⇦Chinese Physics Letters, 2017, Vol. 34, No. 6, Article code 064302⇨ Influence of Change in Inner Layer Thickness of Composite Circular Tube on Second-Harmonic Generation by Primary Circumferential Ultrasonic Guided Wave Propagation * Ming-Liang Li(李明亮)1, Ming-Xi Deng(邓明晰)1**, Guang-Jian Gao(高广健)1, Han Chen(陈瀚)1, Yan-Xun Xiang(项延训)2** Affiliations 1Department of Physics, Logistics Engineering University, Chongqing 401331 2School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237 Received 28 December 2016 ^*Supported by the National Natural Science Foundation of China under Grant Nos 11474361, 11474093 and 11274388.
^**Corresponding author. Email: dengmx65@yahoo.com; yxxiang@ecust.edu.cn Citation Text: Li M L, Deng M X, Gao G J, Chen H and Xiang Y X 2017 Chin. Phys. Lett. 34 064302 Abstract The influence of change in inner layer thickness of a composite circular tube is investigated on second-harmonic generation (SHG) by primary circumferential ultrasonic guided wave (CUGW) propagation. Within a second-order perturbation approximation, the nonlinear effect of primary CUGW propagation is treated as a second-order perturbation to its linear response. It is found that change in inner layer thickness of the composite circular tube will influence the efficiency of SHG by primary CUGW propagation in several aspects. In particular, with change in inner layer thickness, the phase velocity matching condition that is originally satisfied for the primary and double-frequency CUGW mode pair selected may no longer be satisfied. This will remarkably influence the efficiency of SHG by primary CUGW propagation. Theoretical analyses and numerical results show that the effect of SHG by primary CUGW propagation is very sensitive to change in inner layer thickness, and it can be used to accurately monitor a minor change in inner layer thickness of the composite circular tube. DOI:10.1088/0256-307X/34/6/064302 PACS:43.35.-c, 43.25.+y, 43.20.Mv © 2017 Chinese Physics Society Article Text It is known that composite circular tubes have widely been used in chemical, pharmaceutical, food and power engineering industries.[1-4] Generally, a composite circular tube is constituted by two circular tubes of different metals, which are joined together by use of the mechanical or metallurgical technique. In contrast with a single circular tube, a composite circular tube can make better use of optimum performance of the base tube (outer layer) and the clad tube (inner layer). However, it is inevitable that the factor such as severe corrosion or wear of transporting substances during service will result in change (generally, thinning) in inner layer thickness.[1-4] For ensuring the safety of using a composite circular tube, it is of importance to develop a method to accurately monitor the minor change in inner layer thickness. In the last two decades, for accurately assessing minor change in material/structure, the investigations of the nonlinear ultrasonic guided waves have attracted a great deal of attention.[5] Deng first conducted the investigations of second-harmonic generation (SHG) of primary (fundamental) guided wave propagation in solid plates.[6,7] Subsequently, a series of theoretical and experimental investigations have been conducted on the nonlinear ultrasonic guided waves, which further revealed the physical mechanism of SHG of guided wave propagation, and meanwhile presented some effective techniques for measuring the second harmonic generated.[8-11] It has been experimentally verified that the minor change in material properties (e.g., fatigue) could be quantitatively assessed using the acoustic nonlinearity parameter as measured with the second harmonic generated by primary guided wave propagation.[12-14] As a kind of elementary mode of ultrasonic guided wave propagation along the circumference of a circular tube (referred to as circumferential ultrasonic guided wave (CUGW) mode),[15] linear CUGW has provided an efficient means for detecting radial fatigue cracks,[16] pitting corrosion,[17] and longitudinal defects in the circular tube.[18,19] However, it is difficult for the traditional linear CUGW to monitor minor change in material/structure of a circular tube. Based on the theoretical and experimental investigations of nonlinear effect of CUGW propagation in a single layer circular tube,[20,21] it has been experimentally validated that the minor change (early damage) in a tube material can be quantitatively assessed using the acoustic nonlinearity parameter for CUGW propagation.[22] In addition to minor change in the tube material itself, it can be expected that a minor change in geometrical parameter (e.g., inner layer thickness) may also obviously influence the efficiency of SHG of primary CUGW propagation.[23] It is of significance to further understand the corresponding physical process by analyzing the influence of minor change in geometrical parameter on SHG of primary CUGW propagation. Moreover, for ensuring the safety of using a composite circular tube,[1-4] it is necessary to investigate the relationship between the effect of SHG of primary CUGW and minor change in inner layer thickness. This work will focus on analyzing how the effect of SHG of primary CUGW propagation is influenced by minor change in inner layer thickness. The results exhibit a potential for accurately monitoring minor change in inner layer thickness through the acoustic nonlinearity parameter as obtained with the second harmonic by primary CUGW propagation.

