Complex Composite Structures with Integrated Piezoelectric Transducers

Currently, in different industrial fields as transport or aerospace, a research effort is conducted to reduce the structural weight [1, 2]. One of the most promising solutions is the use of composite structures [3], especially the fibres-based composite structures [4], due to their high stiffness, their low mass density and their low damping factor. At the same time, there is an intensification of the operational dynamic environment and an increase of durability requirements [5]. These different expectations seem to be contradictory. One solution to manage these points is to design and manufacture smart composite structures with a fully distributed set of integrated piezoelectric transducers. These structures are able to modify their mechanical properties with respect to their environment (e.g. active vibration control), to interact with other structures (e.g. mechatronic) or with human beings (e.g. Human– Machine Interaction).

Conventionally, to functionalize a mechanical part made of composite, the piezoelectric transducers are glued on the surface of the structure and the power and control electronics are away. Our approach is significantly different. We want to develop a wide distributed network of piezoceramics and integrate them into the heart of the composite. The idea is to protect the transducer elements and their connections and develop some industrial products in “plug and play” mode.

To place these new features in the heart of the structures, it is necessary to integrate the transducers during the manufacturing process of composite structures. To meet this requirement, specific manufacturing methods have been developed in the laboratory IRTES-M3M of the Université de Technologie de Belfort-Montbéliard (UTBM).

Embedding the transducers is however not always sufficient to guarantee the effectiveness of the approach. A specific preliminary design strategy has then been developed. The idea is to be able to predict the final behaviour of the structure. Our approach is based on an experimental approach for determining the useful design parameters [6]. The system architecture allows to establish the needs for experimental characterization. It is necessary to have:

• A characterization of piezoelectric ceramics. It corresponds to the input control of piezoceramics [7].
• A characterization of the fabricated composite. Once the manufacturing process is stabilized, the composite must be fully characterized by a set of tests, which get access to the nominal material parameters and the uncertainty of the manufacturing process.
• A characterization of “integrated” piezoceramics. The idea is to develop a behaviour model, which can evaluate the drift of the electro-mechanical coupling coefficients from the input control data.
• A characterization of the electrical interfaces. The method of electrical connection, particularly between the transducers and the electrical grid, needs to assess the influence of process parameters on the quality of the contacts. Indeed, the electrical connection by welding technology is not used in our manufacturing process [8].
• A characterization of the inter-element coupling. It is necessary to evaluate the mechanical and electrical couplings between the transducers, according to the distance and connection technology, to establish dedicated design rules.

This paper is focused on the way of characterizing a manufactured composite material. Two different methods (Resonalyser Method and Time-of-flight Method) are used to extract the useful composite parameters. The paper is organised as follows. Section 2 shows the experimental setup and introduces both methods, about the theory and the way to use them. In Section 3, the design of experiments is introduced, the results as well as the comparison of the results obtained with both methods are presented. The complementary experiments on a spherical cap are introduced in Section 4. Finally, concluding remarks are given.

Two methods of material parameters characterization parameters (Resonalyser Method and Time-of-flight Method) are operated, for calibrated tests and manufactory process optimization.

For characterization, five plates instrumented with four piezoceramics have been produced. The piezoceramics are positioned at each corner of the plates, as shown in Figure 1. The characteristics of these transducers are given in [7]. In order to exploit the Resonalyser method, the dimensions should according to Poisson’s plates [9] in particular to accurately determine the Poisson ratio. The manufactured material can be considered transversely isotropic. Therefore, the plates should be square. These structures are 298 mm wide and 2 mm thick with a gelcoat of 0.2 mm, as shown in Figure 1. The plates are manufactured in the laboratory IRTES- M3M of the UTBM. This is a laminated composite composed of 6 layers of glass fibres and a polyester resin matrix. The technique of infusion is used as a manufacturing method. The fibre volume ratio is about 35%–40%. At the end, one layer of gelcoat is present on the top surface of the plate. The piezoelectric elements are placed between the first and the second layer. The layer of gelcoat is set as the reference for numbering of the layers.

