Solutions of Non-Classical Nonlinear Acoustics ... - IngentaConnect

ACTA ACUSTICA UNITED WITH Vol. 102 (2016) 341 – 346

ACUSTICA DOI 10.3813/AAA.918950

Solutions of Non-Classical Nonlinear Acoustics Wave Equation in a Conical Bar with Micro-Cracks Xudong Teng1,2) , Xiasheng Guo1,3) , Linjiao Luo1) , Dong Zhang1,4)

1)

2) 3)

4)

Key Laboratory of Modern Acoustics (Nanjing University), Ministry of Education, Institute of Acoustics, Nanjing 210093 College of Electronic and Electric, Shanghai University of Engineering Science, Shanghai 201620, China Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 16802, USA. [email protected] State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100080, China. [email protected]

Summary Non-classical nonlinear acoustic wave (NNAW) plays an important role in the evaluation of micro-cracked solids. In the present work, the characteristics of NNAW in a conical bar embedded with micro-cracks was investigated by modifying the stress-strain relationship with hysteresis and discrete memory characteristics. Considering the variation of the bar’s cross section to be insignificant, a one-dimensional model was developed and the general wave equation was modified accordingly. Analytical analysis of the model was then achieved via the perturbation theory, which showed that accurate prediction of the crack location could be accomplished by analyzing the longitudinal displacement fields across the cracked area. Numerical simulations were carried out to demonstrate the effectivity of the model. The results showed that: (a) the presence of the micro-cracks induced non-classical nonlinear wave distortion, while the longitudinal strain was discontinuous on the two sides of the crack; (b) only the odd-order harmonics were generated, whose displacement amplitudes were highly related to the position and width of the micro-cracked region, and was also dependent on the strength parameter of the non-classical nonlinearity; (c) the displacement amplitudes of the third harmonic components was larger than other higher order harmonics, and could be employed to discern and locate the micro-cracked region. PACS no. 43.25.-x

1. Introduction It has been reported that the mesoscopic elastic properties of micro-damaged structures could exhibit hysteresis at high strain amplitudes (e.g., local strain ε > 10−6 ). This phenomenon makes a major contribution to the nonlinear acoustic responses in nondestructive testing (NDT) [1, 2, 3], and is named as non-classical nonlinearity. The hysteretic characteristics between stress and strain usually cause distinctive wave distortion and harmonic generation locally in the damaged region. In contrast to the classical nonlinearity or propagation nonlinearity, only oddorder harmonic components would be generated, while their amplitudes could be proportional to the square of the fundamental component. Furthermore, the resonant frequency of the whole system could also deviate from its origin as the wave amplitude increases [1, 4, 5]. In order to quantitatively characterize the influence of the non-

Received 05 August 2015, accepted 24 December 2015.

© S. Hirzel Verlag · EAA

classical nonlinearities generated from the micro-cracked area, several promising and powerful methods have been developed. Based on the nonlinear frequency mixing and amplitude-dependent resonance frequency shift, the Nonlinear Elastic Wave Spectroscopy (NEWS) technique has been presented for the detection of micro-cracks [5, 6]. By utilizing signal analysis techniques such as nonlinear convolution, information of the higher order harmonics contributed by the non-classical nonlinearity could be extracted from the acoustical responses, and be further used to detect early stage micro-cracks [2]. A novel time reversal technique was also proposed: the recorded timedomain signals were filtered with a pass-band filter centered at the third harmonic frequency, the cracks could be discerned when time reversing the first wave packet reflected by the defect [7, 8, 9]. However, it should be emphasized that the above-discussed approaches are usually confined to the case of structures of uniform cross-section. For simple structures such as bars, one-dimensional (1-D) models could be adopted to study the wave propagation, while the analytical or quasi-analytical solutions could be derived with the

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Teng et al.: Bar with micro-cracks

Figure 1. The schematic model of a conical bar embedded with micro-cracks.

