Elastic constants measurement of anisotropic Olivier wood plates using air-coupled transducers generated Lamb wave and ultrasonic bulk wave

Ultrasonics 50 (2010) 502–507

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Elastic constants measurement of anisotropic Olivier wood plates using air-coupled transducers generated Lamb wave and ultrasonic bulk wave Souhail Dahmen a, Hassiba Ketata b,*, Mohamed Hédi Ben Ghozlen a, Bernard Hosten c a

Laboratoire de Physique des Matériaux – Faculté des Sciences de Sfax, Tunisia Laboratoire de Physique des Matériaux – Ecole préparatoire aux académies militaires, Tunisia c Laboratoire de Mécanique physique, Université de Bordeaux 1, France b

a r t i c l e

i n f o

Article history: Received 1 January 2009 Received in revised form 27 July 2009 Accepted 17 October 2009 Available online 25 October 2009 Keywords: Anisotropic Olivier wood Lamb wave Inverse problem Elastic constants

a b s t r a c t A hybrid elastic wave method is applied to determine the anisotropic constants of Olive wood specimen considered as an orthotropic solid. The method is based on the measurements of the Lamb wave velocities as well as the bulk ultrasonic wave velocities. Electrostatic, air-coupled, ultrasonic transducers are used to generate and receive Lamb waves which are sensitive to material properties. The variation of phase velocity with frequency is measured for several modes propagating parallel and normal to the fiber direction along a thin Olivier wood plates. A numerical model based mainly on an optimization method is developed; it permits to recover seven out of nine elastic constants with an uncertainty of about 15%. The remaining two elastic constants are then obtained from bulk wave measurements. The experimental Lamb phase velocities are in good agreement with the calculated dispersion curves. The evaluation of Olive wood elastic properties has been performed in the low frequency range where the Lamb length wave is large in comparison with the heterogeneity extent. Within the interval errors, the obtained elastic tensor doesn’t reveal a large deviation from a uniaxial symmetry. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction The anisotropic materials are widely used for many structural applications, the determination of mechanical properties is critical for ensuring reliable performance. The knowledge of complete the elastic stiffness matrix is essential for modelling and evaluating the mechanical behaviour of composite materials under severe loading conditions. Ultrasonic techniques are perhaps the only qualified methods for the non-destructive measurement of all of the elastic constants of such materials [1,2]. Application of these techniques to wood appears to have been pioneered by Hearmon [3]. There is direct and continuing application of these analyses to the design and construction of musical instruments [4–10]. The determination of the elastic properties of different wood species is also of importance in the building industry. Usually, for wood orthotropy is assumed and the measurement of ultrasonic bulk wave velocities is applied to determine its properties [11–13]. The standard techniques utilize an immersion or contact piezoelectric transducers. The test piece must then be coupled to the

* Corresponding author. Address: Institut préparatoire aux études d’ingénieurs de Sfax (BP 1172), route Menzel Chaker km 0.5 3018 Sfax, Tunisia. Tel.: +216 97 588 211; fax: +216 74 246 347. E-mail address: [email protected] (H. Ketata). 0041-624X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2009.10.014

transducers by a liquid (water or gel) to ensure efficient mechanical energy transfer. Not only is this inconvenient and time consuming for industrial application, but, there are many cases for which it is important to eliminate the coupling medium, for instance in presence of high temperatures, porous materials like wood, etc. [14]. Another problem with the contact method when one wants to measure the attenuation, is the strong influence of the coupling agent between transducers and sample. For over two decades, Noncontact techniques such as air-coupled transducers were developed for just reasons [14]. There are now advanced enough to be used in the industry field and it was proved in the past that they are efficient enough to generate Lamb waves in plates and to detect them after propagation [15]. First the system has been tested for anisotropic composite materials made of epoxy matrix reinforced by glass or carbon fibers [15]. We observed that if the plane of propagation coincides with a plane of symmetry of the material, the transducers launch and detect Lamb modes only. If however the plane of propagation does not coincide with a plane of symmetry of the material, then in addition to Lamb modes guided shear horizontal (SH) modes are also excited [16]. The identification of elastic constants Cij from the measurement of phase velocities of Lamb modes were often deduced from the low-frequency plate wave dispersion data [15]. In this frequency range (50–400 kHz), material with hierarchical heterogonous structures like wood [17], can be considered as

