Relative acoustic nonlinearity parameter variation with local defect resonance frequencies and its different harmonics

Abstract

In recent times, nonlinear ultrasonic inspection has become a very reliable technique to characterize micro-scale cracks and small-scale delaminations. In this paper, an analytical framework is developed for the orthotropic composite plate to calculate the variation of relative acoustic nonlinearity parameter, a quantitative measurement of nonlinearity at the delamination region. The analytical investigation is justified with numerical and experimental research using a carbon fiber-reinforced polymer (CFRP) plate with three delaminations. Two incident waves are used for excitation, out of which the first wave frequency is varied as local defect resonance (LDR) frequency (f_d), subharmonic LDR frequency (2f_d), second-order superharmonic LDR frequency (f_d/2), and third-order superharmonic LDR frequency (f_d/3) while the second incident wave is taken as a single periodic frequency. Different delamination sizes, positions, and depths were used to understand the second-order relative acoustic nonlinearity parameter (β₂) with other LDR frequencies as well as its harmonics. It has been observed from analytical, numerical, and experimental results that the relative acoustic nonlinearity parameter is maximum at LDR frequencies. It increases in case of surface level delaminations.

Keywords

Carbon fiber reinforced polymer plate delamination second-harmonic frequency response

Introduction

In recent years, nonlinear ultrasound techniques have been developed rapidly to detect complicated defects like low-velocity impact damages in composite structures. In this technique, the significance of nonlinearity at the defect region is described by the origination of higher harmonics quantified by a relative acoustic nonlinearity parameter (β₂).¹ Two types of nonlinearity, classical and nonclassical, can be observed. Classical nonlinearity appears from material and geometric aspects, whereas contact nonlinearity (arising from both crack and delamination) is generally considered nonclassical nonlinearity.² Inclusion of later types of nonlinearity results leading to early detection and localization of the defect.³

Local defect resonance (LDR) is a nonlinear ultrasound technique that relies on maximizing the clapping and rubbing phenomena at delamination surfaces by exciting at the resonance frequency of the defect.⁴ Selective detection and visualization of defects can be achieved with contact and non-contact excitations.^5,6 Total driving energy can be channelized into the fault by nonlinear LDR excitation. A replacement of high-power ultrasonic thermography is possible by developing this low acoustic power version.⁷ The ultrasonic input energy is converted into heat energy due to resonance vibrations. Corresponding temperature images are represented by the thermosonic Chladni figures.⁸ Wide area non-contact excitations by loudspeakers and inspection by thermosonic and shearosonic techniques are efficient in damage detection.⁹ Furthermore, a fully acoustic version of NDT is found to be possible for comprehensive area inspection in the case of automotive and aerospace structures.¹⁰ The intermodulation terms are generated between the driving and damage resonance frequency shown analytically and experimentally.¹¹ The nonlinear responses such as second and third harmonics are found in the defect region. Corresponding temperature and displacement are obtained at the damage location by performing the nonlinear thermosonic and vibrometer technique.¹² Nonlinear narrow sweep excitation has been implemented based on the LDR concept, which leads to efficient activation of defect surfaces and high heat generation. The nonlinear response images are obtained using an IR camera, and the technique is termed nonlinear ultrasound excited thermography.¹³ It was found from previous discussions that defects can be activated very easily in known LDR frequency. But in actual defects, the determination of the LDR frequency is a very challenging task. So, wideband ultrasonic excitation signals are introduced instead of periodic sweeping signals.¹⁴ Generation of subharmonics and higher harmonics due to nonlinear intermodulation of the driving frequency associated with the resonance frequency at debonding region.¹⁵ The Bicoherence co-efficient and the normalized second-order nonlinear parameter can understand the nonlinear interaction between ultrasonic waves and material damage. The former is found to be better also in the low signal-to-noise ratio.¹⁶ Barely visible impact damages in the composite plate can be imaged using dual excitations with contact piezoelectric transducers known as nonlinear wave modulation thermography.¹⁷ Exciting in-plane LDR frequency claims to provide high sensitivity results for specific geometry of the defects by decreasing measurement time.¹⁸ It is proved both numerically and experimentally that the Bicoherence analysis can detect the LDR frequency at the damage region using single and dual excitation.^19,20 The second harmonic LDR frequency was found from Bicoherence plots. An analytical model is developed to understand the generation of intermodulation terms in dual excitations, validated experimentally.²¹ The nonlinearity parameter is useful to measure the nonlinear behavior of the material, which is expressed by the second harmonics. The Lamb wave low-frequency S₀ mode illustrates different frequency-thickness with a relative acoustic nonlinearity parameter and propagation distance.²² The investigation describes the nonlinear acoustic responses on microstructural damages in metallic structures due to cold rolling and heat treatment processes. The corresponding relation has been developed between the nonlinear acoustic response and grain sizes in the structures. It is recommended that the nonlinear ultrasonic technique can be used for the complete characterization of material degradation.²³ In the similar fashion, the nonlinear behavior of the delaminations with their respective sizes has been presented.²⁴ The higher harmonic responses are not sufficient to find the fundamental frequency. But the second harmonic response provides accurate fundamental frequency, and efficient nonlinearity is located in the flaw region.²⁵ The formation of the second harmonic helps characterize the low-velocity impact damages in composite structures, which is achieved through the phase reversal method.²⁶ The quantitative measurement of nonlinearity for impact damage is implemented by guiding wave mixing.²⁷ However, the behaviour of the relative acoustic nonlinearity parameter has not been extensively studied earlier.