help of Green’s Function Theory and perturbation methods [10, 11, 12]. However, bars with varying cross-sections, such as conical bars, exponential bars and sinusoidal bars, have been widely used in industrial applications, benefiting from their capacities of supporting heavier alternating loads, their optimal structures and their smaller self-weights. In these cases, since the cross-section of a bar varies along its axial direction, the wave propagation should be described using a two-dimensional (2-D) partial differential equation (PDE) with variable coefficients. Considering the localized non-classical nonlinear effects induced by potential micro-cracks, the analytical or quasi-analytical solutions of these problems could be formidable. In the present work, wave motion in a conical bar embedded with micro-cracks was investigated by introducing a modified wave propagation model, in which the stressstrain relationship was locally revised to include hysteresis and discrete memory characteristics of the micro-cracked region. By assuming the variation of the bar’s cross section to be insignificant, the generation of NNAW could be identified from the quasi-analytic harmonic solutions achieved by using the perturbation theory as well as a variable substitution method. Numerical simulations using the finite element method (FEM) were then conducted to validate the effectiveness of the theoretical model. Finally, the localization principle of the micro-cracked area employing the third harmonic characteristics was discussed.

2. Theory and methods A schematic drawing of the model is presented in Figure 1, in which a single micro-cracked area was embedded in a conical bar of length L. The cross-section of the bar S(x) varied along the axial direction (the x-axis), i.e., S(x) = (1+x/L0 )2 S0 , in which L0 and S0 were the ramp constant and the initial cross-sectional area (at the left end, x = 0) of the cross-sectional area, respectively. Considering the variation of the cross-section to be insignificant along the axial direction, the inclination angle of the bar surface, ϕw , satisfied, tan(ϕw ) ∝

d S(x) = O(ε), dx

(1)

in which ε was a small quantity (i.e., 0 < ε 1). The micro-cracked area was located at x = x1 (0 < x1 < L), with its width d much smaller than the bar length

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(d L). Based on the Preisach-Mayergoyz (PM) theory [5, 10, 13, 14], a formula describing hysteretic behaviors was introduced to model the micro-cracked area, so that the local strain responses change alternatively between two state values εo and εc , corresponding the crack states of “open (O)” and “close (C)”, respectively. In light of the above prerequisites, the three-dimensional (3-D) PM theory could be simplified to the 1-D case, expressed as [12, 13, 15, 16, 17] (2) σ = E(1 − βε + δε2 + . . . )ε E 2 2 + α(x) sign(∂t ε)(Δε − ε ) − 2Δε · ε . 2 Here ε = ∂x ξ(x, t) was the longitudinal strain with Δε being its sinusoidal amplitude, E was the Young’s modulus of the bar material. The parameters β and δ were the third and fourth order elastic constants, representing the classical nonlinearity of the material. Variable α(x) was the strength of the non-classical nonlinearity, defined as α(x) =

α; 0;

x1 − d/2 < x < x1 + d/2, otherwise.

(3)

It is noted that non-classical nonlinearity was confined only at the micro-cracked area locally. The first term on the right side of Equation (2) represent the classical nonlinearity, while the second term correspond to its non-classical counterpart. At high strain amplitudes (e.g., ε > 10−6 ), the effect of non-classical nonlinearity could be dominant [2, 3]. Therefore, it was reasonable to ignore the classical nonlinearity in this case (i.e., β = δ = · · · = 0). By applying the stress-strain relationship in Equation (1), the 1-D wave propagation in a conical bar with insignificantly varying cross-section could be described by modifying the wave equation as ∂x S(x)ε + S(x)

α(x) sign(∂t ε)(Δε2 − ε2 ) − 2Δε·ε 2 1 (4) = 2 S(x)∂tt ξ, c

where c = E/ρ was the propagating velocity of longitudinal waves in the bar with ρ being the material density. In Equation (4), the combination of non-classical nonlinearity and the variable coefficient S(x) made the direct analytical solution of this second-order nonlinear partial differential equation (PDE) hard to achieve. Although numerical simulation could also be a candidate to yield accurate solutions, it was incapable of providing adequate physical insights into the nonlinear properties of NNAW. Therefore, the perturbation theory was adopted to obtain the analytical solutions of harmonic wave components, in order that the micro-cracked area could be discerned. 2.1. The linear solution To allow for wave propagation, a forced sinusoidal vibration was applied on one end of the bar whose frequency and amplitude were ω and F0 , respectively, while

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a stress-free boundary condition was adopted for the opposite end. The general solution of Equation (4), ξ(x, t), could then approximately be decomposed into the linear solution ξ0 (x, t) and its perturbation ξ1 (x, t) [13]. Assuming that ξ1 (x, t) was much smaller than ξ0 (x, t), the linear solution could be obtained by solving the linearized form of Equation (4), ∂x S(x)∂x ξ0 (x, t) =