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homogeneous, thus most favourable conditions for reliable velocities measurements. In this work, we apply a hybrid elastic wave method to determine the anisotropic constants of olive wood specimen considered as an orthotropic solid. The method is based on the measurements of the Lamb wave velocities as well as the bulk ultrasonic wave velocities. Air-coupled transducers are applied to obtain the phase velocity dispersion curves of Lamb waves propagating in thin Olivier wood plates. Then the inverse problem for the identification of Cij from the collected phase velocities is applied. In addition, the conventional ultrasonic velocity measurement is utilized to determine the remaining two elastic constants. 2. Calculation of the elastic constants 2.1. Wave propagation theory The theory of wave propagation in orthotropic materials is covered in a number of standard Refs. [1,18–25]. They show that the phase velocity V satisfies the Christoffel cubic equation.

jCik  qV 2 dik j ¼ 0

ð1Þ

In which, Cik = Cijklnjnk is the Christoffel tensor (a function of stiffness matrix Cijkl and of components of unit wave vector nj). q is the density and dik is Kronecker’s delta. This equation may be solved for an arbitrary wave direction to give, in general, three discrete values for velocity in terms of Cijkl and q. Along each symmetry axis of the material, of the three types of waves that can be generated one has its displacement vector parallel to the direction of propagation and is called a longitudinal wave, whereas the other two have orthogonal, transverse displacement vectors and hence are called transverse waves. For a plane wave whose unit vector ~ n lies in a plane containing two of the coordinates but rotates through an angle with respect to the symmetry directions, the stiffness matrix decomposes into a linear and a quadratic factor. The linear factor corresponds to a shear wave. The roots of the quadratic factor correspond to a quasi-longitudinal and a quasi-transverse wave. The ultrasonic technique consists of measuring velocities first along the symmetry directions and then in directions at various angle with the symmetry axis [26]. Let Vij denotes velocity of wave propagating along the xi-axis with polarization in the direction of the xj-axis, then the six diagonal stiffness components can be expressed in matrix form as [27]:

2

C 11 6 4 C 66 C 55

C 66

C 55

3

2

V 211 6 2 4 V 21 V 231

C 22

7 C 44 5 ¼ q

C 44

C 33

V 212 V 222 V 232

V 213 V 223 V 233

3 7 5

ð2Þ

ð3Þ

For example, for the plane P23 formed by the pair axes (x2, x3), equation:

C22 C33  C223  qV 2 ðC22 þ C33 Þ þ ðqV 2 Þ2 ¼ 0

ð4Þ

is used to calculate the Christoffel tenser term C23 and then C23 is determined from:

C 23 ¼ C23 =n2 n3  C 44

Different from semi infinite space, a thin plate preserves two parallel flat boundaries, the elastic wave motions on each surface will interact to produce Lamb waves. The phase velocity of Lamb wave is varied according to the distance between the two boundaries, and therefore, the wave is dispersive. The basic modes of Lamb waves are the symmetric and antisymmetric modes. For the inverse problem, to compute the closest solution to an experimental data, computer program for the calculation of phase velocity dispersion in multi-layered anisotropic was developed. The program was based on the Thomson/Haskell method that was adapted by Hosten and Castaing for the calculation of dispersion curves of anisotropic multi-layered media [15,28,29]. In a plane of symmetry P1i (i = 2 or 3) formed by the pair of axes (x1, xi), where x1 is the normal to the plate and xi is the direction of propagation, the displacements U and stresses vectors r at bottom and top surfaces of the plate are linked by the Thomson/Haskell matrix [15]:

