In everyday language, materials are considered as solid if they have different responses when they are deformed from different initial configurations. This rough definition will be refined all along this chapter to include nonholonomic deformations. For clarity’s sake, let us give an overview of the different models of deformable solids with singularity. A choice of bases and affine connections allows us to classify all possible models.
KeywordsSingularity Density Singularity Distribution Free Energy Function Torsion Tensor Affine Connection
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- 1.Ultrasonic techniques for microcracking detection. For most brittle materials the ultrasonic tech-nique has been developed to characterize the internal degradation by measuring the attenuation of ultrasonic waves rather than the modification of its natural frequencies. Two nonlinear ultrasonic techniques are usu-ally proposed to characterize and monitor the fatigue microcracking damage: acoustic-elastic effects (stress dependence of the attenuation) and higher harmonic generation. The first technique measured the wave attenuation on the basis of Taylor expansion of the sound velocity with respect to the pre-strain level whereas the second technique was used to capture the material degradation by assuming a nonlinear stress-strain law. These methods seemed not suitable for the case of microcracked brittle material, for which the stress-strain law remained linear although with a lower elastic modulus than intact material. For instance, these two methods were not obviously suitable to detect the presence of uniformly distributed microcracks. In the same way, the experimental analysis of elastic wave propagation in microflawed ceramics (pores 1µm) showed strong attenuation and cutoff of frequency, which could not be explained in the light of existing macroscopic models of wave propagation. The cutoff frequency phenomenon exhibited distinct frequency bands with energy transmission (pass bands) and with near-zero energy transmission (stop band with cutoff frequency). Due to the shortness of these microcracks’ characteristic length (1µm to 10µm) compared to the usual wavelength used in ultrasonic techniques, the continuum wave theory has often no sufficient sensitivity to apprehend material degradation at the mesoscopic level. Theoretical models should be developed not only for improving the measurement processes as for ultrasonic inspection techniques but also and mainly for better interpretation of the measured data.Google Scholar
- 2.This frequency equation may be used as a starting point for experimental measurements of the singularity distribution when the density is sufficiently high.Google Scholar
- 3.The dimension of the constants of structure may be defined to be [m-1] (inverse to a length dimension).Google Scholar