In this paper, an analytical expression of relative acoustic nonlinearity parameter (β₂) is derived for flexural wave propagation in a Carbon fiber reinforced polymer (CFRP) plate subjected to dual excitations. The analytical calculation is performed in MATLAB. Subsequently, a numerical study is conducted using explicit dynamic analysis in the ABAQUS platform. Moreover, an experimental setup for the computation of relative acoustic nonlinearity parameter is demonstrated. Finally, the nonlinearity parameter for excitation frequency in the analytical study is reported and further validated with numerical and experimental results. The paper is organized as follows: first, the analytical study is presented, followed by numerical research and fabrication of the CFRP plate. Finally, a description of the experimental setup and results and discussions are provided, followed by conclusions.

Methodology

Analytical investigation

The propagation of the incident wave is used to characterize the defect region in composite material. When the incident wave frequency matches with defect frequency, it produces higher vibrational amplitudes at the defect location. Based on this concept, the analytical LDR frequency can be calculated from the model of the plate containing an FBH as shown in Figure 1⁵

f_{d} = \frac{1.6 d}{r^{2}} \sqrt{\frac{E}{12 ρ (1 - ν^{2})}}

(1)

here f_d = LDR frequency, d = residual thickness, r = radius of delamination, E = Young’s modulus of elasticity, ρ = density,

ν

= Poisson’s ratio. The nonlinear governing equation of flexural wave propagation in a composite plate is expressed as follows²¹

S_{11} \frac{\partial^{4} w_{1}}{\partial x^{4}} + 2 (S_{12} + 2 S_{66}) \frac{\partial^{4} w_{1}}{\partial x^{2} \partial y^{2}} + S_{22} \frac{\partial^{4} w_{1}}{\partial y^{4}} + I_{0} \frac{\partial^{2} w_{1}}{\partial t^{2}} - I_{2} (\frac{\partial^{2} {\ddot{w}}_{1}}{\partial x^{2}} + \frac{\partial^{2} {\ddot{w}}_{1}}{\partial y^{2}}) - q_{1} - q_{2} + q {}_{N L}= 0

(2)

where the generation of nonlinear force is denoted as q_NL at particular coordinates x_d, y_d (Defect location). The two-point forces are denoted as q₁ and q₂, and The _Sij is the bending stiffness matrix. The solution of the above equation for the nonlinear part is given as²¹