Fig. 1. Schematic diagram of the two-dimensional model for CUGW propagation in a composite circular tube.

The schematic diagram of the two-dimensional model used for analyzing CUGW propagation in a composite circular tube is illustrated in Fig. 1, where the materials of the inner layer (layer 1) and the outer one (layer 2) are assumed to be isotropic and homogenous with no attenuation and no dispersion. For a perfect interface, the mechanical boundary conditions at the interface $r=R_{2}$ can be expressed as[24-26] $$\begin{align} P_{rr,1} =\,&P_{rr,2},~P_{\theta r,1}=P_{\theta r,2}, \\ U_{r,1} =\,&U_{r,2},~U_{\theta,1} =U_{\theta,2},~~ \tag {1} \end{align} $$ where $P_{rr}$ and $P_{\theta r}$ are the radial and circumferential components of stress, $U_{r}={\boldsymbol U}\cdot \hat {\boldsymbol r}$ and $U_{\theta}={\boldsymbol U}\cdot \hat {\boldsymbol \theta}$ are the radial and circumferential components of mechanical displacement field ${\boldsymbol U}$, $\hat {\boldsymbol r}$ and $\hat {\boldsymbol \theta}$ are unit vectors along the radial and circumferential directions, respectively, and the subscript number 1 or 2 in $P$ and $U$ corresponds to layer 1 or 2 in Fig. 1. When a primary CUGW mode propagates (with the angular frequency $\omega$ and the order $l$) along the tube circumference shown in Fig. 1, the corresponding displacement field can formally be written as $U_{q,i}^{(l)}(r)\exp [jn^{(\omega,l)}\theta -j\omega t]$, where $i=1$ or 2, $q=r$ or $\theta$, $n^{(\omega,l)}=\omega R_3/{c_{\rm p}^{(\omega,l)}}$ is the dimensionless angular wave number of the $l$th CUGW mode, and $c_{\rm p}^{(\omega,l)}$ is the corresponding phase velocity. The dispersion equation of the $l$th CUGW can be obtained from the mechanical boundary conditions, and can formally be written as[25,27] $$\begin{align} |{\boldsymbol M}(\rho _i,\lambda _i,\mu _i,c_p^{(\omega,l)},\omega,R_j)|=0,~~ \tag {2} \end{align} $$ where ${\boldsymbol M}$ is the coefficient matrix of the boundary condition equations, $\rho _{i}$, $\lambda_{i}$, and $\mu_{i}$ ($i=1$, 2) are the mass density and second-order elastic constants of material of the $i$th circular layer, and $R_{j}$ ($j=1$, 2, or 3) denotes the radius of the composite circular tube. For simplicity, the sinusoidal radial stress component $T_{rr}$ of angular frequency $\omega$ (only existing within the range of circumference angle $\theta\in[0, \eta]$) is applied on the outer surface of the composite circular tube (see Fig. 1), and the CUGW propagation only along the anticlockwise circumference of composite circular tube is considered. According to the modal expansion approach for waveguide excitation, the CUGW field (denoted by ${\boldsymbol U}(r,\theta)$) generated by $T_{rr}$ can formally be written as[20-25] $$\begin{align} {\boldsymbol U}(r,\theta)=\sum\limits_l {a_l} (\theta){\boldsymbol U}^{(l)}(r),~~ \tag {3} \end{align} $$ where ${\boldsymbol U}^{(l)}(r)$ is the displacement field function of the $l$th CUGW mode, and $a_{l}(\theta)$ is the corresponding expansion coefficient. The equation governing the expansion coefficient $a_{l}(\theta)$ can be expressed as[20-25] $$\begin{align} \Big(\frac{\partial}{\partial \theta}-jn^{(\omega,l)}\Big)a_l(\theta)=\frac{f_l^{\rm S} (\theta)+f_l^{\rm V} (\theta)}{4P_{ll}},~~ \tag {4} \end{align} $$ where $P_{ll}$ is the average power flow of the $l$th primary CUGW mode, per unit width along the length of the composite tube. Clearly, the bulk source $f_l^{\rm V} (\theta)$ is zero as no bulk force is applied inside the given composite circular tube. The formal form of surface source $f_l^{\rm S} (\theta)$ is given as $f_l^{\rm S} (\theta)=j\omega \cdot T_{rr}\cdot {\tilde {U}_{r,2}^{(l)} (r)}|_{r=R_3}$,[25] where the superscript $\sim$ means the complex conjugate operation for the corresponding physical quantity. The expansion coefficient of the $l$th primary CUGW mode can formally be written as[25] $$\begin{align} a_l (\theta)=\,&-\frac{T_{rr}\cdot \tilde {U}_{r,2}^{(l)}(r)|_{r=R_{\rm 3}}}{4P_{ll}}\cdot \frac{\omega}{n^{(\omega,l)}}[\exp (-jn^{(\omega,l)}\eta)\\ &-1]\times \exp [jn^{(\omega,l)}\theta].