Many engineering materials behave in an anisotropic manner. Their response to external solicitations depends on the loading direction. A simple but common form of anisotropy is orthotropy. The elastic behaviour of materials having orthotropic symmetry axes (e.g. rolled metal sheets or long-fibre reinforced composites), in a state of plane stress, can be described by Relation (1) between strains and stresses:

In eqn (1), $\left\{\varepsilon_i\right\}$ represents the strain components, $\left\{\sigma_j\right\}$ represents the stress components ( $\left.\mathrm{N} . m^{-2}\right), E_i$ is the Young's modulus in the $i$-direction $\left(\mathrm{N} . m^{-2}\right), v_{i j}$ is the Poisson ratio, and $G_{12}$ is the shear modulus in the $(1,2)$-plane $\left(\mathrm{N} . \mathrm{m}^{-2}\right)$. If the material behaviour is assumed linear, the elastic properties $E_i, v_{i j}$ and $G_{12}$ are also called the 'engineering constants'. Due to the symmetry of the compliance matrix in eqn (1), only four independent engineering constants occur, e.g. $E_1, E_2, v_{12}$ and $G_{12}$.

Recently, an inverse method, called “Resonalyser method”, has been developed to determine the four engineering constants for orthotropic materials [10].

The Resonalyser method is a mixed numerical-experimental method using measured resonance frequencies of freely suspended rectangular plates. Using rectangular plates as a test specimen allows the simultaneous identification of E₁, E₂, v12 and G₁₂. In addition, the obtained elastic material properties are homogenised over the plate surface and hence suitable as input values for finite element models of structures. Moreover, the damages induced by the machining process of the samples is reduced with respect to the specific shaped specimens. The basic principle of the Resonalyser method is to experimentally compare measured frequencies with the numerically computed frequencies of a finite element model of the test plate. This method was introduced in detail in the PhD thesis of T. Lauwagie [11].

In Figure 2, the experimental setup for characterization is presented. A loudspeaker is used as a contactless exciter, with a laser displacement sensor. A function generator (Keithley, 3390) is used, to create an input sine wave in the frequency range of 15 Hz – 185 Hz. The first natural frequencies of the structure are measured, with a conventional manual method based on a digital oscilloscope (Pico Technology, PS 4424).

This material characterization method exploits the ultrasonic wave propagation properties, particularly, the Lamb waves [13]. Such waves have the particularity to spread over long distances in the composite [14]. To generate and capture the wave trains, the piezoelectric transducers integrated into the composite are used. The transducers have a resonant frequency of the radial mode measured in the air around 100 kHz [7]. Once incorporated into the composite by backing effect, the central frequency of this radial mode decreases up to 80 kHz. Then the frequency-thickness product (f.h) is of 0.15 MHz.mm, shown in Figure 3. The phase velocity and the group velocity of the symmetric mode S₀ are then equivalent. So it is possible to measure the S₀ group velocity with using the formulas of extraction of the material parameters developed for the phase velocity [13].

Figure 4 shows the experimental setup to measure the wave trains. A function generator (Keithley, 3390) is used to generate excitation signals via a miniature power amplifier (PiezoDrive, PDM200B). The signals are then captured via a digital oscilloscope (Pico Tech- nology, PS 4424).

At the beginning, the idea is to test the appropriateness of the methods used in relation to our need for characterization and our goal for the preliminary design of adaptive composite structures. A set of assumption is developed to simplify the model. Firstly, for the composite properties, a transversely isotropic model of the homogenised material is considered. The symmetry axis is the thickness axis. For the dimensions of the structures, the assumptions of the theory of Kirchhoff-Love plates are considered [15]. Moreover, the material is assumed in the state of plane stress ($\sigma_3=\tau_{13}=\tau_{13}=0$). Then, with all the assumptions, eqn (1) becomes eqn (2):

The glass fibres are randomly distributed in the plane of the piles. Consequently, the value of Poisson ratio is between the Poisson ratios of the glass fibres ( $v=0.2$ ) and that of the resin $(v=0.4)$. So, for the Time-of-Flight method, the Poisson ratio is around 0.38 . Finally, the mass density of the composite material is around $1,630 \mathrm{Kg} . m^{-3}$ while the mass density of the gelcoat is around $1,100 \mathrm{Kg} \cdot m^{-3}$.

