# Resonant Airborne Acoustic Emission for Nondestructive Testing and Defect Imaging in Composites
## 2 LDR & RACE: Background & Simulation  
In the last decade, the LDR approach has created a great deal of interest and progress in research on the applications of new resonant techniques for NDT and defect imaging in composites [10; 11; 12; 13; 14; 15]. The concept of LDR is based on the fact that material damage is accompanied by a local decrease in stiffness for a certain material mass. By lumping the distributed vibration parameters into equivalent mass and stiffness, the damaged area, therefore, can be modelled as an oscillator with natural frequency \(f_{0}=\frac{1}{2\pi}\sqrt{K_{eff}/M_{eff}}\), where \(K_{eff}\) is the effective material rigidity and \(M_{eff}\) is the vibrating mass of the defect.  
This phenomenology can be readily extended to more rigorous analytics for planar defects (e.g., flat bottomed holes (FBH) or delaminations) in composites [16]. Like in the classical resonance interpretation, the LDR can be then perceived as an outcome of a standing wave formation due to constructive interference of the flexural waves inside the defect which results in the fundamental resonance frequency \(f_{0}=(v/2L)F\), where \(v\) is the wave velocity, \(L\) is the size parameter of the defect and \(F\) is the defect shape factor.  
The flexural wave velocity depends on frequency \(f\) and is known to have a closed form expression [17]:  
\[v=\left(Eh^{2}/12\rho(1-\mu^{2})\right)^{1/4}\omega^{1/2} \tag{1}\]  
where \(E\) is the material Young's modulus, \(\rho\) is its density, \(\mu\) is Poisson's ratio, \(h\) is the plate thickness and \(\omega=2\pi f\).  
At the resonance \(f=f_{0}\) in (1), and after substitution of \(v(f_{0})\) into the above expression, one obtains [16]:  
\[f_{0}=\frac{\pi}{4\sqrt{3}}\frac{h}{L^{2}}F^{2}\sqrt{\frac{E}{\rho(1-\mu^{2})}} \tag{2}\]  
Besides the numerical factor, Equation (2) demonstrates the generic structure of the analytical expressions for LDR frequency. The characteristic square-root term in (2) specifies the dependence of the material properties on the plate bending stiffness, \(D=Eh^{3}/12(1-\mu^{2})\). The general sizing term (\(h/L^{2}\)) indicates a linear increase in the resonance frequency with the depth of the defect, which is reciprocal to the defect area. The calculations from (2) show that for cm-size defects in composites, the LDR frequencies are usually in the kHz-frequency range, while the coupling to ambient air must be accompanied by the airborne acoustic radiation.  
To substantiate the effect of RACE and its connection to LDR, the results of FEM simulation of the resonant behaviour of defects (circular and rectangular FBH) by using COMSOL multiphysics software are shown in Figure 1a-c.  
The simulations in Figure 1a,b are based on Eigenfrequency Analysis in the Structural Mechanics Module (triangular elements, fine mesh, frequency step 10 Hz), that essentially calculates the idealised specimen-vibration patterns at various natural frequencies in vacuum (no acoustic coupling to ambient air). The standing wave LDR of the defects are clearly manifested at certain frequencies: the fundamental LDR frequency corresponds to a half-wave resonance with the vibration node at the defect boundary line. This provides an enclosure of the LDR field: the nodal line along the defect contour \"locks\" the resonant excitation within the damage area, and a strong vibration of the defect nonradiating to the rest of the specimen is observed.  
To include the airborne acoustic coupling of the LDR-induced vibrations, the COMSOL calculations comprised an additional acoustic module, resulting in Figure 1c, which shows a clear airborne radiation from the defect area at the LDR frequency. Since the wavelength in air is substantially smaller than that in solid at the same frequency, the airborne wave field remains similar to a plane wave pattern (near-field zone) at moderate distances from the surface. The RACE signal is, therefore, well-suited for detecting and visualization of defects by revealing their external AE with a distant acoustic wave receiver.