VISCOELASTICITY INFLUENCE ON FREQUENCY DEPENDENCE OF THE ULTRASONIC TRANSMISSION THROUGH PLA TES OF COMPOSITE MATERIALS

Transcription

1 VISCOELASTICITY INFLUENCE ON FREQUENCY DEPENDENCE OF THE ULTRASONIC TRANSMISSION THROUGH PLA TES OF COMPOSITE MATERIALS Mare DESCHAMPS and Bernard HOSTEN Laboratoire de Mecanique Physique U.R.A. C.N.R.S. n 867, Universite de Bordeaux I 351, Cours de la überation, TALENCE Cedex, Franee INTRODUCTION Leaky Lamb waves (llw) have been extensively eonsidered in the direct problem of refleetion of elastie waves from liquid-eoupled orthotropie plates [14 ] and in the inverse problem to determine the elastic properties of eomposites [5} This determination is based on the measurement of the incidenee angle (or frequeney) at fixed frequeney (or fixed incidenee angle) eorresponding to the generation of LLW from the measurement of the amplitude of transmitted (or refleeted) waves through (or by) the sampie. There are several meehanisms by whieh wave propagation in eomposite materials ean be attenuated. Among these are : viseoelasticity of the matrix and diffraetion by the fibers and the porosities. In agreement with experiments, assuming the medium to be homogeneous and taking the whole of these effects into aeeount, the eomposite behavior ean be represented by a eomplex stiffness matrix. As a result, refleetion and refraetion eoefficients of an immersed eomposite plate eome from the interferenees ofheterogeneous plane waves within the solid, whieh strongly depend on the anisotropie attenuation. In the ease of a non absorbing medium, it is weil known that the maximum transmission of a plane wave through an immersed plate leads to a dispersive equation whieh is elose to the solution of II W. In the present worte, inspection of these maximums, for anisotropie and absorbing medium, is done in terms of frequeney dependenee. Review 0/ Progress in Quantitative Nondestructive Evaluation, Vol. 11 Edited by D.O. Thompson and D.E. Chimenti, Plenum Press, New York,

2 tmmw Anisotropie principal plane <I> = 90 c:::j on principal plane c:::::j Quasi-isotropie princi pal plane <I> = 0 Figure 1. Definition of the planes of incidence. IMMERSED ANISOTROPIC SOLID Let us take an anisotropie solid immersed and an incident hannonic plane wave, such that its incidence on the plate is defined by the two angles f) and <p, as shown in figure 1. The thickness of the plate is denoted d. The composite behavior is modelied by the complex stiffness constants c ij, as foliows: for i, j = I to 3 and for i = j = Here w,c~ and 'lij are the angular frequency, the elastic constants and the viscoelastic constants, respectively. (1) The coordinate directions associate to viscoelastie constants (see figure 1) are such that : - direction 1 corresponds to f) ;= 0, for any values of <P, - direction 2 corresponds to f) = 90 and <P =0, - direction 3 corresponds to f) = 90 and <P = 90. REFLECTION AND REFRACTION COEFFICIENTS As usual these coefficients are given by solving the boundary equations written at the two interfaces. However, for anisotropic layers, the propagation within or outside a principal plane must be considered separately. Two quasi-shear waves are generated when the incident plane does not 202

3 correspond to a principal plane. In contrast, if these two planes are identical, only one quasi-shear wave is excited. These coefficients are function of : the incidence, the fluid and solid properties, the thickness and the frequency. This can be written in the following form: (2) Here the subscript r means refracted or reflected, Po and Co are the density and the velocity of wave of the liquid, respectively. P stand for the density of the solid and f represents the frequency. The analytical express ions for Rr are given in references [1-4] for the case of a non absorbing plate and homogeneous waves. For an extensive analysis of this problem to the absorbing case, we refer to the work of reference [6]. In this paper, transmitted and reflected waves result of interferences of heterogeneous plane waves within the plate. In addition, it is noted that a very satisfactory agreement was obtained for the composite under consideration, between experimental and theoretical results. As matter of fact, for any angle of incidence, each wave propagating in the layer has its planes of constant amplitude parallel to the interfaces. In other words, the planes of constant amplitude are different that the planes of constant phase. FREQUENCY DEPENDENCE ON ABSORPTION The damping usually depends, in rather complex manner, upon the frequency. This is due to the great number of physical mechanisms that contribute to the phenomenon. However, ultrasonic losses in many polymers may be adequately described by a linear behavior in terms of frequency. In addition, for any anisotropic media, it is assumed that the attenuation is proportional to the frequency for any angles of incidence and for the three waves generated. Consequently, the values of viscoelastic constants 'lij at any frequency f, may be easily deduced from the values of viscoelastic constants T]Oij at the fixed frequency fo ; that is, in the first order of w 'li/c~, Obviously, this assumption is only available in a frequency range, which depends on the porosity as weil as the composite texture, ie, size of both fiber and ply, matrix/fiber bond, distribution of porosities etc... Nevertheless, it is interesting to note that, for the sampie described in the next section, and for the 1 to 8 MHz frequency range, this behavior agrees very weil with an experimental investigation for various incidence angles (see reference [6] ). On the other hand, even though the plate is absorbing, when using equation (3), simple calculations show that the usual fd product is still an intrinsic parameter representative of the physical phenomena. (3) 203