1 S(x)∂tt2 ξ0 (x, t). c2

where Δε0 = |H (x)|. By applying the Fourier expansion on the term sign(∂t ε0 ), Equation (9) becomes ∂x S(x)

2 · 105

(5)

(6)

where k = ω/c was the wave number. Here a variable substitution was applied by introducing a new variable ψ (x) = u1 (x)R(x), in which R(x) = S(x)/S0 . Equation (6) could hence be simplified as ∂tt2 ψ (x) + k 2 − g(x) ψ (x) = 0,

ξ0 (x, t) = =

ψ (x) cos(ωt) R(x)

S0 B1 cos(kx) + B2 sin(kx) cos(ωt), R(x)

S0 B1 cos(kx) + B2 sin(kx) S(x)

175 sin(ωt) − 91 sin(3ωt) −29 sin(5ωt) − 6 sin(7ωt) − . . . π

1 S(x)∂tt2 ξ1 (x, t) − ∂x S(x)∂x ξ1 (x, t) . c2

(10)

Obviously, only the odd-order harmonics exist in the perturbation solution. Given that the amplitude of the third harmonic was much larger than those of other harmonic components, Equation (10) could be further simplified as ∂x S(x) =

2 −91 sin(3ωt) α(x) −sign H (x) H (x)2 2 105 π

1 S(x)∂tt2 ξ1 (x, t) − ∂x S(x)∂x ξ1 (x, t) , c2

(11)

while the corresponding displacement for the third-order harmonic wave was S0 (3) A cos(3kx) S(x) 1

(3)

ξ1 (x, t) =

(3)

(8a)

and the corresponding longitudinal strain ε0 = ∂x

=

(7)

2 with g(x) = (1/R(x))∂xx R(x). Since the variation of S(x) was small, g(x) could be approximated to zero in this case, i.e., g(x) = 0. Equation (7) thus turns into a secondorder ordinary differential equation (ODE). The linear displacement ξ0 (x, t) could then be given by

− sign H (x) H (x)2

· +2 cos(ωt)

Let ξ0 (x, t) = u1 (x) cos(ωt), Equation (5) could be rewritten as 1 ∂x S(x)∂x u1 (x) + k 2 u1 (x) = 0, S(x)

α(x) 2

+ B1 sin(3kx) sin(3ωt).

(12)

(3)

(3)

To quantify the coefficients A1 and B1 , a set of boundary conditions was defined as follows [17, 18, 19], (3)

(a) ξ1 be continuous across the micro-cracked area [19], cos(ωt)

= H (x) cos(ωt),

(8b)

where H (x) was the amplitude of the stain. It is obvious from Equation (8) that the longitudinal displacement was dependent on both the boundary condition and the variable cross-section, which differed from the cases of uniform cross-section bars.

(3L)

ξ1

(3R)

(x1 , t) = ξ1

α(x) sign(∂t ε0 ) Δε20 − ε20 − 2Δε0 ε0 2 1 = 2 S(x)∂tt2 ξ1 (x, t) − ∂x S(x)∂x ξ1 (x, t) , c

∂x S(x)

∂x

2 −91 sin(3ωt) α(x) sign H (x) H (x)2 2 105 π =

(3L) ∂x ξ1 (x, t)

− d

(3R) ∂x ξ1 (x, t)

x=x1

x=x1

(14)

;

(c) both ends of the conical bar be strain-free for the thirdharmonic component, (3L)

∂x ξ1

(3L)

(9)

(13)

(b) the stresses be continuous across the micro-cracked area,

2.2. The nonlinear solutions The nonlinear term in Equation(4) could be regarded as the perturbation of the linear solution ξ0 (x, t), and was only locally caused by the micro-cracks. The first-order nonlinear solution of the system ξ1 (x, t) could hence be achieved by substituting into Equation (4), obtaining

(x1 , t);

(3R)

(0, t) = ∂x ξ1

(0, t) = 0.

(15) (3R)

Here ξ1 (x1 , t) = ξ1 (x1 , t)x=x1 −d/2 and ξ1 (x1 , t) = ξ1 (x1 , t)x=x1 +d/2 denote the third harmonic displacements on the left and right edges of the cracked area, respectively.