~~ \tag {5} \end{align} $$ Next, the effect of SHG of primary CUGW propagation in composite circular tube will be analyzed. When the $l$th CUGW mode (primary mode) propagates along the tube circumference shown in Fig. 1, within the second-order perturbation, the bulk driving force of double the fundamental frequency ${\boldsymbol F}_{\rm b}^{(2\omega)}={\boldsymbol F}[{\boldsymbol U}^{(l)}]$ inside the composite circular tube, as well as the traction stress tensor of double the fundamental frequency ${\boldsymbol P}_{\rm s}^{(2\omega)}={\boldsymbol P}[{\boldsymbol U}^{(l)}]$ on the interfaces/surfaces of the composite circular tube, can be generated due to the convective nonlinearity and the inherent elastic nonlinearity of solid.[8,20-24,28] According to the modal expansion approach, ${\boldsymbol F}_{\rm b}^{(2\omega)}$ and ${\boldsymbol P}_{\rm s}^{(2\omega)}$ are, respectively, assumed to be the bulk and surface sources for generation of a series of double-frequency CUGW modes that constitute the second-harmonic field (denoted by ${\boldsymbol U}^{(2\omega,l)}(r,\theta)$) of the $l$th CUGW mode, namely,[8,20-24,28] $$\begin{align} {\boldsymbol U}^{(2\omega,l)}(r,\theta)=\sum\limits_m A_m(\theta){\boldsymbol U}^{(2\omega,m)}(r),~~ \tag {6} \end{align} $$ where ${\boldsymbol U}^{(2\omega, m)}$ is the field function of the $m$th double-frequency CGUW generated by ${\boldsymbol F}_{\rm b}^{(2\omega)}$ and ${\boldsymbol P}_{\rm s}^{(2\omega)}$ originated from propagation of the $l$th CUGW mode, and $A_{m}(\theta)$ is the corresponding expansion coefficient. Similar to the process for solving the dispersion equation of primary CUGW, the dispersion equation of double-frequency CUGW mode can readily be obtained by Eq. (2) just using 2$\omega$ instead of $\omega$ and using $c_{\rm p}^{(2\omega,m)}$ instead of $c_{\rm p}^{(\omega,l)}$.[20-24] Here $c_{\rm p}^{(2\omega,m)}$ is the phase velocity of the $m$th double-frequency CUGW mode, and the corresponding dimensionless angular wave number is given by $n^{(2\omega,m)}=2\omega R_3/{c_{\rm p}^{(2\omega,m)}}$. The modal equation governing the expansion coefficient $A_{m}(\theta)$ can readily be obtained through Eq. (4) by substituting $n^{(\omega,l)}$ with $n^{(2\omega,m)}$, $a_l (\theta)$ with $A_m (\theta)$, $P_{ll}$ with $P_{mm}$, $f_l^{\rm V} (\theta)$ with $F_m^{\rm V} (\theta)$, and $f_l^{\rm S} (\theta)$ with $F_m^{\rm S} (\theta)$, respectively, where $P_{mm}$ is the average power flow of the $m$th double-frequency CUGW mode, per unit width along the length of the composite tube. The bulk source $F_m^{\rm V} (\theta)$ (originated from ${\boldsymbol F}_{\rm b}^{(2\omega)}$) and surface source $F_m^{\rm S} (\theta)$ (originated from ${\boldsymbol P}_{\rm s}^{(2\omega)}$) are, respectively, expressed as[8,20-24,28] $$\begin{alignat}{1} F_m^{\rm V} (\theta)=\,&2j\omega \sum\limits_{i=1}^2 \int_{R_i}^{R_{i+1}} \tilde {\boldsymbol U}_i^{(2\omega,m)} (r)\cdot {\boldsymbol F}_{\rm b}^{(2\omega)}dr,~~ \tag {7} \end{alignat} $$ $$\begin{alignat}{1} F_m^{\rm S}(\theta)=\,&2j\omega \sum\limits_{i=1}^2 \{\tilde {\boldsymbol U}_i^{(2\omega,m)} (r)\cdot {\boldsymbol P}_{\rm s}^{(2\omega)} \}\cdot \hat {\boldsymbol r}|_{r=R_i}^{r=R_{i+1}}.