Summary

The carbon epoxy is a composite material extensively used in reason of these particulars mechanical characteristics. It is constituted by an assembly of layers whose the fibers of carbons have all the same orientation. The propagation of acoustics waves in this type of material being the object of several works.

This paper consists to study the water / carbon epoxy interface, with any orientation of the fibers. A complete analysis of the problem of propagation of acoustic waves including a calculation of the reflection-transmission coefficients, for a given orientation of carbon epoxy.

Some results linked to the reflection- transmission like the critical angles, the mode conversion, the new wave and the birefringence will be discussed.

In this study two case will be considered: in the first case we chose a plan of incidence which coincides with a plane of symmetry of the media. In the second case the plan of incidence is out this plan.

1) INTRODUCTION

The propagation, reflection and transmission of acoustic waves in anisotropic materials like composites is a topic research which find some applications in non destructive evaluation (NDE).

The carbon epoxy is a composite material extensively used in reason of its mechanical characteristics. It is constituted by an assembly of layers whose the fibers of carbons have all the same orientation. The propagation of acoustic waves in this type of material being the object of several works[ 1,2,3,4]. The material properties are given in table 1[2].

	C11(GPa)	C12(GPa)	C13(GPa)	C33(GPa)	C44(GPa)	C66(GPa)	r (Kg/m3)
Water	2.19	2.19	2.19	0	0	0	1000
Carbon epoxy	15	7.7	3.4	87	7.8	3.65	1595
Table A1: Tensor of rigidity and density of water and carbon epoxy

The interest of the study of the propagation of acoustic wave in anisotropic media comes especially of the application to the composites materials and to the layers.

To study the phenomenon of reflection- transmission we defined a basis of work (,, ) which (, ) coincides with the interface plane and (, ) with the plane of incidence. The development of the calculations in this basis imposes to convert the tensor of rigidity from the crystallographic basis (, ,) to the basis of work.

is parallel to the fibers and (, ) are two orthogonal vectors of the polar plan [ 2].

Every studied orientation is characterised by two angles a and b. They measure the angular gaps of fibers with the plan of incidence and the interface. a equal to the angle that makes with the projection on the plan of interface. b gives the angle that makes with its projection on this same plan. The vectors and are arbitrary because they belong to the isotropic polar plan. In the present model is taken in the plan of interface (fig.1.).

we designate the angle of incidence by q , and Y the angle of refraction of energy flux vector .

Fig 1: Geometry of configuration of water / carbon epoxy interface with regard to the basis of work.

This work consists to study the water / carbon epoxy interface, with any orientation of the fibbers. The coefficients of reflection and transmission of energy as well as the angle of refraction of energy flux vector versus the incidence angle are computed. The upper media is the water, supposed not viscous, so the incident wave is homogeneous purely longitudinal. This development was applied to different types of interfaces. For each case we calculate the different characteristics parameters of the waves generated by the interface and we present the most representative: the coefficients of reflection- transmission, critical angles, and the "wave new "[ 5,6,7].

In this study two case will be studied: In the first case we chose a plan of incidence which coincides with one of plan of symmetry of the media i.e. a = 0 and b ¹ 0. In the second case the plan of incidence doesn't coincide with a plan of symmetry of the media, i.e. a ¹ 0 and b arbitrary. Such a configuration is obtained by making the fibbers of carbon a rotation a followed by a rotation b.

2) Method of calculation

The hypotheses of basis restraints are those admitted in the theory of the linear elasticity of the homogeneous continuous media in which propagate the acoustics waves. In the theory of dynamic elasticity[ 8], the equations space -temporal writes, in the absence of force of volume:

(2.1)

(2.2)

Where r is the density of the transmitted media, u the field of displacement, C_ijkl the tensor of rigidity and s_ij the field of constraint.

The description of the acoustics waves in term of inhomogeneous plane waves:

(2.3)

where A the complex amplitude of the wave, P is the complex polarization vector, m is the complex slowness vector, w is the angular pulsation of the wave and x is the vector position.

The substitution of the eq (2.1) in the eq (2.2) gives to the equation of Christoffel:

(2.4)

with

(2.5)

Starting from the Snell’s, Fedorov[ 2,9] shows that the slowness vectors of all the incident, reflected and transmitted waves are contained in the plane of incidence and that their projection on the interface is the same. Consequently, in the plane of incidence the slowness vector of every reflected and transmitted wave doesn't admit more than an alone unknown component, the one perpendicular to the interface, the one parallel to the interface being equal to the one of the incident wave.

This unknown component is obtained by nullifying it determinant of Christoffel's tensor:

det(Gik)= 0

(2.6)

Which leads to an equation of degree six which the roots are the six normal components to the interface of each slowness vectors of the six possible waves [10]. Among these six roots, some could be complex who indicate the absence of intersection with the corresponding slowness surfaces. For this type of roots, the slowness vectors correspondents are also complex, this translated by an attenuation of the wave in a certain direction. The nature of this wave is in the general case, inhomogeneous. It appears as soon as the angle of incidence is superior to the critical angle [ 2].