2

3 U1 6U 7 6 i 7 6 7 4 r11 5

r1i

ð5Þ

2

A11 6A 6 21 ¼6 4 A31

A12

A13

A22 A32

A23 A33

3 U1 7 6 A24 76 U i 7 7 76 7 A34 54 r11 5

A41

A42

A43

A44

Bottom

A14

32

r1i

ð6Þ Top

The components of this matrix Aij = f(h, q, Cijkl, SL, ST, PL, PT) depend on the plate thickness h, the density q, the stiffness moduli Cijkl and the components of SL, ST, PL, PT that represent respectively the slownesses and polarisations vectors. L and T superscripts denote quasi-longitudinal and quasi-transverse mode respectively. In a symmetry plane four partial waves are expected, propagating downward and upward in the plate. These vectors are computed from the Christoffel’s equation (Eq. (1)). The expression of Aij can be found elsewhere [29]. Four elastic constants are therefore necessary for modelling the propagation along the direction xi (C1111, C11ii, Ciiii, C1i1i). In the actual study, the plate is placed in air, witch can be considered as a no viscous fluid. Therefore, the boundary conditions at the two surfaces of the plate imply the normal displacement component U1 and the stress components r11 and r1i to be continuous. Moreover, since the acoustic impedance of the air is negligible, compared to that of the tested material plate [16], a simplification that considers that leakage in air is equivalent to that in vacuum, i.e., null, can be made. These simplified boundary conditions lead to the following system of equations [28]:

2

2

3 U1 6U 7 6 i 7 6 7 40 5 0

where the elastic constants have been written in Voig’s contracted notation, in which 11 is replaced by 1, 22 by 2, 33 by 3, 23 by 4, 31 by 5 and 12 by 6. And the three non-diagonal stiffness components have the general form:

C ij ¼ Cij =nk nl  C ii

2.2. Dispersion of Lamb wave in a thin plate

Bottom

32

A11 6A 6 21 ¼6 4 A31

A12

A13

A22

A23

A32

A33

3 U1 7 6 A24 76 U i 7 7 76 7 A34 54 0 5

A41

A42

A43

A44

A14

0

Top

The resolution of this system leads to the dispersion equation [28]:

A31  A42  A32  A41 ¼ 0

ð7Þ

Solutions to the equation are couples, noted (f, V) where f is the frequency, and V is the phase velocity of the guided mode propagating along the direction xi. The Newton–Raphson method has been implemented to solve Eq. (7) as the frequency changes. The purpose of this paper is to utilize the experimental data of Lamb waves velocity propagating along the direction x2 (and x3) to determine inversely the unknown elastic constants: C11, C22, C12, C66 (and C11, C33, C13, C55). Thus, seven out of nine elastic constants can be determined. From Eqs. (2) and (5), The remaining unknown elastic constants C44 and C23, will be determined using a bulk ultrasonic measurements.

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As example, a series of 36 averaged waveforms (30 sweeps) corresponds to the Lamb mode A0 propagating in plane P13, was collected with a step Dx = 2 mm. For each waveform, a time Gaussian-window was used to eliminate the noise. The transmitter is exited by 160 kHz burst, and the angles of the transducers are 18° and 18°. Fig. 3 shows s1(t) and s36(t). The change in phase Dun between two signals sn(t) and sn+1(t) corresponding to adjacent positions xn and xn+1(t) can be obtained by the phase of the transfer function defined by:

x1

fiber direction

x3

x2 Fig. 1. Axes and planes of symmetry in long fiber of the thin plate.

