Nonlinear Properties of Damaged Objects : Application to Crack Detection.

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Nonlinear Properties of Damaged Objects : Application to Crack Detection.

A. Moussatov*, B. Castagnede ^†

*Institut Supérieur d'Électronique du Nord, IEMN - Département ISEB, 41 Boulevard Vauban, 59046, Lille Cedex, France

† Laboratoire d'Acoustique de l'Université du Maine, UMR CNRS 6613 Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France

Abstract. Nonlinear properties have been experimentally evaluated for a number of thick glass plates being damaged by a thermal shock. A direct correlation between the degree of damage and the nonlinear parameter value has been demonstrated (using ratio between second and fundamental harmonics amplitudes). More damage leads to higher nonlinearity. Such nonlinearity for strongly damaged plates increased by the factor 300 compared to the intact ones. Conversion spectrograms have been analyzed and discussed.

For the first time, in cracked solid material, a parametric emitting antenna has been successfully implemented. The non classical anomalous nonlinearity of damaged materials can be effectively used for detection of structural defects, however the nature of this phenomenon still needs understanding.

INTRODUCTION

Many significant achievements have been done in the field of non linear

acoustics to characterize micro-inhomogeneous solids, such as rocks and any materials

having macroscopic inhomogeneities [1]. Various applications have been recently

described dealing with the detection of cracks [2, 3], and the monitoring of materials

fatigue [4]. The non linear methods are considered as much more sensitive to defects

than linear techniques as the coefficient of non linearity for damaged materials is much

higher. Some recent advances have been reported on the diagnostic of micro-cracked

materials [5, 6]. The main assumption lies on the fact that the cracks are much stiffer

than the matrix of the material. High power ultrasonic wave amplitude modulated by a

low frequency signal was used (parametric antenna [7]). This configuration has been

implemented by the authors to characterize the non linear interaction in glass plates

which have been thermally cracked [8]. The present report reviews this work, and

describes some potential applications in the field of diagnostic of engineering

materials. By dealing with a non linear wave equation with a quadratic perturbation

term Γ in the elasticity stress-strain relationship, one can deduce a relative evaluation

of this parameter of non linearity. Details on the analytical treatment and on the

essential data analysis are provided in the next section.

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QUADRATIC COEFFICIENT OF NONLINEARITY

The usual description of the non linear behaviour of a damaged material is done by adding a supplementary term in the linear Hooke's law as written in the form :

σ ^{= M}

0

ε ^{1 +} Γ ε ^{, with} ε ⁼ ^∂ _∂ ^u _x ^{= u} _,x ^, ⁽¹⁾

where σ and ε denotes respectively the applied mechanical stress and strain due to the ultrasonic field, with u standing for the 1D displacement field along coordinate axis x.

The Γ parameter stands for the quadratic coefficient of non linearity, while M ₀ is the mechanical rigidity of the material along the x direction. For undamaged glass the Γ parameter has a value close to 2, but for cracked glass plates with damages having macroscopic size, this parameter should be significantly higher. In equation (1), the perturbation term is much smaller than one ( Γ ε << 1) for undamaged glass plates, because the strain produced by ultrasonic piezoelectric transducers is generally weak, in the range between 10 ^-6 to 10 ^-5 . Accordingly, only in the case where the Γ parameter is high, let say around 10 ² -10 ³ , will the corrective term Γ ε becomes significant. By using the equation of motion, written in the standard form, ρ ^∂ _∂t

²

^u

₂

= ∂ σ

∂x , where ρ stands for the mass density of the solid, the non linear wave equation is then derived in the form :

ρ ^∂ _∂t

²

^u

₂

= M

₀

∂

²

u

∂ x

²

1 + 2 Γ ^∂u _∂x ^. ⁽²⁾

In this equation, M

0

ρ is the speed of ultrasound c along the x direction. One then seek for solutions with plane waves propagating in the x direction, by using the successive approximations method limited to the fundamental wave u ⁽¹⁾ and its second harmonic u ⁽²⁾ , eq. (2) being rewritten u _,