S_{i j} = \frac{1}{3} \sum_{K = 1}^{N} Ψ_{i j}^{(k)} (z_{k + 1}^{3} - z_{k}^{3})

(3)

where reduced stiffness matrix is denoted by ψ_ij^(k) for k^th orthotropic lamina when considering plane stress condition, the number of layers is represented by N. From equation (2), the expression of First and second Mass moment (I0 & I2) inertia followed as²⁸

I_{0} = \sum_{k = 1}^{N} ρ_{0}^{(k)} (z_{k + 1} - z_{k})

(4)

I_{2} = \frac{1}{3} \sum_{k = 1}^{N} ρ_{0}^{(k)} (z_{k + 1}^{3} - z_{k}^{3})

(5)

Figure 1.

The flexural wave is scattered through a plate, which leads to the nonlinear response at FBH.

The boundary condition is considered simply supported. The length and width of the plate are represented as l and b. The displacement and velocity initial conditions are considered as zero. According to first-order perturbation theory, the total displacement considering the quadratic nonlinearity can be expressed as

w_{1} (x, y, t) = w_{2} (x, y, t) + ξ w_{3} (x, y, t) + O (ξ^{2})

(6)

where

w_{2} (x, y, t)

and

w_{3} (x, y, t)

are the displacement corresponding to linear and nonlinear force (q_NL), respectively.

O (ξ^{2})

Is the higher-order term and

ξ

is the nonlinear constant. The linear term in the previous equation couldn’t explain the nonlinearity arising from defect location, which generates the intermodulation terms. So, this part can be neglected, and the last equation can be re-written as

w_{1} (x, y, t) = ξ w_{3} (x, y, t)

(7)

Substituting equations (7) into (2) leads to

\begin{array}{l} ξ S_{11} \frac{\partial^{4} w_{3}}{\partial x^{4}} + 2 ξ (S_{12} + 2 S_{66}) \frac{\partial^{4} w_{3}}{\partial x^{2} \partial y^{2}} + ξ S_{22} \frac{\partial^{4} w_{3}}{\partial y^{4}} + ξ I_{0} \frac{\partial^{2} w_{3}}{\partial t^{2}} - ξ I_{2} (\frac{\partial^{2} \overset{..}{w_{3}}}{\partial x^{2}} + \frac{\partial^{2} \overset{..}{w_{3}}}{\partial y^{2}}) \\ + q_{N L} (ξ, x, y, t) = 0 \end{array}

(8)

From the above equation, the second-order nonlinear transverse load $(q_{N L})$ and displacement $(w_{3})$ can be expressed as

q_{N L} (ξ, x, y, t) = \sum \sum F_{m n}^{I I} (ξ, t) \sin α x \sin β y

(9)

w_{3} (ξ, x, y, t) = \sum \sum X_{m n} (t) \sin δ x \sin γ y

(10)

The nonlinear elastic response at the LDR frequency (f_mn = f_d) is considered to estimate the quadratic nonlinearity as follows²⁹

T_{m n} (ξ, t) = \bar{Q} ξ {[\frac{\cos 2 π f_{1} t - \cos 2 π f_{d} t}{f_{d}^{2} - f_{1}^{2}} + \frac{\cos 2 π f_{2} t - \cos 2 π f_{d} t}{f_{d}^{2} - f_{2}^{2}}]}^{2}

(11)

here

\bar{Q} = {(\frac{Q}{π^{2} a b} \sin δ x_{d} \sin γ y_{d})}^{2}

. Then, substituting equations (9) and (10) into (8) results

ξ X_{m n} [S_{11} δ^{4} + 2 (S_{12} + 2 S_{66}) δ^{2} γ^{2} + S_{22} γ^{4}] + ξ I_{0} {\overset{..}{X}}_{m n} - ξ I_{2} (δ^{2} + γ^{2}) {\overset{..}{X}}_{m n} + T_{m n} = 0

(12)