3. Materials and methods 3.1. Lamb wave measurements The coordinate axis is chosen such that the axis x1 is normal to the plate surfaces and the x3 direction is aligned with the fibers direction for the olive wood thin plate. The density of the plate is 932 kg/m3 and the thickness is 7 mm (Fig. 1). Two cases are considered, one is for Lamb wave propagating along the fibers direction, the other is for a Lamb wave propagating normal to the fibers direction. Fig. 2 presents the test bed used to measure the phase velocities of Lamb waves. It is noted that olive wood behaves differently for both configurations during Lamb experiment. Two air-coupled capacitive transducers are placed at the same side of the tested sample. The wide frequency bandwidth of these transducers allows modes to be excited in the band of 50–400 kHz. The frequency exited zone for a particular mode is controlled within this bandwidth by the frequency spectrum of the electrical excitation sent to the transmitter. The angular excitation zone is fixed by the angle and the directivity of the transducers. Because of the low velocity of sound (0.5–3 mm) in air and the large diameter of the transducers (45 mm), their angular aperture is very narrow thus making it easy to be mode selective [15]. A detailed description on this has been published by Castings and Cawley [30]. However, the size of these transducers, implies that the investigated material zone is large ( 30  30 cm2). This method is not convenient to detect small defects but appropriate to follow the change in the properties of a material on a widespread zone [15]. The phase velocity of plate modes has been measured by improving the technique described by Castaings and Hosten [31] which is itself based on the well-known spectrum method [32]. The two transducers are oriented such that a pure mode is generated and detected. The receiver is then moved step by step alongside the plate, over a distance D with a step Dx. For each of its positions xn, the signal sn(t) is captured, the setup being controlled by a computer.

  FT½snþ1 ðtÞ Snþ1 ðf Þ ikðxnþ1 xn Þ e ¼ FT½sn ðtÞ Sn ðf Þ 

ð8Þ

where FT is the Fourier transform operator, t is the time, f the frequency and k the wave number of the guided mode. To increase the accuracy of the measurement of this change in phase, a series of signals sn(t) is captured and the average change in phase Du = k(xn+1  xn) is estimated for each frequency f of the spectrum, from the slope of the curve u versus x. A fast Fourier transform routine and a slope estimation routine are used to extract this change in phase Du with step Dx directly from the measured waveforms. Then, the phase velocity can be calculated for any frequency using the following basic relation:

V ph ¼

2pf Dx D/

ð9Þ

Fig. 4 shows the frequency spectrum of the wave signal in Fig. 3. The propagation direction of the Lamb wave is along the fibers direction. The result shows that the maximum frequency of the received signal is around 160 kHz. A similar result was obtained for each wave signal sn(t). Utilizing Eqs. (8) and (9) the dependence of the phase velocity of the Lamb wave on the frequency is shown in Fig. 5. This experiment was repeated several times, and the repeatability was seen to be excellent. The reproducible nature of the results indicated that the hierarchical heterogonous structures of wood were not a problem. To capture the whole of the waveforms, it is necessary to duplicate the same procedure for various angles. Typically 3–4 angles are necessary and since the properties of the material are supposed unknown, it is good precaution to use a small translation step to satisfy the Shannon’s theorem condition [15]: Dx  2k, where k is the wave length of Lamb mode. In the case of the angular distance between the excitation angles for two different Lamb modes are much smaller, two modes are detected. To separate those modes p , where k and k are their wave number [33]. we chose D > jk12k 1 2 2j Generally, a distance D of about 40–100 mm with a step Dx comprised between 1 and 2 mm was chosen to perform these measurements.

Fig. 2. Experimental set-up.

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S. Dahmen et al. / Ultrasonics 50 (2010) 502–507

(a)

6

(b)

2 1.5

4

1 Amplitude (mv)

Amplitude (mv)

2

0

-2

0.5 0 -0.5 -1

-4

-1.5

-6 0.2

0.4

0.6 Time (µs)

0.8

1

-2 0.2

0.4

0.6 Time (µs)

0.8

1

Fig. 3. Averaging time corresponding to mode A0 generated and detected by the air-coupled transducers and propagating in the plane of symmetry P13: (a) s1(t) and (b) s36(t).

e ¼ Fð PÞ

0.45 0.4

m  X

e K  Vð PÞj e V K

2

ð10Þ

K¼1

Amplitude

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 130

140

150

160 170 Frequency (KHz)

180

190

Fig. 4. The frequency spectrum of the waves signals in Fig. 3: (_______) FFT of s1, (- - - - - -) FFT of s36(t).

e ¼ P þ @P ¼ ½p þ @p  is the best estimated vector solution where P i i plus an error vector oP. V is a function of the unknown components that is given by the solution of the Eq. (7). The true elastic constants were then determined using the simplex method [34,35]. h i The upper bounds of the relative errors r ¼ Dppi for each of i those parameters are then computed from the insensitivity matrix [36]. In practice, this computation leads to overestimated errors, but at least the results can be given with a great confidence in the error zone. A statistical analysis by assuming a near Gaussian distribution of error is used as an estimate of the uncertainty in the respective stiffness [37].

