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Physics Procedía 70 (2015) 467 - 470

2015 International Congress on Ultrasonics, 2015 ICU Metz

Ultrasonic determination of the elastic constants of epoxy-natural

fiber composites

C.A. Meza Valenciaa *, J.F. Pazos-Ospinaa, E.E. Francob, Joao L. Ealoa, D.A. Collazos-Burbanoa, G.F. Casanova Garciaa

a Universidad del Valle,School of Mechanical Engineering, Calle 13 # 100-00, 760032, Cali-Colombia b Universidad Autónoma de Occidente, Calle 25 # 115-85, 76001000, Cali-Colombia

Abstract

This paper shows the applications ultrasonic through-transmission technique to determine the elastic constants of two polymer-natural fiber composite materials with potential industrial application and economic and environmental advantages. The transversely isotropic coconut-epoxy and fique-epoxy samples were analyzed using an experimental setup which allows the sample to be rotated with respect to transducers faces and measures the time-of-flight at different angles of incidence. Then, the elastic properties of the material were obtained by fitting the experimental data to the Christoffel equation. Results show a good agreement between the measured elastic constants and the values predicted by an analytical model. The velocities as a function of the incidence angle are reported and the effect of the natural fiber on the stiffness of the composite is discussed. ©2015The Authors.PublishedbyElsevierB.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ICU 2015

Keywords: Ultrasonic through-transmission, composite materials, vegetable fibers, elastic constants, Christoffel's equation.

1. Introdution

Ultrasonic non destructive evaluation (US-NDE) is an important approach in the characterization and evaluation of mechanical parts. Ultrasonic waves are related to properties of the propagation medium. Therefore, the measurement of acoustical quantities, such as propagation velocities and attenuation, provides information about the elastic properties of the material. The main advantage of US-NDE is its non destructive and non invasive nature, and the flexibility to be adapted to different applications (Balasubramanian (1996)).

The elastic characterization of anisotropic material is an application were the US-NDT has proved useful. The measurement principle is the variation on the wave velocities in the different directions of the anisotropic material. In this way, the measurement of ultrasonic phase velocity along predetermined directions of high symmetry and the comparison with the wave propagation theory permits the estimation of the stiffness tensor (Markham (1969)). However, ultrasonic contact methods require samples cut in various directions to allow access to all constants. This samples are difficult to manufacture or even impossible in some cases. The through transmission technique, i.e. transmission of bulk waves through thin immersed samples at different incidence angles, is best suited to measure

* Corresponding author. Tel.: +57-3212133. E-mail address: carlos.meza@correounivalle.edu.co

1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Scientific Committee of ICU 2015

doi:10.1016/j.phpro.2015.08.287

the stiffness tensor, specially for laminated materials, such as fiber-reinforced composites (Franco (2011); Hosten (2001)).

Fig. 1: (a) scheme of the measurement setup and (b) coordinate system in the sample.

The immersion through-transmission method is based on the measurement of the time-of-flight of the wave with and without placing the sample between transducers. This time is used to calculate the phase velocity and the refraction angle in the sample (Balasubramanian (1996)). By using mode conversion at a liquid-solid interface, one quasi longitudinal and two quasi transverse bulk waves can be generated and transmitted in numerous directions in the solid (Adamowski (2007)). The determination of elastic constants from a set of bulk ultrasonic wave phase velocities in an arbitrary direction of a measured sample of composite material is based on the Christoffel's equation (Adamowski (2007)). By inverting this equation, the elasticity constants can be determined from a suitable set of experimental velocities at various directions.

In this work, the ultrasonic through-transmission technique is employed for the characterization of composites of polymer (epoxy) and natural fibers: fique (Furcraea andina) and coconut (Cocos nucifera). Probable applications of these materials are light panels, with interesting thermal and acoustic isolation characteristics, for the automotive and aeronautical industries. Values of the propagation velocities and the estimated elastic constants are reported.

2. Theoretical background

The through-transmission technique is based in the measurement of the time-of-flight of the ultrasonic waves in a transmission-reception test. Fig. 1a and 1b show the experimental setup and the placement of the coordinate axes in the sample, respectively. When the sample is rotated, three different waves are transmitted into the sample: a quasi-longitudinal and two quasi-shear. The velocity of these three waves are different, then they refract in a different angle. The velocity can be calculated by the following expression (Balasubramanian (1996)):

VP(8r )

1 \2 2Mcos6i /A^ 2

where At is the time delay of the wave with respect to a reference time provided by the case without sample, VW is the wave velocity in water, 6i and 6r are the incident and refracted angles respectively, and d is the thickness of the sample.

The elastic constants of the anisotropic material and the propagation velocities can are related by the Christoffel's equation (Balasubramanian (1996); Markham (1969)):

|r0- - 6ijpV2U 0 (2)

where rij = Cijkmnknm is called Christoffel's tensor, Cijkm is the stiffness tensor, nk and nm are components of the vector of direction wave propagation, ôij is the Kronecker delta, V is the experimental wave velocity and p is the density. By using a fitting or optimization algorithm, it is possible to find the elastic constants (Cj which satisfy the equation 2 (Deschamps (1995)).

A transversely isotropic material have a symmetry plane in which the elastic properties are the same in both directions, but different in the other one. The stiffness tensor of this material can be expressed as showed in equation (3) (Hyer (1998)).