For the numerical model, the Comsol Multiphysics software is used. Firstly, a Poisson’s plate model is implemented, with a set of initial parameters (E₁, E₂, v₁₂ and G₁₂). Secondly, this model is connected to Matlab, with a specific program outputting the eigenfrequencies of the plate. Then, the experimental eigenfrequencies of the five plates are measured. Only the first five modes of vibration are considered. The nominal values of each frequency are in Table 1. At last, an optimisation loop is implemented to find the optimal material parameters by minimising the error between the experimental and numerical eigenfrequencies. The results are shown in Table 2, and the optimisation process is shown in Figure 5. Moreover, a correlation step is necessary to identify and match the numerical and experimental eigenshapes.

To avoid issues related to the determination of the centroid of the S₀ symmetrical lamb wave train, the excitation signal is a pulse and the duration of the flight time is determined by the rising edges of the pulse and the received signal. Only the paths along the diagonals of the plates are operated in this study to limit the boundary parasitic effects. The group velocity obtained is around 3,380 m/s. The measured phase velocity being related to the mechanical properties [13], by using eqn (3), it is possible to determine a Young’s modulus $\mathrm{E}=14.9 \mathrm{GPa} \pm 15.1 \%$.

The Young’s modulus values obtained from both methods are in quite good agreement for the considered structures. The relative difference is around 20%, which seems reasonably low considering the various excitation types and frequency domains used in the two techniques. The Time-of-Flight method will be mainly considered in our future research, since it is simple and quick to set up and external device requirements are limited. Of course, there is still some difference between both results. Among others, the difference in characterization scale explains a part of the discrepancies. The Resonalyser method is an approach based on the modal responses of the structure. The obtained values are global. The Time-of-Flight method is based on the propagation of guided waves along a particular path. Consequently, the meas- ured values are local and more prone to variability in the material properties. Finally, it should be emphasized that, it would be interesting to perform a calibration of the shear modulus.

To test the validity of the Young’s modulus obtained, an experiment has been carried out to predict, in a preliminary design phase, the vibration behaviour of a spherical cap. This structure is made of the same composite material as the Poisson’s plates. The results obtained are then compared with the measurements made on a prototype. The experimental setup is depicted in Figure 6.

The numerical model of the spherical cap has been developed under Comsol Multiphysics software. The first two natural modes of vibration have been used for the comparison. The results from the numerical model and experiments are presented in Table 3.

As shown in Figure 6, the use of draining nets during the manufacturing process creates a more flexible structure for specific natural modes, in particular for numerical Mode 2. Moreover, the thickness of the structure is far from being uniform over the entire structure and thus creates significant changes in stiffness. This probably explains the observed difference between the numerical and experimental eigenfrequencies. As this is the preliminary design phase, the deviation remains quite acceptable. Furthermore, according to this study, the choice of the characterization method has not a significant impact on the difference between the numerical and experimental results.

In this article, two non-invasive methods for identification of composite parameters have been used and compared, the Resonalyser method and the Time-of-Flight method. These data are used to find reliable parameters for the preliminary design of adaptive composite structures. Poisson’s plates are fabricated and characterized via these methods. In the first approach, a set of modelling assumptions for the material behaviour is formulated to simplify the identification of a few parameters while demonstrating the difficulties and the interest of the approach. The impact of variability of parameters due to the manufacturing process is very important on the final response of the structure. But this impact is difficult to quantify during the prelimi- nary design phase because the specific manufacturing steps are not known. A natural question arises: is it necessary to accurately know the material parameters as this design stage? The answer is negative in most cases. Indeed, a numerical model allowing to coarsely simulate the behaviour of the structures may be sufficient. Of course, the detailed design phase needs accurate material parameters and consequently accurate identification processes.