8000

Phase velocity (m/s)

7000

A1

6000

S1

5000 4000

S0

3000 2000

A0

1000 0

0

0.5

1

1.5 Frequency (Hz)

2

2.5

3 x 10

5

Fig. 5. Phase velocities of Lamb modes in olive wood in plane P13: anti-symmetric Lamb modes, A0 and A1, symmetric Lamb modes, S0 and S1. (_______) computed velocities; () experimental data.

3.2. Procedure of Cij identification and precision of results 3.2.1. Inversion scheme The inverse problem consists in deducing a vector P composed of pi ði 2 ½1; . . . ; nÞ unknown elastic moduli from the data. For propagation of Lamb waves in a plane of symmetry, this vector is either P = {C11, C22, C66, C12} or P = {C11, C33, C55, C13} for propagation plane P12 or P13, respectively. e K¼1;...;m , with m  n, denote the m experimental values Let V relating to Lamb wave velocities. Then the inverse problem cone of the pi parameters that minisists in finding the best estimate P mize the quadratic, positive form:

3.2.2. Inversion results The experimental phase velocities for modes A0, S0 and A1 propagating along the fibers direction are presented on Fig. 5. The optimization process gives the values of elastic constants C11, C33, C13 and C55 with the uncertainties (Table 1) and these values are used to compute the velocities shown on Fig. 5. We note that the experimental results are in accordance with the theoretical prediction. The A0 mode was found very sensitive to the shear moduli C55. In fact, this mode was the easiest to generate and to detect since it is the most coupled to the air. As result C55 was determined with relatively high accuracy. The measured phase velocities for modes A0 and S0 propagating normal to the fibers direction was utilized to determine inversely the elastic constants C11, C22, C12 and C66 (Table 2) and these values are used to compute the velocities shown on Fig. 6. Again, very good agreement is obtained between the experimental data and the computed data Fig. 6, thus giving a good confidence in the Cij values. Note that the optimized C11 value is not very different from those calculated previously. Table 3 contains the all stiffness determined by only one optimization. Moreover the two planes P13 and P12 are treated simultaTable 1 Elastic properties of the olive wood thin plate identified from the plane P13 and their associated uncertainties. C33 (GPa) C11 (GPa)

10.7 ± 0.25 4.21 ± 0.10

C13 (GPa) C55 (GPa)

2.52 ± 0.17 1.12 ± 0.01

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S. Dahmen et al. / Ultrasonics 50 (2010) 502–507

Table 2 Elastic proprieties of the olive wood thin plate identified from the plane P12 and their associated uncertainties. 4.56 ± 0.34 4.66 ± 0.45

C12 (GPa) C66 (GPa)

3.17 ± 0.24 0.91 ± 0.06

Table 3 Elastic proprieties of the olive wood thin plate identified from the planes {P13 U P12}. C33 C11 C22 C55

(GPa) (GPa) (GPa) (GPa)

10.90 4.29 4.36 1.12

C66 (GPa) C13 (GPa) C12 (GPa)

0.99 2.65 2.89

3500 3000 2500 2000

S0

1500 1000

A0

500 0

0

2

4

6

8 10 12 Frequency (Hz)

14

16

18 x 10

4

Fig. 6. Phase velocities of Lamb modes in olive wood in plane P12. (_______) computed velocities; () experimental data.

neously and alike. As expected, within the intervals of errors, the stiffnesses are equal to those calculated previously. The average values of the inversely determined constants are shown in Table 4. The remaining unknown elastic constants C44 and C23, will be determined using bulk ultrasonic measurements.