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From Equations (12)–(15), the amplitudes of sinusoidal (3) and cosinoidal components in ξ1 could be determined as  S0 cos(3kx1 )−λ sin(3kx1 ) (3L) R(x1 ), A1 = 2d S(x1 ) λ6kS0 + Sx (0) 1 (3L) (3L) = γA1 , B  1 6 (16) 1 S0 (3R) A1 = d 3 S(x1 ) 6kS0 cos(3kx1 ) + Sx (0) sin(3kx1 ) · R(x1 ), λ6kS0 + Sx (0)  (3R) (3R) B1 = −λA1 , where 6kS(L) tan(3Lk) + Sx (L) Sx (0) , , γ= tan(3Lk)Sx (L) − 6kS(L) kS0 91 R(x) = sign H (x) α Sx (x)H (x)2 105π

λ=

+ 2S(x)H (x)Hx (x) . It is clearly observed here that the displacement fields of the third harmonic component were highly related to the location and width of the micro-cracked area as well as the strength of the non-classical nonlinearity α(x). Therefore, it could also be expected that localization and assessment of the cracked area using the non-classical nonlinearity be feasible. Other higher-order harmonic components (e.g., 5th, 7th, . . . ) could be calculated with the same protocol likewise.

3. Results and discussion 3.1. NNAW responses In order to validate the obtained solutions and localize the cracked area, numerical simulations were conducted with finite element method (FEM) using a commercial software (Comsol Multiphysics v4.3a, COMSOL Inc., Palo Alto, CA, USA). The related parameters were chosen as: bar length L = 0.2 m, density ρ = 7.8·103 kg/m3 ,longitudinal wave velocity c = 6·103 m/s, cross sectional ramp constant L0 = 10 m, initial cross-sectional area S0 = 2.5 · 10−3 m2 , and the non-classical nonlinearity strength α = 2000. A micro-cracked area with the width d = 2 mm was located (1) at x1 = 0.17 m or (2) at x1 = 0.03 m, although the parameters d and x1 were assumed unknown when trying to localize and assess the cracks with the detected third harmonics. A sinusoidal excitement of F0 = 105 N and ω0 = 2π · 16 kHz was applied on the left end of the bar. With a spatial interval of 0.5 mm, the time-dependent wave motions were recorded along the axial direction in both theoretical analysis and FEM simulations. Figure 2 compares the longitudinal displacement amplitudes of the total response for a crack located at x1 = 0.17 m, ξ(x, t), between the analytical calculations (solid line) and simulation results (dashed line). A good agreement was found at x < 0.13 m, but the numerical results

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Figure 2. Comparison of the longitudinal displacement amplitude ξ(x) = |ξ(x, t)| for a crack located at x1 = 0.17 m between analytical and numerical solutions.

started to deviate from the theoretical predictions as x further increased. This deviation might be contributed by two factors: (1) the FEM simulations were based on a 2-D plane strain model where the shear components of displacement and strain/stress had to be included, inducing a quantitative error when compared with the 1-D theoretical model; (2) the discretization in the FEM model might not be optimized both in time and in space domains. The third harmonic components were then extracted from the recorded signal series by applying an FIR digital filter algorithm (central frequency: 48 kHz, bandwidth: 2 kHz). Figures 3a and 3b show the displacement distribu(3) (3) tion of ξ1 for micro-cracks at x1 = 0.03 m and ξ1 for micro-cracks at x1 = 0.17 m, respectively. The numerical results of the third harmonic components were found to be of the similar oscillation waveform as the analytical solu(3) tion predicted, and ξ1 always kept continuity across the micro-cracked area. In addition, the amplitudes of the third harmonics are observed dependent on the location of the crack, being much smaller than those of the total response in Figure 2. It demonstrated that the NNAW component was only a perturbation of the linear wave motion. Figure 4 illustrates the frequency spectra of the longitudinal displacement at the crack center, ξ(x, t), for the single micro-cracks zone at x1 = 0.03 m or x1 = 0.17 m, respectively. The amplitude peaks are generated only when the frequency was odd times of the fundamental frequency 16 kHz, (i.e., 16, 48, 80 kHz, . . . ). This further confirms the presence of NNAW. It is also noted that the third and the fifth harmonics were more significant when compared with other harmonics. 3.2. Assessment of the cracked area According to Equation (16), the displacement coefficients (3R) (3L) of the third harmonic A1 and A1 are highly related to the location and width of the micro-cracked area. Therefore, the location x1 and width d of the micro-crack zone could be evaluated through the displacement coefficients