The above equation can be simplified as in the following equation

{\overset{..}{X}}_{m n} + 4 π^{2} f_{m n}^{2} X_{m n} = - \bar{T_{m n}}

(13)

where

\bar{T_{m n}} = A^{I I} {[\cos 2 π f_{1} t (f_{d}^{2} - f_{2}^{2}) + \cos 2 π f_{2} t (f_{d}^{2} - f_{1}^{2}) + \cos 2 π f_{d} t (f_{1}^{2} + f_{2}^{2} - 2 f_{d}^{2})]}^{2}

and

A^{I I} = \frac{\bar{Q}}{M_{m n} (f_{d}^{2} - f_{1}^{2}) (f_{d}^{2} - f_{2}^{2})}

The above inhomogeneous ordinary differential equation can be solved to obtain displacement equation as

w_{3} (x, y, t) = \sum \sum (V_{1} + V_{2} + V_{3} + V_{4,5} + V_{6,7} + V_{8,9}) \sin δ x \sin γ y

(14)

where individual terms are as follows

\begin{array}{l} V_{1} = \frac{A^{I I} (f_{2}^{2} - f_{d}^{2}) (1 + \cos 4 π f_{1} t)}{8 π^{2} (4 f_{1}^{2} - f_{m n}^{2})} \\ V_{2} = \frac{A^{I I} (f_{1}^{2} - f_{d}^{2}) (1 + \cos 4 π f_{2} t)}{(4 f_{2}^{2} - f_{m n}^{2})} \\ V_{3} = \frac{A^{I I} (f_{1}^{2} + f_{2}^{2} - 2 f_{d}^{2}) (1 + \cos 4 π f_{d} t)}{8 π^{2} (4 f_{d}^{2} - f_{m n}^{2})} \\ V_{4,5} = \frac{A^{I I} (f_{d}^{2} - f_{2}^{2}) (f_{d}^{2} - f_{1}^{2})}{4 π^{2} f_{m n}} [\frac{(f_{1} \pm f_{2}) {\cos 2 π (f_{1} \pm f_{2} - f_{m n}) t - \cos 2 π (f_{1} \pm f_{2} + f_{m n}) t}}{{(f_{1} \pm f_{2})}^{2} - f_{m n}^{2}}] \\ V_{6,7} = \frac{A^{I I} (f_{d}^{2} - f_{1}^{2}) (f_{1}^{2} + f_{2}^{2} - 2 f_{d}^{2})}{4 π^{2} f_{m n}} [\frac{(f_{2} \pm f_{d}) {\cos 2 π (f_{2} \pm f_{d} - f_{m n}) t - \cos 2 π (f_{2} \pm f_{d} + f_{m n}) t}}{{(f_{2} \pm f_{d})}^{2} - f_{m n}^{2}}] \\ V_{8,9} = \frac{A^{I I} (f_{d}^{2} - f_{2}^{2}) (f_{1}^{2} + f_{2}^{2} - 2 f_{d}^{2})}{4 π^{2} f_{m n}} [\frac{(f_{1} \pm f_{d}) {\cos 2 π (f_{1} \pm f_{d} - f_{m n}) t - \cos 2 π (f_{1} \pm f_{d} + f_{m n}) t}}{{(f_{1} \pm f_{d})}^{2} - f_{m n}^{2}}] \end{array}

(15)

where f₁ is variable frequency, i.e. subharmonic LDR frequency (2f_d), second-order superharmonic LDR frequency (f_d/2), third-order superharmonic LDR frequency (f_d/3) or LDR frequency (f_d); and f₂ is taken as fixed single periodic frequency (SPF) for all cases. V₁-V₃ represents the classical terms, whereas V_4,5-V_8,9 indicates nonclassical or intermodulation terms. Now, the first amplitude (A₁) and second amplitude (A₂) are derived from equation (14) as

\begin{array}{l} A_{1} = \frac{A^{I I} (f_{d}^{2} - f_{2}^{2}) (f_{1}^{2} + f_{2}^{2} - 2 f_{d}^{2})}{2 π^{2} f_{1} (f_{1} + 2 f_{d})} \\ A_{2} = \frac{A^{I I} (f_{2}^{2} - f_{d}^{2})}{8 π^{2} (4 f_{1}^{2} - f_{m n}^{2})} \end{array}

