Nonlinear generation of a zero group velocity mode in an elastic plate by non-collinear mixing.

Zero Group Velocity (ZGV) modes are peculiar guided waves that can exist in elastic plates or cylinders, and have proved to be very sensitive tools in characterizing materials or thickness variations with sub-percent accuracy at space resolutions of about the plate thickness. In this article we show theoretically and experimentally how such a mode can be generated as the sum-frequency interaction of two high amplitude primary waves, and then serve as a local probe of material non-linearity. The solutions to the phase matching condition, i.e. condition for a constructive non-linear effect, are obtained numerically in the mark of classical, quadratic non-linearity. The coupling coefficients that measure the transfer rate of energy as a function of time from primary to secondary modes are derived. Experiments are conducted on an aluminum plate using piezo-electric transducers and a laser interferometer, and explore the interaction for incident symmetric and anti-symmetric fundamental Lamb modes. In an experiment operated without voltage amplifier we demonstrate that the resonant nature of these ZGV modes can be leveraged to accumulate energy from long excitations and produce detectable effects at extraordinarily low input power even in such weakly non-linear material.

Assessment of elastic coefficients of child cortical bone using resonant ultrasound spectroscopy.

The assessment of the anisotropic elastic properties of non-pathological child cortical bone remains a challenge for the biomechanical engineering community and an important clinical issue. Resonant ultrasound spectroscopy (RUS) can be used to determine bone stiffness coefficients from the mechanical resonances of bone specimens. Here, a RUS protocol was used on 7 fibula specimens from children (mean age 14 ±â€¯3 years) to estimate the whole elastic stiffness tensor of non-pathological child cortical bone considered as orthotropic. Despite a small number of sample, results are consistent with this hypothesis, even if a trend towards transverse isotropy is discussed. Indeed, the average values of the 9 independent stiffness coefficients obtained in this study for child bone are: C11 = 16.73 ±â€¯0.19 GPa, C22 = 16.19 ±â€¯0.12 GPa, C33 = 24.47 ±â€¯0.30 GPa, C44 = 4.14 ±â€¯0.08 GPa, C55 = 4.16 ±â€¯0.07 GPa, C66 = 3.13 ±â€¯0.05 GPa, C12 = 10.14 ±â€¯0.20 GPa, C13 = 10.67 ±â€¯0.27 GPa, C23 = 10.25 ±â€¯0.14 GPa.

Rayleigh wave harmonic generation in materials with depth-dependent non-linear properties.

Non-linear Rayleigh waves are investigated theoretically in materials having inhomogeneous third order and homogeneous second order elastic properties. Compared to the complete homogeneous case, the constants coupling the harmonics turn out to depend on frequency. The sensitivity and selectivity to the three Murnaghan's constants is discussed considering mono- and bi-chromatic inputs in the mark of the quasi-linear approximation. Numerical examples are given for depth profiles characteristic of dislocation densities in shot-peened metals. The predicted low-frequency scales are consistent with observation recently reported.

Transient 3D elastodynamic field in an embedded multilayered anisotropic plate.

The aim of this paper is to study the ultrasonic response to a transient source that radiates ultrasonic waves in a 3D embedded multilayered anisotropic and dissipative plate. The source can be inside the plate or outside, in a fluid loading the plate for example. In the context of Non-Destructive Testing applied to composite materials, our goal is to create a robust algorithm to calculate ultrasonic field, irrespective of the source and receiver positions. The principle of the method described in this paper is well-established. This method is based on time analysis using the Laplace transform. In the present work, it has been customized for computing ultrasonic source interactions with multilayered dissipative anisotropic plates. The fields are transformed in the 2D Fourier wave-vector domain for the space variables related to the plate surface, and they are expressed in the partial-wave basis. Surprisingly, this method has been very little used in the ultrasonic community, while it is a useful tool which complements the much used technique based on generalized Lamb wave decomposition. By avoiding mode analysis - which can be problematic in some cases - exact numerical calculations (i.e., approximations by truncating infinite series that may be poorly convergent are not needed) can be made in a relatively short time for immersed plates and viscoelastic layers. Even for 3D cases, numerical costs are relatively low. Special attention is given to separate up- and down-going waves, which is a simple matter when using the Laplace transform. Numerical results show the effectiveness of this method. Three examples are presented here to investigate the quality of the model and the robustness of the algorithm: first, a comparison of experiment and simulation for a monolayer carbon-epoxy plate, where the diffracted field is due to a source located on the first free surface of the sample, for both dissipative and non-dissipative cases; second, the basic configuration of an aluminum plate immersed in water has been chosen to study wave propagation in ZGV (Zero Group Velocity) conditions; finally, a 2D plate consisting of 8 stacked carbon-epoxy layers immersed in water is treated, with a source located inside the plate, distributed in depth and extending over four layers.