Acoustic methods of land mine detection rely on the vibrations of the top plate of the mine in response to sound. For granular soil (e.g., sand), it is expected that particle size will influence the mine response. This hypothesis is studied experimentally using a plate loaded with dry sand of various sizes from hundreds of microns to a few millimeters. For low values of sand mass, the plate resonance decreases and eventually reaches a minimum without particle size dependence. After the minimum, the frequency increase with additional mass includes a particle-size effect. Analytical continuum models for granular media applied to this problem do not accurately capture the particle-size effect. In addition, a continuum-based finite element model (FEM) of a two-layer plate is used with the sand layer replaced by an equivalent elastic layer. For a given thickness of the sand layer and corresponding experimental resonance, an inverse FEM problem is solved iteratively. The effective Young's modulus and bending stiffness of the equivalent elastic layer that match the experimental frequency are found for every layer thickness. Smaller particle sizes are shown to be more compliant in bending. The results clarify the importance of particle size on acoustic detection methods.
The scattering of elastic waves in a medium with damage from microcracking is discussed. A generalized tensor-based approach is used such that the results are coordinate free. The influence of damage from penny-shaped microcracks within a homogeneous medium is considered. The microcracks are assumed to be randomly oriented and uniformly distributed. Explicit expressions are derived for the attenuation of longitudinal and shear elastic waves in terms of the statistics damage parameter and the effective elastic moduli of the medium. The derivation is based upon diagrammatic methods. The problem is formulated in terms of the Dyson equation, which is solved for the mean field response within the limits of the first-order smoothing approximation. The attenuations are given here in a direct way. The longitudinal and shear attenuations are discussed in terms of their frequency dependence and damage dependence. In particular, the effective elastic stiffness of statistical distribution of microcracks and the example results are discussed. The attenuations are shown to scale with the square of the damage parameter for low frequency.
The analysis of the dynamic behavior of the micro- cantilevers employed in atomic force microscopy (AFM) is often limited to linear or weakly nonlinear behavior without damping. Finite element simulations are used here to study the cantilever dynamics outside of these restrictions. The nonlinear contact mechanics between the AFM tip and the material surface are modeled using the JKR model with different damping. This model is most appropriate for AFM cantilevers that are most compliant than the specimen. The focus is on the contact case in all analyses to simplify the problem. Thus, the AFM cantilever tip is assumed to remain in contact with the specimen surface at all times during the motion. Applications for both weakly and strongly nonlinear behavior are examined. The properties of the vibration, the influence of different initial loads and different damping models on the behavior, like nonlinear shifts of the resonance frequencies, the eccentricity and asymmetry of the amplitude, of the nonlinear vibration are calculated by FEM. The numerical analysis shows that the eccentricity and the asymmetry of the amplitude are more sensitive to the change of damping and the contact stiffness than the resonance frequencies. The response of the cantilever and the evaluation of elastic properties of the sample can be studied appropriately using this model.
To investigate nanoscale mechanical behavior, new approaches using dynamic modes of the atomic force microscope cantilever are being developed. One method, atomic force acoustic microscopy (AFAM), measures cantilever resonances in the acoustic frequency range to obtain elastic-property information. We describe quantitative AFAM measurements and compare them to results from techniques like surface acoustic waves and instrumented indentation. With AFAM we examined a niobium film using two separate calibration samples and two cantilever geometries. Depending on the cantilever type we found M=105-114 GPa, in good agreement with literature values of M=116-133 GPa for bulk niobium and M=120 GPa obtained with surface acoustic waves. We also obtained AFAM values of M=54-81 GPa for the indentation modulus of an aluminum film. In comparison, literature values for bulk aluminum are M=76-81 GPa, while other results on the same film yielded M=78-85 GPa. To understand the results more thoroughly, we compare two methods of AFAM spectrum analysis. The analytical approach assumes a cantilever of uniform rectangular cross-section while the finite-element model accounts for spatial variations in cantilever dimensions. The same data are interpreted with the two approaches to better understand measurement uncertainty and accuracy.
Recent atomic force microscopy research has focused on dynamical methods in which AFM probes are vibrated while in contact with a specimen during scanning. The nonlinear tip-sample interactions can induce nonlinear features into the dynamic response. Nonlinear responses observed experimentally include the DC shift (or lift-off) and primary response softening as well as the development of subharmonics and superharmonics. Here, this problem is formulated in terms of a nonlinear boundary value problem which is solved using the method of multiple scales. The main result of this analysis is the amplitude-frequency relation for all vibration modes. The nonlinear normal modes are comprised of terms representing the softening effect of the resonance, the static offset, and harmonics. The softening effect on the primary response is shown to be a function of the particular vibration mode as expected. The contact mechanics model used here is restricted to Hertzian contact, but can be generalized to more complex models. Results of the primary response for various excitations are presented. The amplitude-frequency behavior is dependent on the linear contact stiffness, the forcing amplitude, and contact damping. It is also shown that the modes have a differing sensitivity to the nonlinearities present in the contact.
Advancements in atomic force microscopy have led to the development of new measurement techniques that take advantage of the different vibration modes of the cantilevers. Each vibration mode has a different sensitivity to the variations in surface stiffness. The cantilever interacts with the sample surface through the tip of the cantilever. This interaction is approximated as a linear spring such that linear vibration theory may be used for analysis. This simplification restricts the results to experiments involving low amplitude excitations. For imaging, a single vibration mode is selected for feedback control. The image contrast is directly controlled by the modal sensitivity of the cantilever. Low-stiffness cantilevers have typically been unusable for imaging of stiff materials because of the lack of sensitivity of the first flexural mode. In this article, a closed form solution of the modal sensitivity for flexural vibration modes is derived for cantilevers with constant cross-sections. For cantilevers with other shapes, an approximate solution is developed using the Rayleigh-Ritz method. For given nominal values of surface and AFM cantilever properties, the appropriate mode for highest contrast may be predicted.
The propagation and scattering of high-frequency ultrasound in concrete is discussed. Frequencies above 100 kHz have wavelengths short enough for sensitivity to microcracking. However, the heterogeneous composition of concrete causes the ultrasound at such frequencies to scatter considerably. Theoretical descriptions of the scattering attenuations based on a stochastic wave equation are discussed. These expressions require information about the two-point spatial correlation function. The form for this function is proposed and confirmed experimentally. Finally, ultrasound diffusion experiments are discussed. In the limit of many scattering events, the ultrasonic energy density in circular cylinders of concrete is shown to evolve in accordance with a one-dimensional diffusion equation. The ultrasonic diffusivity was measured experimentally over the frequency range of 100-900 kHz. Theoretical descriptions of the diffusivity are in accord with the experimental values. Such frequencies are well above typical frequencies used for concrete inspection. Thus, it is anticipated that the use of these higher frequencies will result in new techniques for characterizing material properties and damage in concrete structures.
Many versions of dynamic atomic force microscopy have been proposed for imaging specimens. All of these methods rely on the relative motion between the atomic force microscope (AFM) tip and the specimen surface. These techniques are used to extract quantitative information about the surface stiffness with high resolution. These techniques utilize the dynamic response of the cantilever, specifically in terms of the higher-order cantilever modes. These techniques rely on tip-sample mechanics models in order to determine material properties. The implications of the different models on the interpretation of AFM images is discussed. In particular, the effect of adhesion on these measurements is discussed.