(16)

Relative acoustic nonlinearity parameter for second harmonic (β₂)

The generation of the second harmonic frequency response is very significant to characterize the material nonlinearity. So a relative acoustic nonlinearity parameter (β₂) is described by the ratio of the second harmonic frequency response amplitude to the square of first harmonic response amplitude as follows^22,24,25

β_{2} α \frac{A_{2}}{A_{1}^{2}}

(17)

where A₁ and A₂: amplitudes of fundamental and second harmonic frequency, respectively.

Numerical investigation

The numerical study on a multi-delaminated CFRP plate model (Figure 2(a)) was performed by the explicit dynamic analysis in ABAQUS- 6.14 software. Twenty layers were used for modeling the CFRP plate with a 7 mm thickness (Figure 2(a)). The size of each layer is considered as 190 × 170 × 0.35 mm³, the same as the fabricated one (Figure 2(a)). Different sizes of delaminations are created between the first and second layers (Ø 19 mm), fifth and sixth layers (Ø 22 mm), 10th and 11th layers (Ø 24 mm), respectively. The positions of all delaminations are similar to the delamination positions of the actual CFRP plate, as shown in Figure 2(a). The depth of 19, 22, and 24 mm delaminations are 0.7, 1.75, and 3.5 mm, respectively. The properties of the CFRP plate, mentioned in Table 1, are determined based on the rule of mixture concept.³⁰ Subsequently, a fixed time increment step was created for performing the explicit dynamic analysis using an element-by-element time increment estimator—the total period considered as 0.001 s. Later, the interaction is created between the surfaces of the layers at the delamination region by considering the frictionless interaction property. All adjacent layers of the CFRP plate are fixed by providing the tie constraints, except the delamination region. Further, the pinned boundary conditions are provided at the plate edges. The mesh attributes are assigned to all layers independently. A 10-noded modified quadratic tetrahedron (C3D10 M) mesh is created. The element thickness over the delamination and the rest of the plate is selected as 2.215 mm and 3 mm, respectively. Two transmitters and one receiver node are created over the surface of the plate, as shown in Figure 2(b).

Figure 2.

(a) Schematic diagram and (b) Meshed model of carbon fiber reinforced polymer plate.

Table 1.

Properties of the CFRP composite material.

Material	Density(Kg/m³)	Modulus of elasticity(E) in GPa	Poisson’s ratio
Epoxy(LV556)	1200	33	0.35
Carbon fibre	1546	136	0.29
CFRP (rule of mixture)	1478	106	0.29

The delamination is activated by providing the dual signals as input to the plate through the transmitter nodes. One input signal in the form of a SPF is considered, which is kept fixed for all delamination cases. Another input signal in the form of variable frequencies, i.e. LDR frequency (f_d), subharmonic LDR frequency (2f_d), second-order superharmonic LDR frequency (f_d/2), and third-order superharmonic LDR frequency (f_d/3), are considered.

The force amplitude at both the transmitters is taken as 40 N. First, the LDR frequency (f_d) is calculated analytically from equation (1) for each delamination. Later, analytical LDR frequency (f_d) is verified using the steady-state analysis.^19,20

Finally, the output data are extracted from the receiver node (over the delamination) and processed in MATLAB to obtain the frequency spectra and relative acoustic nonlinearity parameter (β₂). The corresponding results are presented in the results and discussions section.