⁽²⁾

_tt = c

²

u _,

⁽²⁾

_xx + 2 Γ ^u _,

⁽¹⁾

_xx ^u _,

⁽¹⁾

_x . The formal relation between the amplitude A ₁ of the fundamental wave (at frequency ω ) and A ₂ of its second harmonic (at 2 ω) takes finally the following form (see [8] for further details) :

20 log Γ

^D

Γ

^V

^{= A}

²

V

– A

₁^V

– A

₂^D

– A

₁^D

, (3)

where all the above spectral amplitudes (both on fundamental and second harmonic

waves) are expressed in dB. This procedure is also available to determine the ratio of

the coefficients of non linearity for two plates having different level of damages, i.e. to

rank the amount of observed damages.

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EXPERIMENTAL PROCEDURES AND RESULTS

The coefficient of non linearity has been obtained with the experimental set-up which is described on Fig. 1. It is a standard configuration with a pump wave generated by a piezoelectric transducer at 100 kHz with adequate power electronics.

This pump wave is low frequency amplitude modulated, in the range of a few kHz.

The transmitted signal is then probed with a different ultrasonic transducer which is properly mounted on the opposite edge of the plate. Rectangular 18 mm thick glass plates have been used. Some of the plates were damaged through a thermal process.

The received continuous acoustical waves were captured at the same time on a LeCroy oscilloscope and on a FFT vector signal analyzer.

30

-50 -40 -30 -20 -10 0 10 20

500

0 100 200 300 400

FFT spectrum with modulation

dB

kHz

15

-45 -40 -30 -20 -10 0 10

500

0 100 200 300 400

FFT spectrum with modulation

dB

kHz

a)

b)

2 1 3 4

5

6

7

8

9

IEEE 488 bus

Wave Generator Power

amplifier Glass plate

Oscilloscope

FFT analyzer

Computer Receiver

Preamplifier

Transmitter

FIGURE 1. Experimental set-up

Amplitude spectra at high level for low frequency modulated signal in a) a virgin glass plate, b) an highly damaged glass plate (showing numerous cracks).

FIGURE 2.

Some recorded spectra are shown on Fig. 2, for the case of LF modulated signals, for a virgin (undamaged) and strongly cracked plate. On the recorded spectra, the so-called "cascade process", shows up a significant increase of the harmonic rays when the material is damaged. For the undamaged plate, the amplitudes of the harmonics are almost unchanged versus output voltage. On the other hand, the damaged plate demonstrates a dramatic increase of the same harmonics. Moreover, low frequency demodulation on the damaged plate at high amplitude level is clearly seen (with a very strong demodulated peak), resulting in dramatic changes of the detected temporal signal. The relative increase of the second harmonic enables to determine Γ , the quadratic coefficient of non linearity (as explained in the previous section). A significant example is shown with the spectrum data gathered on Table 1.

It can be shown that the evaluation of Γ is independent on the transducer driving voltage. As stated in Eq. (3), one obtains the ratio of the coefficient of non linearity between two plates. Here, 20 log Γ

^D

Γ

^V

^{= 50 dB} ^⇒ Γ

^D

^{= 300} Γ

^V

. Because for virgin

raw glass Γ

(4)

is around 2, one deduce that the most damage glass plate tested here has Γ

^D^§

With other plates with less damages (than plate # 8), we found a smaller Γ ^(D) ^/ Γ ^(V) ratio indicating an intermediate quadratic coefficient of non linearity (see [8] for details).

Level A1 (plate # 8) A2 (plate # 8) A1 (plate # 9) A2 (plate # 9) ΓΓ((88) ) / / ΓΓ((99))

+ 40 dB - 38,6 dB - 70, 5 dB - 4,0 dB - 88,2 dB + 52,3 dB

+ 30 dB - 48,4 dB - 92,2 dB - 14,4 dB - 108,0 dB + 49,8 dB

TABLE 1. Calculation of the ratio between the quadratic coefficient of non linearity for a damaged glass plate (# 8) and a virgin plate (# 9) from the amplitude spectra measurements with eq. (3).