3.3. Identification of the remaining unknown elastic constants with contact technique Measurements of the ultrasonic velocities were made using a transmission technique with the transducers in direct contact with the specimens. The receiver and transmitting piezoelectric ceramic transducers were identical at 0.5 and 1.0 MHz for longitudinal and transversal wave. The diameter of the transducers is 3 cm. They were connected to a Panametrics analyser and the output signal was fed to an oscilloscope from which the transmission time was read. If the test sample is thick enough, it is a simple matter to measure the travel time of a wave generated at one face and received at the other face. The dimensions of the sample and the mass density can be measured in the normal way. The shear modulus was computed by using the average of two transverse waves of x2 and x3 directions (Eq. (2)):

C 44 ¼

q 2

V 223 þ V 232



ð11Þ

To measure the phase velocities V23 and V32 it was necessary to prepare samples having x2 and x3 a normal direction. This dimension was selected so that the shear waves would give an adequate signal in wood. The C23 remaining moduli can be measured with the aid of Eq. (5) and by sending waves propagate in diagonal direction preferably in the symmetry plane P23. To do that, it is necessary to cut the sample at a precise angle (for instance 45°) from axis of symmetry x2 and x3 and place contact transducers. The two remaining elastic constants C44 and C23 measured with contact technique are 1.05 GPa (with accuracy about 12%) and 2.08 GPa (with accuracy >50%) respectively.

Table 4 Elastic proprieties of the olive wood thin plate identified from the average values of the inversely determined constants. C33 C11 C22 C55

A1

4000 Phase velocity (m/s)

C11 (GPa) C22 (GPa)

4500

(GPa) (GPa) (GPa) (GPa)

10.8 4.35 4.51 1.12

C66 (GPa) C13 (GPa) C12 (GPa)

0.95 2.59 3.03

4. Identification of the symmetry medium On combining the constants elastics recovered by Lamb wave with those values measured by the ultrasonic bulk wave method, the nine elastic constant of the olive wood are determined as:

2

4:35 3:03 2:59 0

6 3:03 6 6 6 2:59 ½C ij  ¼ 6 60 6 6 40 0

0

0

0

0

0

2:08 10:8

0

0

0

0 0

0 0

1:05 0 0 0 1:12 0

0

0

0

4:51 2:08

0

3 7 7 7 7 7 GPa 7 7 7 5

ð12Þ

0:95

These results show that plane P12 which is normal to the fibers is approximately an isotropic plane since C22  C11, C44  C55, C23  C13 and 2C66 + C12  C22. In fact, there is significant difference between C23 and C13 because the two elastic constants were measured by two different techniques. Both experimental approaches lead to consistent results. 5. Conclusion The anisotropic elastic constants of olive wood, considered as an orthotropic solid, have been determined using a combination of the Lamb wave and ultrasonic bulk wave. Dispersion of Lamb wave velocities of many modes were measured by using air-coupled, single-sided system. These measurements were combined with an inverse algorithm to obtain seven out nine elastic constants of the material (C11, C22, C33, C12, C13, C55, C66). The two remaining unknown elastic constants (C23, C44) were obtained using the measured bulk ultrasonic wave velocity. Based on the results of this study, the unidirectional nature of the olive wood has been recovered. In the future, the Contact less technique could be used to the identification of wood viscoelastic properties. Acknowledgment The research was supported by ‘‘Ministry of higher education, scientific research and technology in Tunisia”. References [1] R.F.S. Hearmon, An Introduction to Applied Anisotropic Elasticity, Oxford University Press, Oxford, 1961. [2] M.J.P. Musgrave, Crystal Acoustics, Holden Day, San Francisco, 1970. [3] R.F.S. Hearmon, The Elasticity of Wood and Plywood, His Majesty’s Stationery Office, London, 1948. [4] J.C. Schelleng, Requirements for sounding board material, Catgut Acoust. Soc. Newsl. 11 (1969) 18–22.

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