Experimental investigation

In this section, the experimental work has been presented for justification of analytical and numerical work. A CFRP plate with 20 layers, each having a thickness of 0.35 mm and 0/90^o fiber orientation, is fabricated using the hand layup process. The epoxy resin (LY556) and the hardener (HY951) in a 1:1 weight ratio are used to fabricate the composite plate. In order to introduce delaminations, Teflon tape of desired diameters (19 mm, 22 mm, and 24 mm) are used between the first-second, fifth-sixth, and tenth-eleventh layers, respectively. The respective coordinates of 19 mm size delamination are considered x = 130 mm and y = 100 mm from the left bottom corner of the plate. Similarly, the co-ordinates of 22 mm delamination is considered as x = 35 mm, y = 130 mm; and the co-ordinates of 24 mm delamination is considered as x = 140 mm, y = 30 mm, as shown in Figure 2(a). The CFRP plate is cured for 24 h at room temperature after the fabrication process is completed. Later, the plate is trimmed from all four sides to remove the excess material, and a plate of dimension 190 × 170 × 7 mm³ is obtained. Further, piezoelectric patches (SP-5A) are glued over the top surface of the plate in respective positions, as shown in Figure 3(b).

Figure 3.

(a) Schematic diagram of the experimental setup, (b) carbon fiber reinforced polymer plate with PZTs-Transmitter’s, (c) Experimental setup.

Subsequently, experiments are performed by using NI-PXI system (signal generator: NI PXIe-5413―PXI1Slot2 and PXI1Slot4, oscilloscope: NI PXIe-5170R―PXI1Slot3), Laser Doppler Vibrometer (Polytech GmbH), voltage amplifier (WMA 300 from Falcon systems), BNC (Bayonet Neill-Concelman) cables, SMA (SubMiniature version A) cables, piezoelectric patches (SP-5A) and CFRP plate, as shown in Figure 3(a) and (c).The NI-PXI system consists of two waveform generators, one oscilloscope, and a computer system with NI-PXI software. The Laser Doppler Vibrometer contains a laser head (OFV 300) and controller. The controller’s velocity output is fed to the oscilloscope of the NI-PXI system (Figure 3(a)).

The activation of the delamination is similar to the case of a numerical study (Figure 3(a)). The first input signal generated from the function generator (10 V_pp) gets amplified using the amplifier (50 V_pp) and passes to the plate through a piezoelectric patch (Figure 3). The second input signal is directly connected to PZT. The Laser Doppler Vibrometer is aligned perpendicular to the delamination surface.

Frequency spectra for each experiment are plotted for determining variation in nonlinearity parameter. Finally, the analytically and numerically determined LDR frequencies (f_d) are compared with experiments by exciting the plate using chirp signals.^19,20

Results and discussions

Frequency spectra and corresponded nonlinearity parameter descriptions are are presented in this section. Three different size delaminations such as 19, 22, and 24 mm are used in the CFRP plate, as shown in Figure 2. The excitation frequencies of three delaminations are presented in the Table 2, which is obtained by providing the chirp signals based on the analytical LDR frequency to particular size of delamination. It can be observed that the analytically calculated values are highest in all cases. This happens due to the assumption of clamped boundary conditions for the LDR frequency equation, which changes in supported boundary conditions in numerical and experiment cases. The SPF (f2) is taken as 277 kHz for all delamination cases.

Table 2.

Variable frequency for different sizes of delamination.

Variable frequency		Delamination diameter (in mm)
Variable frequency		19	22	24
LDR frequency (f_d) (in kHz)	ANA	31.72	59.15	99.41
	NUM	29.9	59.1	98.5
	EXP	27.5	60.0	95.0
Subharmonic LDR frequency (2f_d) (in kHz)	ANA	63.44	118.31	198.82
	NUM	59.8	118.2	197
	EXP	55.0	120	190
Second order superharmonic LDR frequency (f_d/2) (in kHz)	ANA	15.86	29.57	49.70
	NUM	14.95	29.55	49.25
	EXP	13.75	30.00	47.5
Third order superharmonic LDR frequency (f_d/3) (in kHz)	ANA	10.57	19.71	33.13
	NUM	9.96	19.7	32.83
	EXP	9.16	20.0	31.66

Frequency spectrum

Figure 4(a)–(c) represents the frequency spectra obtained from the analytical, numerical, and experimental investigation when the subharmonic LDR frequency (2f_d) excitation corresponds to 19 mm delamination is provided. The second harmonic (4f_d) has been identified in analytical, numerical, and experimental results, as shown in Figure 4. Similarly, the second harmonics have been identified for all the cases, and corresponding relative acoustic nonlinearity parameter (β₂) variations are presented in the following sub-section.