4 COMPOSITE MATERIAL BEHA VIOR Although we do not give experimental results in the present paper, the numerical calculations are made for values corresponding to areal composite, which is a stack of unidirectional fibers plies bound in an epoxy matrix. In the coordinate system used in figure I, the fiber direction is given by 3. Due to the orthorhombic symrnetry, nine elastie and viscoelastie constants are required to describe the anisotropic behavior of this solid. These constants are reported in the following table. Table 1 Viscoelastic properties of the composite. Elastie constants in GPa. 1 C91 C~2 c~~ C92 c9~ c~~ c~ C~ Viscosities in GPa.).ls at MHz. TlIJ Tl22 Tl33 Tl 12 Tl 13 Tl23 Tl44 Tl C~ 3.32 Tl66 0.ü09 These values are obtained, at the fixed frequency fo = MHz, by using methods described in references [7,8,91 whieh deal with similar sampies. Finally, note that the densities p, Po, and the velocity Co are such that: Po = 1 glcm 3, p = 1.5 glem 3 and Co = 1500 mls. Unidireetional fiber eomposites are of interest for aeousticians, beeause they present quasiisotropie properties ( C~ 1 elose to C~2 ) in the plane perpendieular to the fibers direction and high anisotropic propenies in the plane whieh contains the fibers ( C~3 much greater than C~ 1 ). In addition, at the usual frequency, the homogeneity assumption is generally available. DISPERSION EQUA TION OF LAMB WA VES It is not the purpose of this study to deal with theoretieal Lamb waves in anisotropie and absorbing media. For that, the generalisation of the work presented in referenee [10] should be studied. Our interest is only focused on the modeling of praetieal eases, for instanee, when using LLW method to solve the inverse problem in order to reeover elastie eonstants. Typieally, the experimental results of these method are based on the measuring of the maximum/minimum of transmissionlreflection. These coeffieients depend strongly on the wave damping, as shown in figure 2. Effectively, these eurves represent, at fixed incidenee angles, the theoretieal variation of the transmission eoeffieient versus the produet fd, in both elastic (dashed line) and viscoelastie (solid line) eases. lt is notieeable that the ealeulations are made using viseoelastie properties given at the frequeney MHz, and extrapolating them to the whole frequeney range with the help of relation (3). 204

5 O ".' " I / "I',',I J 'tl' f ", f " I ", ~ ~ I j, :/: ',,'! fd (MHz rnrn) L-~~~~~~-L----~--~ Fig.2 Transmission coefficients in an out of symmetry plane of incidence versus fd : Elastic plate. --: Viscoelastic plate. f1 ", j I, 205

6 Elastie plate o Absorbing plate 8 c 40 ~ 35 0.: 'ö 30 4> ~ 25 -< o fd (MHz mm) Fig.3. Angular position of the maximum transmission versus fd (Quasi-isotropie plane 4J = 0 ) 40 g ~ 35 0 ;.: 30 o.., ~ 25 -< Elastie plate o Absorbing plate 10 5 o fd (MHz mm) Fig.4. Angular position of the maximum transmission versus fd. (Out of symmetry plane 4J = 45 ) 206