CONCLUSIONS AND PERSPECTIVES

Thermally cracked glass plates have been studied with intense ultrasonic fields.

By using a non linear wave equation which includes a quadratic perturbation term, measurements of the coefficient of non linearity Γ has been achieved for various damaged glass plates. It has been shown that the value of such coefficient is directly related to the damages produced in the glass plates.

ACKNOWLEDGEMENTS

This work has been supported in the frame of DGA contract N° 00.34.026. We are thankful to Engineer Eric Pleska for authorizing the publication of the present work.

REFERENCES

1. Guyer, R. A., and Johnson, P. A. , Nonlinear mesoscopic elasticity : evidence for a new class of materials, Physics Today 52, 30 (1999).

2. Sutin, A.M. , and Nazarov, V.E. , Non linear acoustic methods of crack diagnostics, Radiophysics and Quantum Electronics, 38, 109-120 (1995).

3. Zaitsev, V. , Sutin, A.M. , Belyaeva, I.Y. , and Nazarov, V.E. , Nonlinear interaction of acoustical waves due to cracks and its possible usage for cracks detection, J. Vibration & Control, 1, 335-344 (1995).

4. Nagy, P.B. , Fatigue damage assessment by nonlinear ultrasonic material characterization, Ultrasonics, 36, 375-381 (1998).

5. van den Abeele, K.E.A. , Johnson, P.A. , and Sutin, A.M. , Non linear elastic wave spectroscopy (news) techniques to discern materials damage, Part I: Non linear wave modulation spectroscopy (NWMS), Res. Nondestr. Eval., 12, 17-30 (2000).

6. Zaitsev, Y. , and Sas, P. , Elastic moduli and dissipative properties of microinhomogeneous solids with isotropically oriented defects, Acta Acustica, 86, 216-228 (2000).

Nonlinear Properties of Damaged Objects : Application to Crack Detection.

Nonlinear Properties of Damaged Objects : Application to Crack Detection.

A. Moussatov*, B. Castagnede †

*Institut Supérieur d'Électronique du Nord, IEMN - Département ISEB, 41 Boulevard Vauban, 59046, Lille Cedex, France

† Laboratoire d'Acoustique de l'Université du Maine, UMR CNRS 6613 Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France

For the first time, in cracked solid material, a parametric emitting antenna has been successfully implemented. The non classical anomalous nonlinearity of damaged materials can be effectively used for detection of structural defects, however the nature of this phenomenon still needs understanding.

INTRODUCTION

Many significant achievements have been done in the field of non linear

acoustics to characterize micro-inhomogeneous solids, such as rocks and any materials

having macroscopic inhomogeneities [1]. Various applications have been recently

described dealing with the detection of cracks [2, 3], and the monitoring of materials

fatigue [4]. The non linear methods are considered as much more sensitive to defects

than linear techniques as the coefficient of non linearity for damaged materials is much

higher. Some recent advances have been reported on the diagnostic of micro-cracked

materials [5, 6]. The main assumption lies on the fact that the cracks are much stiffer

than the matrix of the material. High power ultrasonic wave amplitude modulated by a

low frequency signal was used (parametric antenna [7]). This configuration has been

implemented by the authors to characterize the non linear interaction in glass plates

which have been thermally cracked [8]. The present report reviews this work, and

describes some potential applications in the field of diagnostic of engineering

materials. By dealing with a non linear wave equation with a quadratic perturbation

term Γ in the elasticity stress-strain relationship, one can deduce a relative evaluation

of this parameter of non linearity. Details on the analytical treatment and on the

essential data analysis are provided in the next section.

