were selected basing on the authors’ experience and by numerically exploring several solutions; neither parametric studies nor optimization analyses have been performed. The names “Stop Hole”, “Ellipse”, “Double T” and “Circle” refer to the geometry of the pores as visible in Fig. 42.1, but different considerations were made on the selection of the shapes: The “Ellipse” was chosen since, based on the previous work of Bertoldi, Taylor et al. [6, 8], it was demonstrated that it’s possible to tune the Poisson’ ratio by the variation of the aspect ratio (the ratio between the two major axis) of a regular patterns of ellipsis. However, the modification of the aspect ratio of the ellipsis introduces in the structures very sharp tips leading to high stress concentration factors and allowing the samples to be easily prone to plasticity and failures. The other two proposed solutions, the “Stop Hole” and the “Double T”, were then designed in order to limit this detrimental effect. The “circle” sample was fabricated and included in the study to provide a non-auxetic benchmark during the investigation. The first part of the investigation was to characterize the structure by calculating numerically the effective Poisson’s ratio from the finite element simulations. The results are reported in Table 42.1: 42.3 Experimental Procedures and Methods To accomplish the eigenmode analysis, the specimens need to be clamped and to be excited close to their natural frequencies. This goal was attained by designing a customized loading fixture modifying one of the two clamps of an Hemo fix-point vice (Fig. 42.2) to include a Thor Labs, PZS001 piezo actuator driven by an E-500 High Power (80 W) amplifier and a piezo-controller system manufactured by Physik Instrumente. The additional support was made by aluminium and designed in order to load the samples along the longitudinal dimension of the sample, thus limiting the out-of-plane components of the displacement. Both the designing and the fabrication processes were supported and verified by means of finite element simulations. During the testing operation, two different speckle interferometers were utilized to catch the intensity images of the speckle produced on the sample surfaces; among all the different possibilities, we used one optical set-up sensitive to the in-plane displacement and a second one partly sensitive to the out-of-plane component of the displacement [6, 9, 10]. The optical set-up, shown in the left and in the central part of Fig. 42.3, is relatively simple: the laser beam generated by a NdYg laser (Cobolt Samba, wavelength ¼532 nm, power ¼300 mW) is halved by a cubic beam splitter and then expanded and driven along the two branches by using other standard optical equipment; in particular, one of the two beams is reflected by a mirror moved by a piezo-actuator in order to perform the phase-shifting. This is a fundamental step to quantitative measuring the displacement of the samples since at least three images are required to calculate the phase. Among all the possible available solutions, we favored the 5-image Hariharan phase-shifting algorithm [11] due to its superior performance in error compensation and precision, compared to other classical methods. The speckle field was recorded using a CCD digital mono camera (Allied Vision Pike F421, resolution ¼2048 2048, pixel size 7.4 μm) using time-averaged speckle interferometry. The “time-averaged method” [12, 13], is a particular way to perform the numerical integration of the speckle fields directly using the camera CCD; if the frequency of the object recorded is higher than the shutter, during a single acquisition the sensor of the camera “physically” integrates the speckle patterns (having a varying light intensity during the acquisition period) and straight provides the Bessel fringes connected with the displacement of the object. In the right side of Fig. 42.3, a basic scheme of the optical setup is presented for the interferometer sensitive to the in-plane displacement but the second configuration is also very similar since the two set-ups only differ in the orientation of the diffusing surface (the NPR sample) while no other changes are required in the remaining optical equipment. By rotating the specimen with respect to a vertical axis, the sensitivity vector becomes a simultaneous function of the in-plane and the out-of-plane components of the displacement. In the case the sample is perfectly parallel to the CCD camera (as showed in the three frames of Fig. 42.3), only the planar component of the displacement vector is obtained. The main advantage of using an interferometer partly sensible to the out-of-plane displacement is that, since the transverse motion of the samples is large, a normal interferometer sensitive to the out-of-plane displacement could easily run into de-correlation problems due to their high sensitivity. Table 42.1 Effective Poisson’s ratio of the tested samples Poisson’s ratio Base material (stainless steel) Circular (b) Stop hole (c) Ellipsis (d) Double T (e) þ0.30 þ0.281 0.671 0.605 0.832 42 Numerical and Experimental Eigenmode Analysis of Low Porosity Auxetic Structures 337
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