Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

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700 ROBERT SHORTER, JOHN D. SMITH, VINCENT A. COVENEY AND JAMES J. C. BUSFIELD Crosshead Load cell Instron frame Steel plate Flat fixture Table tennis ball Figure 7. A schematic of the experimental setup used to measure the compression behaviour of the table tennis balls. is derived the elastic behaviour. These properties were measured both in tension and then in three point bending. Both techniques delivered similar values for the tensile modulus of 2.2 MPa which compares very well with the value deduced from the compression experiment. Stresses taken from the model at buckling show that the maximum stresses were well below the yield stress measured at 60 MPa on a dumbbell shaped tensile test piece also cut from the table tennis ball shell. Figure 8 plots the force versus displacement (on the appropriate dimensionless axes) from an axisymmetric model of the initial buckling of the table tennis ball with the modulus data in the model being RF/Eh 3 14 12 10 8 6 4 2 0 Test Model 0 1 2 3 4 5 Figure 8. Finite element axisymmetric model compared to experimental compression of a table tennis ball of radius 20 mm and wall thickness 0.4 mm, FEA prediction uses a modulus of 2.2 MPa. ε
AXIAL COMPRESSION OF HOLLOW ELASTIC SPHERES 701 taken from the DMTA tests and the geometry from the measured results. This is compared with an experimentally measured data set for the compression of a table tennis ball. It is clear that the general behaviour is well predicted by the FEA approach, with the initial buckling displacement, εb being predicted accurately. The small oscillations observed in the model response are due to small discrete changes in the number of elements that are in contact with the rigid surfaces throughout the model. A separate model, not shown, with a realistic amount of plasticity introduced in the model, had no influence on this initial buckling phenomenon. Next a series of different Expancel spheres were compressed using a UMIS 2000 nano-indentation machine. The machine was calibrated using the incorporated video microscope display and the X, Y coordinates set at a reference point. A small number of microspheres were placed on a microscope slide and were dispersed using distilled water, which was allowed to evaporate. The microscope slide was placed onto a specimen plinth using dental wax, which was melted at approximately 40 ◦ C and allowed to cool and set. The plinth was then placed in the nano-indention machine and initially single microspheres were located using the video microscope display. Clearly it was important to compress just a single sphere. The indenter tip was 1 mm in diameter and made from a ruby, it was initially placed above the microsphere and allowed to stabilise for thirty minutes. The indentation was achieved by moving the indenter tip in the −Z direction by approximately 50% of the microsphere diameter. The process was repeated for a number of microspheres of differing measured diameters. Figure 9 shows the typical behaviour of the initial elastic compression of one of the microspheres with a radius of 24 µm. Assuming a Poisson’s ratio of 0.3 as in [Nakamura et al. 2004], from the buckling point it is possible to identify the wall thickness to be 0.38 µm ± 0.02 µm and from the force at the half way to buckling point the modulus was deduced as being 3.4 MPa. This gives a good indication of the values that we can expect for the wall thickness and the modulus. This will be Force / mN 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 Test Model 0 0 0.5 1 1.5 2 2.5 3 Displacement / µm Figure 9. Experimental compression of a single microsphere (R = 20.24 µm). The predicted curve has wall thickness 0.4 µm, modulus 3.4 MPa and Poisson’s ratio 0.3.
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