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Sound waves define a sequence of compression and rarefaction and is the means by which acoustic energy propagates in an elastic medium. The number of re- peated sequences per unit time is called the frequency which can be used to classify the sound waves. The ability of the human ear to detect sound is used as the threshold limits along the frequency axis, where the sound wave frequency lower and upper limits are called infrasound and ultrasound, respectively. Ul- trasonics (from ultrasound) is a term which refers to the application of sound waves in transporting mechanical energy at a frequency greater than the upper threshold limit (typically considered to be 20 kHz).

Ultrasonic waves found a great interest in the last years in the Structural Health Monitoring (SHM) field, especially in composites, for the identification of cracks, delaminations, interfacial debondings, fibre fractures and breakages, matrix crack- ing [59, 60, 61]. The general concept is that an ultrasonic wave, travelling in thin walled structures, propagates undisturbed as long as no obstacles are present on its path, whereas, if any of the above-mentioned damage appears, it is typically reflected, diffracted, and also mode conversions might occur. The obstacle or discontinuity can also be represented by a structural component that is part of the structure, such as a rivet hole, a stiffener or a change in thickness.

Over structural health monitoring ultrasonic guided waves offer a variety of applications in fields such as non-destructive testing [62, 63] and material charac- terization [64]. About the last, in fact, the ultrasonic tests allow to characterize materials meticulously and mostly non destructively, providing an accurate deter- mination of the elastic constants. However wave propagation in composite struc- tures is more complex respect to an isotropic structure due to the nature of het-

Wave types in solids Particle Vibrations

Longitudinal Parallel to wave direction

Transverse (Shear) Perpendicular to wave direction Surface - Rayleigh Elliptical orbit - symmetric mode Plate wave - Lamb Component perpendicular to surface (extensional wave) Plate wave - Love Parallel to plane layer, perpendicular to wave direction

Stoneley Wave guided along interface

Table 3.4: Some wave types in solids

erogeneity of the constituents, inherent material anisotropy and the multi-layered construction, which introduces many interesting wave phenomena: a directional dependence of wave speed, a difference between phase and group velocity of the waves, wave skewing, and so on. Since many parameters influence the wave prop- agation an understanding of the nature of the waves in composites is required. In this sense, propagation characteristics of Lamb waves in a unidirectional flax fibres reinforced PE laminate, with emphasis on group velocity, were investigated experimentally and numerically in order to characterize the material properties of the structure and validate the one obtained from static test. Before to discuss about numerical and experimental evaluation of the panel under investigation a brief introduction to ultrasonic waves, with a special emphasis to Lamb waves, is needed.

In solids, sound waves can propagate in four principle modes that are based on the way the particles oscillate. Sound can propagate as longitudinal waves, shear waves, surface waves, and in thin materials as plate waves. Some types of waves in solids is reported in Table 3.4. The particle movement responsible for the propagation of longitudinal and shear waves is illustrated below.

In longitudinal waves, the oscillations occur in the longitudinal direction or the direction of wave propagation. Since compressional and dilational forces are ac- tive in these waves, they are also called pressure or compressional waves. They are also sometimes called density waves because their particle density fluctuates as they move. Compression waves can be generated in liquids, as well as solids be- cause the energy travels through the atomic structure by a series of compressions and expansion (rarefaction) movements.

Figure 3.6: Longitudinal wave

to the direction of propagation. Shear waves require an acoustically solid material for effective propagation, and therefore, are not effectively propagated in materi- als such as liquids or gasses. Shear waves are relatively weak when compared to longitudinal waves. In fact, shear waves are usually generated in materials using some of the energy from longitudinal waves.

Figure 3.7: Transverse wave

Surface (or Rayleigh) waves travel the surface of a relatively thick solid material penetrating to a depth of one wavelength. Surface waves combine both a lon- gitudinal and transverse motion to create an elliptic orbit motion as shown in the image and animation below. The major axis of the ellipse is perpendicular to the surface of the solid. As the depth of an individual atom from the surface increases the width of its elliptical motion decreases. Surface waves are generated when a longitudinal wave intersects a surface near the second critical angle and they travel at a velocity between .87 and .95 of a shear wave.

Figure 3.8: Rayleigh wave

Plate waves are similar to surface waves except they can only be generated in materials a few wavelengths thick. Lamb waves are complex vibrational waves, a combination of longitudinal and vertically polarized shear bulk waves propagating in a direction parallel to the plate throughout the thickness of the material. Propagation of Lamb waves depends on the density and the elastic material properties of a component. They are also influenced a great deal by the test frequency and material thickness. Lamb waves are generated at an incident angle in which the parallel component of the velocity of the wave in the source is equal to the velocity of the wave in the test material. Lamb waves will travel several meters in steel and so are useful to scan plate, wire, and tubes. The theory of Lamb waves has been fully documented in a number of textbooks [65],

[66]. Lamb waves are commonly seen in plate like structures. They consist of two basic modes: an extensional or symmetric (S0) mode that often appears at

higher velocity but lower amplitude waves preceding flexural or asymmetric (A0)

mode (Figure 3.10). For instance, the Figure 3.11 reports the aforementioned waves (longitudinal, flexural, Lamb and Rayleigh) for a 1-mm thick aluminium plate. Note that axial and flexural waver are only low-frequency approximations of the Lamb wave S0 and A0 modes. The Rayleigh wave is the high-frequency

asymptote of the Lamb wave S0 and A0 modes [67].

Figure 3.9: Zero-order symmetric (a) and asymmetric (b) Lamb waves in plate

Figure 3.10: Early arriving symmetric mode and later asymmetric one