Keywords: plate structures, imperfections detection, dynamics, Rayleigh waves.

This paper is part of a set of publications related to the development of mathematical models aimed to simulate the dynamic input and output of experimental nondestructive tests in order to detect structural imperfections. The structures to be considered are composed by steel plates of thin thickness. The imperfections in these cases are cracks and they can penetrate either a significant part of the plate thickness or they can be superficial micro cracks or imperfections. The first class of cracks is related to structural safety and the second one is more connected with the structural protection to the environment, particularly if protective paintings can be deteriorated. Two mathematical groups of models have been developed. The first group tries to locate the position and extension of the imperfection of the first class of imperfections, i.e. cracks. Bending Kirchhoff thin plate models belong to this first group and they are used to this respect. The another group of models is dealt with membrane structures under the superficial Rayleigh waves excitation. With this group of models the micro cracks detection is intended.

The Kirchhoff plate model has been applied to a simple test case, namely to a square thin plate simply supported along its boundary. In the case of a sane plate, i.e. without cracks, the different natural frequencies and modes can be analytically found. For cracked plates a numerical solution based on the Finite Element method was used. Also an error estimation of this numerical technique was obtained by using as comparison the exact results of the sane plate computed by the analytical solution. Therefore, in this way a similar degree of accuracy was achieved through all the computations.

From the above analysis, in the application of the Kirchhoff plate model to the detection of cracks, it has been observed that the differences between the natural frequencies of the non cracked and the cracked structures are very small. Also modes vectors comparison using different norms, like and are not reliable tools to detect structural imperfections, because this comparison may depends on the crack position and the excited mode. However, geometry and crack position can be identified quite accurately if this comparison is carried out between first derivatives (mode rotations) of the natural modes are used instead.

Finally, in relation with the analysis of the superficial cracks existence the use of Rayleigh waves seems to be very promising. A actual test aimed to detect a micro crack, with a depth of the order of one millimeter, in a thin plate has been simulated by the mathematical membrane model. The plate was subjected to a set of travelling Rayleigh waves. The geometry and the penetration of the micro crack could be mathematically detected very accurately as in the actual test. However, the mathematical and numerical treatment of the generation of these Rayleigh waves present serious complexities. Some of them could be handled by dividing the plate response into two terms: the pseudo-dynamic and the dynamic responses. Other difficulties were related with the numerical techniques used. Particularly it should mention difficulties due to the dispersion problems appearing during the analysis by finite differences along the time domain and the ones related to the computation of a large number of finite elements on the spatial coordinates needed in this model.

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