~~ \tag {8} \end{alignat} $$ Considering the initial condition for generation of the double-frequency CUGW, i.e., $A_{m}(\theta)=0$ when $\theta=0$, the expansion coefficient of the $m$th double-frequency CUGW mode can be formally given by[8,20-24,28] $$\begin{align} A_m (\theta)=\,&A_m \frac{\sin (\Delta n\theta)}{\Delta n}\exp (j\Delta n\theta)\times \exp [jn^{(2\omega,m)}\theta], \\ A_m =\,&[F_m^{\rm S} (\theta)+F_m^{\rm V} (\theta)]/4P_{mm},~~ \tag {9} \end{align} $$ where $\Delta n=[n^{(\omega,l)}-n^{(2\omega,m)}/2]$ is used to describe the degree of dispersion (i.e., the difference of phase velocity between the $l$th primary and $m$th double-frequency CUGWs). It is found that $A_{m}(\theta)$ in Eq. (9) may grow linearly with the circumferential angle $\theta$ when $\Delta n=0$ or $\Delta n \approx 0$ (i.e., the phase velocity matching is satisfied), and its formal solution is given by $A_{m}(\theta)=A_m \exp [jn^{(2\omega,m)}\theta]\times \theta$.[8,20-24] For this case, the second-harmonic field ${\boldsymbol U}^{(2\omega,l)}(r,\theta)$ (mainly dependent on the $m$th double-frequency CUGW mode) accumulates along the circumferential direction. It should be noted that the solution given by Eq. (9) is valid when the condition of second-order perturbation approximation is applicable. To highlight the influence of change in inner layer thickness (relative to its initial thickness $d_{0}$) on SHG by primary CUGW propagation, for simplicity, the material properties of the composite circular tube in Fig. 1, as well as its outer layer thickness, are assumed to be unchanged in the analysis process. For the initial case where the inner layer thickness is $d_{0}$, an appropriate mode pair (e.g., the $l$th primary and $m$th double-frequency CUGW modes), as well as the driving frequency $f$, can be selected to ensure $A_m \ne 0$ and $\Delta n=0$ or $\Delta n \approx 0$ (i.e., $c_{\rm p}^{(\omega,l)}=c_{\rm p}^{(2\omega,m)}$). Thus the clear second-harmonic signal (mainly dependent on the $m$th double-frequency CUGW mode) can be observed.[7,20-24] Theoretically, the influence of change in inner layer thickness on SHG of primary CUGW propagation is reflected in the following aspects. First, change in inner layer thickness will influence the dispersion relation of CUGWs. For the mode pair selected, the condition $\Delta n=0$ or $\Delta n \approx 0$ will no longer be satisfied when a change in inner layer thickness takes place. This will remarkably influence the efficiency of SHG by the $l$th primary CUGW propagation (reflected through the factor ${\sin (\Delta n\theta)}/{\Delta n}$ in Eq. (9)). Secondly, the magnitude of inner layer thickness is directly associated with the acoustic field for the $l$th primary CUGW mode.[7,20-24] This will also influence the magnitude of $A_{m}$ because the source functions $F_m^{\rm V} (\theta)$ and $F_m^{\rm S} (\theta)$ in Eqs. (7) and (8) are proportional to the square of amplitude of primary CUGW mode.[7,20-24] The given composite circular tube in Fig. 1 from the inner layer to the outer one in order is assumed to be copper and steel, and its initial radii are set to be $R_{1}=130$ mm, $R_{2}=132$ mm, and $R_{3}=140$ mm. The initial thickness of the inner tube is $d_{0}=R_{2}-R_{1}=2$ mm. Due to corrosion and/or wear, change in the inner layer thickness of the composite circular tube is generally manifested by the increase of $R_{1}$, and correspondingly the thinning inner layer thickness varies from $d_{0}$ (initial thickness) to $d$ (current thickness). The relative change rate $e=\Delta d/d_{0}$ is defined to describe the degree of change in the inner layer thickness, where $\Delta d=d_{0}-d$.