The selection of the three physically acceptable waves is made on energetic criterion, given by Fedorov the reflected and transmitted waves must have their energy flux vector oriented respectively toward the reflected and transmitted media[ 11].

3) Orientations of carbon epoxy

Fig 2.a and fig2.b give a section of slowness surfaces of carbon epoxy for different types of orientations; the upper is the slowness surface of the water. Its allows to identify the invariant component of slowness vector. Two types of orientation were envisaged: (0, b) and (a, b) orientations.

Fig 2a: Slowness surface of carbon epoxy for (0,65) orientation.	Fig 2b: Slowness surface of carbon epoxy for (20°,20°) orientation.

3.1) Orientation (0,b)
The (0, b) orientation corresponds to the axis of high symmetry i.e. the fibbers of carbons, lie in the incidence plan. The incident wave being purely longitudinal, only the QL (quasi-longitudinal) and QTV (quasi-transverse vertical)) modes will be excited. For different value of b, we present the variations of the coefficients of reflection- transmission in energy as function of the angle of incidence fig 3.(a,b,c,d). Also the angles of refraction of the transmitted waves are given by fig4.(a,b,c,d).

Fig 3 (a,b,c,d): Reflection - transmission coefficients in energy versus incidence angle.

Fig 4 (a,b,c,d): Refraction angle of flux energy versus incidence angle.

3.2) Orientation (a,b)
This orientation corresponds to the axis of the fibbers is out the plan of incidence. In this case The QTH (quasi-transverse horizontal) mode is now excited.

The same families of figures that previously are presented fig5.(a,b,c,d).

Fig 5(a,b,c): Reflection - transmission coefficients in energy versus incidence angle.

4) Numerical results

In this paragraph we comment the two families of curve for the two types of orientation. The phenomenon of "wave new", birefringence and interference will be examined.

Some facts linked to the anisotropy is obtained by varying the incidence from 0° until 90°.

This development neglected the dissipation of energy (real constant rigidity), consequently the incident energy finds again, for every incidence, divided between the reflected and the transmitted waves. The conservation of energy appears clearly on the representative curves of reflection- transmission coefficients versus the incidence angle (fig. 3), (fig.5) and (fig.7).

Fig 6(a,b,c): Refraction angle of flux energy versus incidence angle.

Fig 7(a): Reflection - transmission coefficients in energy versus incidence angle.

Further, in orientation (0,b) we notes that the angular zone of existence of the " wave new " under the homogeneous state depends with b , its varies from 1.8° for b = 50° until 5° for b = 65.5° (fig3.c).

The energy transported by this "wave new" is relatively important in this zone, its reaches 60% of the incident energy. In the literature the energy transported by the "wave new" is in generally negligent [2].

Figure.8 shows the refraction angle of the vector flux of energy versus the incident angle. Its can been seen the " wave new " and the QTV wave refracts in the same angle to the neighbourhood of b = 60°. A phenomenon of interference in this zone will be produced. The obtention of this phenomenon stays tributary of the energy guided by every wave and their polarization.

Fig 8(b): Refraction angle of flux energy versus incidence angle.

By elsewhere the variations of the angle of refraction of energy flux vector versus the incident angle for b > 63°, shows that the QL waves and QTV refracts in the same direction. The speeds of these two modes are different (Fig.9-10.), consequently This situation doesn't seem to produce a phenomenon of interference.

Fig 9: Refraction angle of flux energy versus incidence angle.

An other aspect resides in the angular gap between the QL and QTV waves, this phenomenon called birefringence. The birefringence between the two modes is important even under the normal incidence. The variation of birefringence as a function of b angle is presented in figure11. It is clear that the birefringence is appreciable to the least variations of b . However the variations of the angle a is without effect on the birefringence because the interface being the same i. e coincides with the plan meridian containing the fibbers.

Fig 11: Birefringence (L-TV) versus angle b, under normal incidence

The (0, b) orientation reveals that the reflected energy neighbourhood of the normal incidence varies from 0.3 to 0.6 when b varies from 0° to 90°. In these conditions, the plan of the interface coincides with the meridian plan, then the QTH mode stays no excited.

5) Conclusion

In the developed formalism, the reflection- transmission of homogeneous plane wave incident on an interface between fluid / anisotropic solid are presented. Numerical examples of the excitation and propagation of acoustics waves are illustrated for a given orientation of transmission media. The characteristics of the reflected and transmitted waves are presented: Coefficients of reflection transmission in energy, refraction of energy of flux vector, critical angles etc....

The birefringence where the angular gap between the QL mode, QTV and QTH, null when the direction of the fibbers is in the plan of interface, could reach some angles relatively elevated when this same axis is not in this plane.

Besides, the surface slowness, associate to the QTV mode of the carbon epoxy reveals the existence of "wave new" which transport about 60% of the incident energy. This result is observed when the other transmitted waves become evanescent.

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Acknowledgements

Our acknowledgements are addressed to the Foundation Humboldt Alexander (Germany), for the Grant of materials to the Department of Physics of the Faculty of the Sciences of Sfax (Tunisia), permitted the realisation of this work.

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