Issue 37

M. Vieira et alii, Frattura ed Integrità Strutturale, 37 (2016) 131-137; DOI: 10.3221/IGF-ESIS.37.18 133 2 2 2 2 x u E t u       (1) where u is the displacement, t is time, E is the Young Modulus, x is the associated coordinate system and  is the specific mass of the material. The mathematical solution of Eq. 1 is:              l ct sen l xn cos A )x(u   0 (2) where A 0 is the generic amplitude of vibration, l is the bar length and c is the wave propagation speed. Eq. 2 represents the axial displacement along the generic bar, respective to a certain n th mode, while Eq. 3 represents the natural non-damped frequency,  n for the n th mode.    E l n n  (3) The solution found at Eqs. 2 and 3 are equally valid for torsional modes, replacing the Young Modulus by the Shear Modulus of a certain material. Solving the equation for a cylindrical bar with a length of 250 mm (and noting that for this solution, the diameter of the bar does not affect the final results), the following results are obtained for the first five modal non-damped frequencies for the axial and torsional directions for common construction steel (E=200 GPa,  =7800 kg/m3): Axial frequencies Torsional frequencies n (rad/s) Hz n (rad/s) Hz 0 0.0 0.0 0 0.0 0.0 1 63632.3 10127.4 1 40244.6 6405.1 2 127264.6 20254.8 2 80489.2 12810.3 3 190896.9 30382.2 3 120733.8 19215.4 4 254529.2 40509.6 4 160978.4 25620.5 Table 1 : First five modal frequencies for axial and torsional directions. Obviously, and due to the differences found between the Young and Shear Modulus, the results are lagged by a certain ratio. This implies that, for a cylindrical shape, the first axial mode will have a significative different frequency value from the respective first torsional mode. If a cylindrical shaped specimen is pretended, this raises a relevant problem to the creation design. One could think, from the analysis of Tab. 1, that the second longitudinal mode (n=2) and the third torsional mode (n=3) could be used to design a specimen to produce biaxial testing, since frequencies are very close together. Still, this solution is of little application since, because of the shapes of these two modes, axial and shear stresses on specimen would be higher on different locations. Other combinations of modes were considered, but the final design consisted of a specimen that possesses its first axial mode (n=1) and its third torsional mode (n=3) at the same frequency. This is achieved by designing a cylindrical specimen with three throats, as seen in Fig. 2: Figure 2 : 2D representation of the developed specimen.