Fig. 2. Dispersion curves for primary and double-frequency CUGWs in the given composite circular tube.

The material parameters of composite circular tube are listed in Table 1. The dispersion curves of CUGWs in the composite circular tube with the initial radii are calculated, as shown in Fig. 2. The curves of the $l$th primary and the $m$th double-frequency CUGW modes cross at point D$_{0}$, which means that the phase velocity matching is satisfied ($c_{\rm p}^{(\omega,l)} =c_{\rm p}^{(2\omega,m)}=5.132$ km/s) at the driving frequency $f=0.152$ MHz given by the vertical dotted line $V$. When the $l$th primary CGUW mode (point D$_{0}$) propagates along the circumference, the effect of SHG takes place and the corresponding second-harmonic field (i.e., ${\boldsymbol U}^{(2\omega,l)}(r,\theta)$ in Eq. (6)) can be decomposed into a series of double-frequency CUGW modes (denoted by points D$_{0}$, D$_{1}$, D$_{2}$, and D$_{3}$ in Fig. 2). Based on the above analysis, it is known that ${\boldsymbol U}^{(2\omega,l)}(r,\theta)$ in Eq. (6) is mainly dependent on the double-frequency mode (point D$_{0}$) due to the fact that $c_{\rm p}^{(\omega,l)}=c_{\rm p}^{(2\omega,m)}$ will accumulate along the circumference, and the contribution of other double-frequency modes (e.g., D$_{1}$, D$_{2}$, and D$_{3}$) to ${\boldsymbol U}^{(2\omega,l)}(r,\theta)$ will be negligible due to phase velocity mismatching.[20-22]

