure 15 shows the corresponding cross-sectional area ratio $A_2/A_1=\alpha$ (or A₂ = αA₁). The received signal of the reflected stress at t₁ corresponds to L_m in Figure 15. The location of the first cross-sectional area variation, L₁, is calculated as L_m + L_s.
• The third part of the signal obtained is the reflected stress ${\sigma }_{T3}^p$ at a time interval t₂ from the second stress peak to the third stress peak. From this signal, A₃ and L₂ can be determined as follows: the first peak reflected stress ${\sigma }_{T3}^p$ leads to a reflected ratio of ${\sigma }_{T3}^p/{\sigma }_I^p$. Based on Figure 21, the corresponding cross-sectional area ratio is $A_3/A_2=\beta$ with A₂ = αA₁ and $A_3=\alpha \beta A_1$. The received signal of the reflected stress at t₂ corresponds to l₂ in Figure 21.

Figure 20. The cross-sectional area (A₂, A₃) and length (L₁, l₂) were determined based on the stress signal measured at x = L_s.

Figure 20. The cross-sectional area (A₂, A₃) and length (L₁, l₂) were determined based on the stress signal measured at x = L_s.

As shown in Figure 22 and

Table 7. Comparison of calculated length (L₁, l₂) and cross-sectional areas (A₂, A₃) using Rayleigh–Love theory with real lengths and real cross-sectional areas.

No.	Length L1			Length l2			Cross-Section A2			Cross-Section A3
	L1calculated (m)	L1real (m)	Error (%)	l2calculated (m)	l2real (m)	Error (%)	A2calculated (cm2)	A2real (cm2)	Error (%)	A3calculated (cm2)	A3real (cm2)	Error (%)
1	1.094	1.1	0.57	0.035	0.01	250.20	5.531	3.534	56.49	4.018	7.069	43.16
2	1.108	1.1	0.73	0.103	0.05	106.94	3.742	3.534	5.87	7.914	7.069	11.95
3	1.108	1.1	0.73	0.105	0.1	5.06	3.392	3.534	4.02	9.252	7.069	30.89
4	1.108	1.1	0.73	0.191	0.2	4.49	3.392	3.534	4.02	7.400	7.069	4.69
5	1.108	1.1	0.73	0.259	0.25	3.79	3.392	3.534	4.02	7.390	7.069	4.55
6	1.108	1.1	0.73	0.311	0.3	3.66	3.392	3.534	4.02	7.376	7.069	4.34
7	1.108	1.1	0.73	0.414	0.4	3.47	3.392	3.534	4.02	7.233	7.069	2.32
8	1.108	1.1	0.73	0.509	0.5	1.88	3.392	3.534	4.02	7.122	7.069	0.75

, the signals are received at the sensor position with various length of l₂ (area reduction zone). As the length of l₂ increases, the time t₂ to receive the reflected signal also increases. The main wavelength λ of the stress impulse of duration T_i is equal to

$$\lambda =c{T}_i=c\frac{2L}{c_{\mathrm{s}} }=2L\left(\frac{c}{c_{\mathrm{s}} }\right),$$

(38)

where c is the wave speed of the semi-infinite rod material and c_s is the wave speed of the striker rod material.

From Equation (38), we know that the wavelength λ can be less than 2L when the striker rod material has a higher Young’s modulus than the semi-infinite rod material. When c = c_s, the wavelength λ is equal to 2L. It is known from the NDT theory that the defect of size l₂ can be identified only by wavelength λ if the impact wave is smaller than l₂, i.e., λ = 2L < l₂. The damaged length l₂ is determined as

$$l_2=\frac{1}{2}{t}_{2}c.$$

(39)

When l₂ is less than or equal to 2L (2 × 0.1 m = 0.2 m), the received signal is noisy due to the effects of lateral inertia and Poisson’s ratio; this is indicated in Table 7 with a large error (>5%), making it difficult to accurately determine l₂. This result highlights the significant influence of inertia and Poisson’s ratio on stress wave propagation in the rod, which causes noise in the signal when the damaged length l₂ is small, thus complicating the determination of changes in length and cross-sectional area. From Equation (38), it is known that the wavelength λ of the detecting incident wave is affected by the striker rod length L and the wave speed ratio of the rod and striker materials. The minimum defect length l₂ that can be identified is equal to the wavelength λ of the incident wave. The result is confirmed through the comparison in Table 7 of the calculated lengths (L₁, l₂) and cross-sectional areas (A₂, A₃) using the Rayleigh–Love theory with the real lengths and real cross-sectional areas.

The examples presented above aim to determine the change in cross-sectional area and its position in a rod based on the Rayleigh-Love rod theory. The model remains theoretical; it assumes the material to be linearly elastic, homogeneous, isotropic, and uniformly varying in cross-section, which may differ from real-world conditions. This theoretical work, however, establishes a foundation for understanding the behavior of stress wave propagation in Rayleigh–Love rods with abrupt variations in the cross-sectional area, considering lateral inertia and Poisson’s ratio. This understanding is crucial for developing accurate nondestructive testing methods applicable in real-world scenarios. Based on this research, further developments will be pursued with more complex material models and intricate geometric changes. This approach will enhance the applicability of our method for nondestructive testing (NDT) of rod-type structural elements under more realistic conditions.

5. Conclusions

This paper presents a detailed study of stress wave propagation in a Rayleigh–Love rod characterized by a sudden change in the cross-sectional area after a distance from the impacted end of the rod. Based on the analytical solution by Yang et al. [16], the analytical solutions of the transmitted and reflected stresses in a Rayleigh–Love rod with sudden cross-section variation were obtained. This study highlights the influence of a sudden cross-sectional variation on wave behavior, triggering reflections and transmissions at the interface of discontinuity. The role of Poisson’s effect is emphasized, demonstrating how it modulates wave behavior in the rod with discontinuities. Examples solved using the FEM verify the correctness of the modeling and numerical algorithm in terms of the analytical results of this study.

In addition to the forward analysis of Rayleigh–Love wave propagation in rods impacted by a striker rod with the same impedance as the rod to be inspected, an impact-echo-type NDT method is proposed to assess the condition of rod-type structural components with sudden cross-sectional area changes within a cover medium based on the measured signal at the measurable zone of the rod to be inspected.

The investigation results show that when determining the cross-sectional area using the σ_R/σ_I and σ_T3/σ_I ratios, the decision to consider or not consider Poisson’s ratio has little influence on the final ratio (Figure 15 and Figure 21). Including Poisson’s ratio leads to a similar rate of increase in amplitudes in the incident and reflected stress waves. As a result, the reflected ratio ${\sigma }_R^p/{\sigma }_I^p$ remains nearly the same, regardless of whether Poisson’s ratio is accounted for or not. However, in practical applications, the first peak ${\sigma }_I^p$, ${\sigma }_R^p$, and ${\sigma }_{T3}^p$ values can be easily determined through measured signals. The technique proposed in this study can be easily applied in the field to conduct nondestructive evaluation of the location L₁, extension l₂, and ratios of cross-sectional area changes (A₂/A₁, A₃/A₂) of a defect in a rod within a cover medium.