Koen Van Den Abeele

Micro-damage diagnostics using nonlinear elastic wave spectroscopy (NEWS)

Nonlinear elastic wave spectroscopy (NEWS) represents a class of powerful tools which explore the dynamic nonlinear stress–strain features in the compliant bond system of a micro-inhomogeneous material and link them to micro-scale damage. Hysteresis and nonlinearity in the constitutive relation (at the micro-strain level) result in acoustic and ultrasonic wave distortion, which gives rise to changes in the resonance frequencies as a function of drive amplitude, generation of accompanying harmonics, nonlinear attenuation, and multiplication of waves of different frequencies. The sensitivity of nonlinear methods to the detection of damage features (cracks, flaws, etc.) is far greater than can be obtained with linear acoustical methods (measures of wavespeed and wave dissipation). We illustrate two recently developed NEWS methods, and compare the results for both techniques on roofing tiles used in building construction.

Damage assessment in reinforced concrete using spectral and temporal nonlinear vibration techniques

Both linear and nonlinear (amplitude-dependent) acoustical experiments are performed on a reinforced concrete (RC) beam in which damage is gradually induced by means of static loading tests. At different levels of damage, a complete experimental modal analysis (EMA) is carried out, assuming the structure to behave linearly. The analysis in terms of modal curvatures indicates a gradual reduction of the bending stiffness along the beam. Strong amplitude dependence of the linear dynamic behavior is observed as damage increases. After each loading step, measurement of resonant frequencies and damping ratios as function of vibration amplitude are performed, both using a frequency domain technique and a time domain technique. The nonlinearity is quantified as function of the damage. We compare the results of the linear and nonlinear techniques, and value them against visual damage and local bending stiffness.

On the quasi-analytic treatment of hysteretic nonlinear response in elastic wave propagation

Microscopic features and their hysteretic behavior can be used to predict the macroscopic response of materials in dynamic experiments. Preisach modeling of hysteresis provides a refined procedure to obtain the stress–strain relation under arbitrary conditions, depending on the pressure history of the material. For hysteretic materials, the modulus is discontinuous at each stress–strain reversal which leads to difficulties in obtaining an analytic solution to the wave equation. Numerical implementation of the integral Preisach formulation is complicated as well. Under certain conditions an analytic expression of the modulus can be deduced from the Preisach model and an elementary description of elastic wave propagation in the presence of hysteresis can be obtained. This approach results in a second-order partial differential equation with discontinuous coefficients. Classical nonlinear representations used in acoustics can be found as limiting cases. The differential equation is solved in the frequency do...

Three-dimensional finite element simulation of closed delaminations in composite materials.

Early stage delaminations in composite materials tend to be closed at rest. Inspection with traditional linear ultrasonic techniques generally fails to diagnose and locate such imperfections. However, if undetected and left untreated, incipient defects may gradually grow within the material and eventually lead to failure of the component. Kissing bonds or clapping contacts inherently demand a non-linear diagnostic method, applying a finite excitation amplitude that is able to overcome an activation threshold to open and close the contact. In order to obtain a better understanding and analysis of the macroscopic non-linear behavior that can be observed at the component level, we developed and investigated the results of a finite element model for a composite material containing a single circular delamination. The model makes use of local node splitting and the non-linear constitutive behavior is implemented by means of spring-damper elements at the delamination interface. The results of this parametric study allow a better insight in the behavior of the excited delamination in experimental conditions, including the appearance of localized subharmonics and harmonics of the excitation frequency. Based on the developed model, two different detection and localization techniques (using either a single frequency or a sweep excitation) were demonstrated to determine position, shape, depth and orientation of one or multiple delaminations.

Elastic pulsed wave propagation in media with second‐ or higher‐order nonlinearity. Part I. Theoretical framework

A theoretical model is presented that describes the interaction of frequency components in arbitrary pulsed elastic waves during one‐dimensional propagation in an infinite medium with extreme nonlinear response. The model is based on one‐dimensional Green’s function theory in combination with a perturbation method, as has been developed for a general source function by McCall [J. Geophys. Res. 99 (B2), 2591–2600 (Feb. 1994)]. A polynomial expansion of the equation of state is used in which four orders of nonlinearity in the moduli are accounted for. The nonlinear wave equation is solved for the displacement field at distance x from a symmetric ‘‘breathing’’ source with arbitrary Fourier spectrum imbedded in an infinite medium. The perturbation expression corresponds to a higher‐order equivalent of the Burgers’ equation solution for velocity fields in solids. The solution is implemented numerically in an iterative procedure which allows one to include an arbitrary attenuation function. Energy conservation ...