Figure 4.

Frequency spectra of 19 mm delamination excited with Subharmonic local defect resonance frequency (2f_d) – (a) Analytical results, (b) Numerical results, (c) Experimental results.

Relative acoustic nonlinearity parameter (β₂) variation

Figure 5(a)–(c) represents the variation of relative acoustic nonlinearity parameter (β₂) with variable frequency in analytical, numerical, and experimental results. The variable frequencies are LDR frequency (f_d), subharmonic LDR frequency (2f_d), second-order superharmonic LDR frequency (f_d/2), and third-order superharmonic LDR frequency (f_d/3). The variation of the nonlinearity parameter (β₂) is similar for analytical, numerical, and experimental results (Figure 5(a)–(c)) in all delamination. Also, it can be observed that the relative acoustic nonlinearity parameter (β₂) is maximum when the three delaminations are excited with their respective LDR frequency (f_d). The variation of the nonlinearity parameter (β₂) is small in three delamination cases when excited with the rest of the excitation frequencies such as f_d/2, f_d/3, 2f_d.

Figure 5.

(a–c) Relative acoustic nonlinearity parameter (β₂) variation with variable frequency − Analytical, numerical and experimental observations.

It should also be noted that the nonlinearity parameter is less in the case of the deeper defect (24 mm) compared to shallow one (19 mm). It might be happens due to stretching effect is more at nearest surface under the plane stress condition. Also, observed that the nonlinearity is reduced with increasing the residual thickness at the LDR frequency. Moreover, the wavelength (λ) is an influencing parameter to characterize the material nonlinearity.¹ It found that quarter wavelength (λ/4) is low for deeper defects compare to the rest of the defects. However, the stretching effect will be more pronounced in case of shallow defects. It also understands that more energy is pumped towards higher harmonic generation (due to more nonlinearity) is required in the case of a deeper defect compared to a shallow defects at LDR frequency.

Conclusions

An analytical investigation has been presented for estimating the measurement of nonlinearity at the delamination region of the orthotropic composite plate by the relative acoustic nonlinearity parameter (β₂). Subsequently, the analytical investigation is validated with numerical and experimental research on CFRP plates having three delaminations. Two incident waves are used to stimulate the defect present in the plate. The first incident signal is provided as variable frequency such as LDR frequency (f_d), subharmonic LDR frequency (2f_d), second-order superharmonic LDR frequency (f_d/2), and third-order superharmonic LDR frequency (f_d/3). The second incident signal is provided as a SPF for all cases. It found that the relative acoustic nonlinearity parameter (β₂) variation is slight in the case of second, third-order superharmonic LDR frequency and subharmonic LDR frequency excitations for analytical, numerical, and experimental study. It also observed that the relative acoustic nonlinearity parameter variation is maximum for respective LDR frequency (f_d) excitation. Furthermore, the maximum relative acoustic nonlinearity parameter is observed for shallow defects than the deeper defects. The higher harmonic imaging for defects is better suited at LDR frequency, and this fact can be used in the future for higher harmonic imaging of defects.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Tanmoy Bose

Notation

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Relative acoustic nonlinearity parameter variation with local defect resonance frequencies and its different harmonics

Abstract

Keywords

Introduction

Methodology

Analytical investigation

Relative acoustic nonlinearity parameter for second harmonic (β2)

Numerical investigation

Experimental investigation

Results and discussions

Frequency spectrum

Relative acoustic nonlinearity parameter (β2) variation

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Notation

References

Relative acoustic nonlinearity parameter for second harmonic (β₂)

Relative acoustic nonlinearity parameter (β₂) variation