QUADRATIC COEFFICIENT OF NONLINEARITY

The usual description of the non linear behaviour of a damaged material is done by adding a supplementary term in the linear Hooke's law as written in the form :

σ = M

ε 1 + Γ ε , with ε = ∂ ∂ u x = u ,x , (1)

where σ and ε denotes respectively the applied mechanical stress and strain due to the ultrasonic field, with u standing for the 1D displacement field along coordinate axis x.

u

= ∂ σ

∂x , where ρ stands for the mass density of the solid, the non linear wave equation is then derived in the form :

ρ ∂ ∂t

u

= M

∂

u

∂ x

1 + 2 Γ ∂u ∂x . (2)

In this equation, M

ρ is the speed of ultrasound c along the x direction. One then seek for solutions with plane waves propagating in the x direction, by using the successive approximations method limited to the fundamental wave u (1) and its second harmonic u (2) , eq. (2) being rewritten u ,

tt = c

u ,

xx + 2 Γ u ,

xx u ,

x . The formal relation between the amplitude A 1 of the fundamental wave (at frequency ω ) and A 2 of its second harmonic (at 2 ω) takes finally the following form (see [8] for further details) :

20 log Γ

Γ

= A

– A

– A

– A

, (3)

where all the above spectral amplitudes (both on fundamental and second harmonic

waves) are expressed in dB. This procedure is also available to determine the ratio of

the coefficients of non linearity for two plates having different level of damages, i.e. to

rank the amount of observed damages.

EXPERIMENTAL PROCEDURES AND RESULTS

The coefficient of non linearity has been obtained with the experimental set-up which is described on Fig. 1. It is a standard configuration with a pump wave generated by a piezoelectric transducer at 100 kHz with adequate power electronics.

This pump wave is low frequency amplitude modulated, in the range of a few kHz.

The transmitted signal is then probed with a different ultrasonic transducer which is properly mounted on the opposite edge of the plate. Rectangular 18 mm thick glass plates have been used. Some of the plates were damaged through a thermal process.

The received continuous acoustical waves were captured at the same time on a LeCroy oscilloscope and on a FFT vector signal analyzer.

dB

kHz

dB

kHz

a)

b)

FIGURE 1. Experimental set-up

Amplitude spectra at high level for low frequency modulated signal in a) a virgin glass plate, b) an highly damaged glass plate (showing numerous cracks).

FIGURE 2.

It can be shown that the evaluation of Γ is independent on the transducer driving voltage. As stated in Eq. (3), one obtains the ratio of the coefficient of non linearity between two plates. Here, 20 log Γ

Γ

= 50 dB ⇒ Γ

= 300 Γ

. Because for virgin

raw glass Γ

is around 2, one deduce that the most damage glass plate tested here has Γ

With other plates with less damages (than plate # 8), we found a smaller Γ (D) / Γ (V) ratio indicating an intermediate quadratic coefficient of non linearity (see [8] for details).

Level A1 (plate # 8) A2 (plate # 8) A1 (plate # 9) A2 (plate # 9) ΓΓ((88) ) / / ΓΓ((99))

A. Moussatov*, B. Castagnede ^†

σ ^{= M}

ε ^{1 +} Γ ε ^{, with} ε ⁼ ^∂ _∂ ^u _x ^{= u} _,x ^, ⁽¹⁾

^u

ρ ^∂ _∂t

^u

1 + 2 Γ ^∂u _∂x ^. ⁽²⁾

ρ is the speed of ultrasound c along the x direction. One then seek for solutions with plane waves propagating in the x direction, by using the successive approximations method limited to the fundamental wave u ⁽¹⁾ and its second harmonic u ⁽²⁾ , eq. (2) being rewritten u _,

_tt = c

u _,

_xx + 2 Γ ^u _,

_xx ^u _,

_x . The formal relation between the amplitude A ₁ of the fundamental wave (at frequency ω ) and A ₂ of its second harmonic (at 2 ω) takes finally the following form (see [8] for further details) :

^{= A}

^{= 50 dB} ^⇒ Γ

^{= 300} Γ

With other plates with less damages (than plate # 8), we found a smaller Γ ^(D) ^/ Γ ^(V) ratio indicating an intermediate quadratic coefficient of non linearity (see [8] for details).