[C ] =

C11 C12 C12 0 0 0

C12 C22 C23 0 0 0

C12 C23 C22 0 0 0

0 0 0 C44 0 0

0 0 0 0 C55 0

0 0 0 0 0 C55

For a composite made of transversely-isotropic reinforce fibers into an isotropic matrix, the elastic constants can be estimated by the analytical model called rule of mixtures (Tsai (1998)). This model provide the elastic properties of the composite from the elastic properties and volume fractions of the fibers and the matrix.

3. Materials and Methods

The measurement system is composed by a an oscilloscope (TDS2014C, Tektronix, USA) used as digitalizing device, pulse/receiver (5072PR, Olympus NDT, USA), two ultrasonic transducers of 4 MHz and 0 = 24 mm (C4024F -RL1, Mana Instruments), a mechanical system for location of the traducers and the automatic rotation of the sample and a desktop computer for data storage and processing.

The samples were made by hand Fig 2. using a specially designed mold for the positioning of the natural fibers. The fiber were disposed in some layers in a specific direction in order to have an unidirectional reinforcement. Then, the epoxy resin was poured in the mold and cured for 24 hours. Finally, the samples were machined to improve the parallelism.

Fig. 2: Epoxy-natural fiber composites

4. Results

Initially, the property extraction method was tested using an aluminum sample. This material is isotropic and has well-known elastic properties. The propagation velocities were measured for incident angles from 0° to 60° and the velocities calculated from the waveforms acquired, using the correlation approach for the determination of the delays. Then, the fitting algorithm, implemented using the optimization toolbox of Matlab, was used to calculate the stiffness matrix. The obtained results was in agreement with theory and from the fitted constants it was possible to obtain the Young's and Poisson's moduli of the material. This test allowed us to validate the correct implementation of the property extraction method. Results with the aluminum sample are not shown.

The composite samples were tested using the same procedure used with the aluminum sample. Fig. 3 shows the experimental and theoretical velocities as a function of the incident angle. The longitudinal (VL) and shear (Vqs ) wave velocities are reported for both epoxy-coconut and epoxy-fique samples in two different planes: 31 and 32. Because the orientation of the fibers, plane 31 is anisotropic and 32 is isotropic. In the anisotropic plane, the value of VL increases in the epoxy-fique case and is almost constant in the epoxy-coconut case when the incident angle increases. This behavior can be a consequence of the difference of stiffness between the matrix and the reinforcement: higher than that of the matrix for the fique fiber and lower for the coconut. The behavior obtained in the plane isotropic is similar to that obtained with the aluminum sample.

Table 1 shows the theoretical and experimental values of the elastic constants obtained for both anisotropic samples. It can be seen a good agreement, with deviations below 8.5% and 16% in the epoxy-coconut and epoxy-fique cases, respectively.

£ 1600

-VL (Predited) ...................VQS (Predited) Experimental

20 30 40 50 Incidence Angle (degree)

>> 1800

2 1600

-VL (Predited)

...................VQS (Predited)

Experimental

20 30 40 50 Incidence Angle (degree)

Fig. 3: Wave velocities as a function of the incidence angle for (a) and (b) epoxy-coconut fiber composite in planes 31 and 32, respectively, and (c) and (d) epoxy-fique fiber composite in planes 31 and 32, respectively.

Table 1: Elastic constants of the composite samples obtained by ultrasound and comparison with the theoretical results provided by the rule o mixtures.

C11 C33 C44 C66 C12

Epoxy-coconut Theoretical 7.1925 6.3848 1.4933 1.8120 3.3982

Experimental 7.0675 6.2816 1.5563 1.9657 3.6586

Epoxy-fique Theoretical 7.4682 5.3732 1.1088 1.8541 3.1557

Experimental 7.5198 5.4854 1.2847 1.6878 3.6703

5. Conclusion

The ultrasonic through-transmission technique was implemented and used for the successful characterization of epoxy-natural fiber composites. The values of the measured elastic constants are in agreement with those obtained using an analytical model, where the deviation was less than 16% .

References

Adamowski, J., Andrade, M., Franco, E., Buiochi, F., 2007. Ultrasonic through-transmission characterization of fiber reinforced composite using a

large aperture receiver. Proceedings of the International Congress on Ultrasonics, Vienna. Balasubramanian, K., Whitney, S.,1996. Ultrasonic through-transmission characterization of thick fiber-reinforced composites. NDT & E International 4, 225-236.

Deschamps, M., Bescond, C.,1995. Numerical method to recover the elastic constants from ultrasound group velocities. Ultrasonics 33, 205-211. Franco, E., Meza, J., Buiochi,F., 2011. Measurement of elastic properties of materials by the through-transmision technique.Dyna 78,59-64. Hosten, B., 2001. Ultrasonic through-transmission method for measuring the complex stiffness moduli of composite materials. Handbook of elastic

properties of solids, liquids and gases, 67-86. Tsai, S., 1998. Composites Design. Think Composites, Ohio, Paris and Tokyo, pp.69. Hyer, M., 1998. Stress analysis of fiber-reinforced composites materials. McGraw-Hill,Boston, pp. 109. Markham, M.,1996. Measurement of the elastic constants of fibre composites by ultrasonics. Composites 1,145-149.