**Table 1.** Material parameters of composite circular tube.
Material	Mass density	Longitudinal wave	Shear wave	3rd order elastic constants (GPa)[29]
Material	$\rho $ (10$^{4}$ kg/m$^{3}$)	velocity $c_{\rm L}$ (km/s)	velocity $c_{\rm T}$ (km/s)	$A$	$B$	$C$
Copper	0.89	4.66	2.26	$-$1480	150	$-$320
Steel	0.78	5.85	3.32	$-$720	$-$230	180

Next, analysis will be focused on the influence of change in the inner layer thickness on the effect of SHG for the selected primary mode (point D$_{0}$) and double-frequency mode (point D$_{0}$). For the driving frequency $f=0.152$ MHz in Fig. 2, the curves of the phase velocities of both the $l$th primary and the $m$th double-frequency CUGWs versus the relative change rate $e$ of the inner layer thickness are shown in Fig. 3. Clearly, the phase velocity matching condition, $c_{\rm p}^{(\omega,l)}=c_{\rm p}^{(2\omega,m)}$, is no longer satisfied when the minor change in the inner layer thickness takes place.

Fig. 3. Influences of $e$ on phase velocities of primary and double-frequency mode selected.

It can be found from Fig. 3 that when the relative change rate $e$ is within 7% (i.e., decrease in the inner layer thickness is within 0.14 mm), the phase velocity $c_{\rm p}^{(\omega,l)}$ of the $l$th primary CUGW mode almost linearly increases with $e$, and its maximum relative change rate (defined by $[c_{\rm p}^{(\omega,l)}|_{d}-c_{\rm p}^{(\omega,l)}|_{d_0}]/c_{\rm p}^{(\omega,l)}|_{d_0}$) is 0.68%. According to Eqs. (3) and (5), for the surface excitation source $T_{rr}$ of angular frequency $\omega$ (distributed uniformly along circumferential direction), Fig. 4 presents the curve of normalized amplitude of $a_l(\theta){\boldsymbol U}^{(l)} (r)/T_{rr}|_{r=R_3}$ of the $l$th primary CUGW (point D$_{0}$) versus the relative change rate $e$, which is independent of the circumference angle $\theta$. It is found that when the relative change rate $e$ is within 7%, the normalized amplitude of the $l$th primary CUGW (point D$_{0}$) varies within a range of 2.7%, and additionally the relationship between the normalized amplitude of $a_l(\theta){\boldsymbol U}^{(l)}(r)/T_{rr}|_{r=R_3}$ and $e$ is not monotonic. Obviously, the linear property of CUGW propagation (e.g., change in phase velocity or amplitude) is not suitable for accurately monitoring the minor change in inner layer thickness.

Fig. 4. Curve of normalized amplitude of $a_l(\theta){\boldsymbol U}^{(l)}(r)/T_{rr}|_{r=R_3}$ ($l$th primary CUGW, point D$_{0}$) versus $e$.

Based on Eqs. (6)-(9), under different $e$, the curves of the displacement amplitudes of the $m$th double-frequency CUGW (point D$_{0}$) at $r=R_{3}$ versus circumferential angle $\theta$ are shown in Fig. 5, where $U^{(l)}$ is the amplitude of the $l$th primary CUGW at $r=R_{3}$. Clearly, with the increase of $e$, the efficiency of SHG of primary CUGW propagation gradually falls off due to the phase velocity mismatching between the primary and double-frequency CUGWs (see Fig. 3). Considering the fact that the ratio of $A_{\rm 2f}/A_{\rm 1f}^2$ is a measure of the acoustic nonlinearity parameter relative to its initial value and is proportional to the absolute acoustic nonlinearity parameter, for simplicity, $A_{\rm 2f}/A_{\rm 1f}^2$ (denoted by $\beta$) can directly be used as the acoustic nonlinearity parameter for material/structure evaluation.[30-32] Because this investigation uses the specified CUGW mode pair (point D$_{0}$ in Fig. 2), $A_{\rm 1f}$ and $A_{\rm 2f}$ are, respectively, the amplitudes of the primary and double-frequency CUGW modes (point D$_{0}$) at the driving frequency given by the vertical dotted line $V$ in Fig. 2. Based on the calculated results shown in Figs. 4 and 5, for a specific circumferential angle $\theta$, $A_{\rm 1f}$ and $A_{\rm 2f}$ of the primary and double-frequency CUGWs, as well as the acoustic nonlinearity parameter $\beta$, can readily be determined for different relative change rates $e$. Figure 6 shows the curve of normalized acoustic nonlinearity parameter $\beta$ versus the relative change rate $e$ at the circumferential angle $\theta=4$ rad, which falls off sensitively and monotonically with the increase of $e$. Obviously, for an appropriate circumferential angle $\theta$ (e.g., $\theta=4$ rad), $\beta$ of primary CUGW propagation is more sensitive and suitable for monitoring the minor change in the inner layer of the composite circular tube.