Laboratory study of linear and nonlinear elastic pulse propagation in sandstone

Linear and nonlinear elastic wave pulse propagation experiments were performed in sandstone rods, both at ambient conditions and in vacuum. The purpose of these experiments was to obtain a quantitative measure of the extremely large nonlinear response found in microcracked (i.e., micro‐inhomogeneous) media like rock. Two rods were used, (1) a 2‐m‐long, 5‐cm‐diam rod of Berea sandstone (with embedded detectors) used in previously published experiments and (2) a somewhat smaller 1.8‐m‐long, 3.8‐cm‐diam rod. In the earlier experiments, wave scattering from the embedded detectors was a critical problem. In most of the experiments reported here, this problem was avoided by mounting accelerometers directly to the outside surface of the rod. Linear results show out of vacuum attenuations varied from 1.7 Np/m at 15 kHz (Q=10) for the large rod to 0.4 Np/m at 15 kHz (Q=55) for the small rod; attenuations for the small rod in vacuum were much less, typically about 0.15 Np/m at 15 kHz (Q=150). Wave velocities ranged...

Multi-mode nonlinear resonance ultrasound spectroscopy for defect imaging: An analytical approach for the one-dimensional case

A nonlinear version of the resonance ultrasound spectroscopy (RUS) theory is presented as an extension of the RUS formalism to the treatment of microdamage characterized by nonlinear constitutive equations. General analytical equations are derived for the one-dimensional case, describing the excitation amplitude dependent shift in the resonance frequency and the generation of harmonics resulting from the interaction between bar modes due to the presence of either localized or volumetrically distributed nonlinearity. Solutions are obtained for classical cubic nonlinearity, as well as for the more interesting case of hysteresis nonlinearity. The analytical results are in excellent quantitative agreement with numerical calculations from a multiscale model. Finally, the analytical formulas are exploited to infer critical information about damage position, degree of nonlinearity, and width of the damage zone either from the shifts in resonance frequency occurring at different excitation modes, or from the shif...

A nodal discontinuous Galerkin finite element method for nonlinear elastic wave propagation

A nodal discontinuous Galerkin finite element method (DG-FEM) to solve the linear and nonlinear elastic wave equation in heterogeneous media with arbitrary high order accuracy in space on unstructured triangular or quadrilateral meshes is presented. This DG-FEM method combines the geometrical flexibility of the finite element method, and the high parallelization potentiality and strongly nonlinear wave phenomena simulation capability of the finite volume method, required for nonlinear elastodynamics simulations. In order to facilitate the implementation based on a numerical scheme developed for electromagnetic applications, the equations of nonlinear elastodynamics have been written in a conservative form. The adopted formalism allows the introduction of different kinds of elastic nonlinearities, such as the classical quadratic and cubic nonlinearities, or the quadratic hysteretic nonlinearities. Absorbing layers perfectly matched to the calculation domain of the nearly perfectly matched layers type have been introduced to simulate, when needed, semi-infinite or infinite media. The developed DG-FEM scheme has been verified by means of a comparison with analytical solutions and numerical results already published in the literature for simple geometrical configurations: Lambs problem and plane wave nonlinear propagation.

Three component time reversal: Focusing vector components using a scalar source

In acoustics, it is known that, for a given response signal at an arbitrary location, reciprocity and time reversal (TR) can be used to focus high levels of acoustic energy at that position. In solid media, elastic waves generally induce different disturbances in three directions. In this paper, both experimental and numerical wave propagation results for solid materials demonstrate the ability to use a scalar source, a three component detector and the reciprocal TR process to selectively focus each of the different vector components, either individually or collectively. The principle is explained from an analytical point of view. The numerical and experimental study demonstrates excellent temporal and spatial focalization. Applications of the selective vector component focusing can be found in damage imaging techniques using both linear or nonlinear ultrasonic waves.

Elastic pulsed wave propagation in media with second‐ or higher‐order nonlinearity. Part II. Simulation of experimental measurements on Berea sandstone

The theoretical 1‐D wave propagation model described in Part I is applied to laboratory data from dynamic propagating wave experiments on a 2‐m‐long cylindrical rod of Berea sandstone as previously reported by Meegan et al. [J. Acoust. Soc. Am. 94, 3387–3391 (1993)]. Using the iterative procedure, good agreement is obtained limiting model parameters up to cubic anharmonicity (i.e., two nonlinear terms proportional to β and δ in the stress‐strain polynomial expansion). Both the data and simulations illustrate that nonlinear response is likely to occur even at extremely small strains (order 10−7). As generally expected for disordered materials, the resulting values for the nonlinear parameters are several orders of magnitude larger than those for intact (uncracked, noncompliant) materials. The values obtained for the dynamic nonlinearity parameters are discussed in relation to commonly obtained static and resonance results which suggest the need to include more complicated phenomena such as hysteresis in th...