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(3)

Figure 3. The amplitude of the third harmonic displacement ξ1 as a function of spatial location (a) x1 = 0.03 m, (b) x1 = 0.17 m.

of the third harmonics, (3L)

(3R)

ξ1 =

− A1 A 1 , atan 1(3L) (3R) 3k −B B 1

1

d =

1 (3L) A 2 1

(17)

S(x1 ) λ6kS0 + Sx (0) . S0 cos(3kx1 ) − λ sin(3kx1 ) R(x1 )

Similarly, the crack location x1 could also be determined from the displacement coefficients of the 5th harmonics, (5L)

(5R)

− A1 A 1 , x1 = atan 1(5L) (5R) 5k B1 − B1

(18) (5L)

(5L)

(5R)

(5R)

A1

(3L)

= −3.716 · 10−8 ,

(3L)

= −7.393 · 10−11 ,

(3R)

B1

A1

= −3.137 · 10−7 ,

B1

(3R)

= −2.288 · 10−7 ,

where A1 and B1 (or A1 and B1 ) denote the amplitudes of the sinusoidal and cosinoidal components of the 5th harmonics on the right (or left ) side of the single micro-crack zone, respectively. Equations (17) and (18) indicate that the nonlinear solutions contain all information concerned with the microcracked area. The detailed information of the crack could thus be predicted. With the aforementioned parameters, the third harmonic displacement coefficients for the microcracks located at 0.17 m were:

Figure 4. Frequency spectrum of the longitudinal displacement ξ(x, t) at x1 = 0.03 m or x1 = 0.17 m, respectively.

while those of the fifth harmonics were: A1

(5L)

= 6.559 · 10−8 ,

A1

(5R)

= −4.173 · 10−9 ,

(5L)

B1

= 7.829 · 10−11 ,

(5R)

B1

= −7.244 · 10−9 .

Substituting the above coefficients into Equations (17) and (18), locations of the crack was then estimated. However, since the inverse tangent function have multiple solutions, several possible solutions x1 of Equation (17) were obtained, i.e., 0.045 m, 0.107 m and 0.17 m. Similarly, five solutions of Equation (18) were also found to be 0.02 m, 0.06 m, 0.09 m, 0.13 m and 0.17 m. Finally, Figure 5 demonstrates the possible locations of the micro-cracks obtained from Equation (17) and Equation (18) with grayscale maps. The dark regions in these

Figure 5. The possible locations of the micro-cracked area determined from (a) the third harmonic coefficients, and (b) the fifth harmonic coefficients.

maps indicates possible locations of the micro-cracks. By considering the results in Figures 5a and 5b simultaneously, the only overlap at x1 = 0.17 m could then be discerned. It is concluded that the exact position of the microcracked area must be determined through the combined

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data concerning the3th , and5th harmonics or other higherorder components.

4. Conclusions The micro-cracked area in a conical bar leads to the nonclassical nonlinear stress-strain relationship when a large strain is exerted, and high order harmonics could be generated in the acoustic responses. Compared with the classical nonlinearity, the NNAW is only associated with the localized crack other than the material properties. The generated odd-order harmonics could then be used to evaluate the location and size of the micro-cracked area. In the case of a conical bar with its cross-section varying insignificantly, the 2-D NNAW equation is approximated into a 1-D partial differential equation with variable coefficients. By using the perturbation theory as well as a variable substitution method, the wave propagation model is converted into an ordinary differential equation with constant coefficients. The nonlinear solution of the model is then obtained with the help of Fourier expansion. Through analytical and numerical evaluations, it has been demonstrated that both the location and size of the crack could be uniquely determined. Further studies will extend this method into the evaluations of multiple cracks in a conical bar. Acknowledgement This work is co-funded by the National Basic Research Program of China (Grant No. 2011 CB707900), the National Natural Science Foundation of China (Grant Nos. 81127901, 81227004, 11374155, 11174141, 11274170, 11474001, and 11474161), the National High-Tech Research and Development Program 863 (Grant No. 2012 AA022702). References


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