7 ... u c:: lastic plate 0 Ab orbing plate <I> = 90 0 l -0 'ü c öo c «.l 30 a m! 0 0 Cl o.0 oe fd (MHzmm) Fig.5. Angular position of the maximum transmission versus fd. (Principal plane <I> = 90 ) Therefore, it is of interest to inspect the angular position of the maximum of the transmission, for instance, for various frequency-thickness products, whether the damping is taking into account or not. To this end, on changing the incident plane, three typical cases are analysed, which deal with two principal planes (figures 3 and 4) and one out of the symmetry plane (figure 5). These curves show the angular position of the maximum transmission for the absorbing ( 0 ) and non absorbing plates (.). When the two results are superimposed, the symbol ( 0 ) masks the other one. It should clearly be noted that the influence of the damping depends on both incidence angles and fd product. As an example, for e = 0 and any fd values, the position of maxima are almost identical. In contrast, at fixed angle of incidence, the differences between elastic and viscoelastic responses increase versus the fd product. Keeping in mind that the attenuation increases with the propagation distance, i.e., with the angle of incidence in this particujar case, and with the frequency, these resujts are not surprising. On the other hand, damping effects are at their maximum near critical angles, as can be seen specially in figures 4 and 5, for e = 8 and e = 13, respectively. This reinforces the fact that these particujar areas do not have to be used to solve the inverse problem. Unfortunately, as pointed out in reference [11] for non absorbing plates, it is the only region where LLW are sensitive to some elastic constants. 207

8 Finally, note that the attenuation, in certain cases, makes the detection of LLW impossible. Note also that, conceming the reflection coefficient, similar results and comments are available. CONCLUSION In conclusion, this study clearly shows the influence of attenuation on the reflection and transmission by an anisotropic plate. Consequently, generation and propagation of LLW can be affected by this phenomenon, as it was seen for particular angular areas and/or at high frequencies. In references [1-3], the importance ofbounded beams effects are pointed out for elastic composites. In the present paper, damping in anisotropic material is dealt with in the framework of plane wave theory. Obviously, taking these two effects into account, leads to a successful agreement between both theoretical and experiment results, as shown in reference [12]. However, we think that, at frequencies beyond 1 MHz, the attenuation of the wave remains the major effect which precludes the use of LLW in certain cases. REFERENCES A.H.Nayfeh, D.E.Chimenti " Ultrasonic wave reflection from liquid-coupled orthotropic plates with application to fibrous composites". J. Appl. Mecha., 55, (1988). 2 A.H.Nayfeh, D.E.Chimenti " Ultrasonic leaky waves in asolid plate separating a fluid and vacuum". J. Acous. Soc. Am., 85, (1989). 3 A.H.Nayfeh, D.E.Chimenti " Elastic wave propagation in fluid-loaded multiaxial anisotropic media". 1. Acous. Soc. Am., 89, (1991). 4 S.K.Datta, A.H.Shah, R.L.Bratton, T.Chakraborty "Wave propagation in laminated composite plates".1. Acous. Soc. Am.,83,(6), , (1988). 5 M.R.Karim, A.K.Mal and Y.Bar-Cohen "Inversion of leaky Lamb wave data by simplex algorithm".1. Acous. Soc. Am.,88,(1), (1990). 6 M.Deschamps, B.Hosten "Viscoelastic effects on the ultrasonic waves reflection and transmission by an orthotropic plate". Submitted to 1.Acous. Soc. Am. 7 B.Hosten, M.Deschamps, B.R.Tittmann "Inhomogeneous wave generation and propagation in lossy anisotropic solids. Application to the viscoelastic characterization of composite materials". 1. Acous. Soc. Am., ßb (5), (1987). 8 S.Baste, B.Hosten. Evaluation de la matrice d'elasticite des composites orthotropes par propagation ultrasonore en dehors des plans principaux de symetrie. Revue de Physique Appliquee. 25 (Fevrier 1990). 9 B. Hosten. "Reflection and transmission of acoustic plane waves on an immersed orthotropic and viscoelastic solid layer". 1. Acous. Soc. Am., 89 (6), (1991). 10 V.Dayal, V.Kinra "Leaky Lamb Waves in an anisotropic plate.i: An exact solution and experiments" 1. Acous. Soc. Am.,.8i (6), (1989). 11 S.I.Rokhlin, D.E.Chimenti " Reconstruction of elastic constants from ultrasonic reflectivity data in a fluid coupled composite plate". Review ofprogress in QNDE, Vol. 9b (1990). 12 P.Chevee, M.Deschamps "Non specular effects of reflection and refraction at a viscoelastic anisotropic layer". To be published in proceedings of Ultrasonic International