Fig. 5. Curves of the displacement amplitudes the double-frequency CUGW (point D$_{0}$) versus $\theta$ for different $e$: (a) radial component and (b) circumferential component.

Fig. 6. Normalized acoustic nonlinearity parameter $\beta$ versus $e$ at the circumferential angle $\theta=4$ rad.

In conclusion, using a modal analysis approach for waveguide excitation, the formal solution for the second harmonics of CUGW propagation in a composite circular tube has been obtained within the second-order perturbation approximation. It has been found that the efficiency of SHG by CUGW propagation is closely associated with the inner layer thickness, when a specific CUGW mode pair is selected. Change in inner layer thickness of the composite circular tube will sensitively and monotonically influence the efficiency of SHG through influencing the degree of phase velocity matching and the acoustic field of primary CUGW mode selected. The theoretical analyses and numerical simulation results indicate that the efficiency of SHG of primary CUGW propagation is very sensitive to change in the inner layer thickness, and the acoustic nonlinearity parameter as obtained with the second harmonic by primary CUGW propagation can be used to accurately monitor the minor change in the inner layer of the composite circular tube. References The Basic Research on Numerical Simulation of Bimetal Composited T-Tube through Hydraulic BulgingTheoretical and Experimental Study of Bimetal-Pipe HydroformingModern Materials: Bimetallic TubesOptimization procedures for GMAW of bimetal pipesReview of nonlinear ultrasonic guided wave nondestructive evaluation: theory, numerics, and experimentsSecond-Harmonic Properties of Horizontally Polarized Shear Modes in an Isotropic PlateCumulative second-harmonic generation of Lamb-mode propagation in a solid plateAnalysis of second-harmonic generation of Lamb modes using a modal analysis approachOn the existence of antisymmetric or symmetric Lamb waves at nonlinear higher harmonicsSecond harmonic generation in composites: Theoretical and numerical analysesTime-domain analysis and experimental examination of cumulative second-harmonic generation by primary Lamb wave propagationAssessment of accumulated fatigue damage in solid plates using nonlinear Lamb wave approachCumulative second-harmonic analysis of ultrasonic Lamb waves for ageing behavior study of modified-HP austenite steelImpact Damage Detection in Composite Structures Considering Nonlinear Lamb Wave PropagationGuided circumferential waves in layered cylindersReview of Progress in Quantitative Nondestructive EvaluationUltrasonic Circumferential Guided Wave for Pitting-Type Corrosion Imaging at Inaccessible Pipe-Support LocationsOn Circumferential Disposition of Pipe Defects by Long-Range Ultrasonic Guided WavesExperimental Observation of Cumulative Second-Harmonic Generation of Circumferential Guided Wave Propagation in a Circular TubeAssessment of accumulated damage in circular tubes using nonlinear circumferential guided wave approach: A feasibility studyCharacterization of surface properties of a solid plate using nonlinear Lamb wave approachFinite-amplitude waves in isotropic elastic platesEvaluation of fatigue damage using nonlinear guided wavesFatigue damage assessment by nonlinear ultrasonic materials characterizationAssessment of material damage in a nickel-base superalloy using nonlinear Rayleigh surface waves

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