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

698 ROBERT SHORTER, JOHN D. SMITH, VINCENT A. COVENEY AND JAMES J. C. BUSFIELD
ε b
3
2.8
2.6
2.4
2.2
2
h/R
0.1
0.05
0.02
0.01
0 0.1 0.2 0.3 0.4 0.5
ν
ε b
3
2.8
2.6
2.4
2.2
2
ν
0.49
0.40
0.30
0.20
0.10
0.00
0 0.02 0.04 0.06 0.08 0.1
h / R
Figure 5. Left: Snap-through buckling point εb versus Poisson’s ratio ν for a range of
thickness-to-radius ratios (h/R) (left), and versus h/R for a range of values of ν (right).
The two parts of Figure 5 plot the dependence of the buckling instability displacement on the Poisson’s
ratio of the material and also for a wide range of ratios of wall thickness to sphere radius. Pauchard and
Rica [1998] approach as given by (A-1) suggests that each of the various different geometries should
superimpose to a single master curve when the buckling point is plotted against Poisson’s ratio. Clearly
this is not the case as there is a small but clear geometric dependence as well. Pauchard and Rica
suggested a value of about 2 for εb and this is comparable with the values that we have derived for the
thinnest spheres. It is not clear from their paper but it appears likely that their approach is only valid
for very thin shells. A principal aim of this work is to produce curves to allow the wall thickness to be
measured from a simple measure of the buckling displacement provided that the Poisson’s ratio for the
material is known.
The second aim of this paper is to use the buckling force to deduce the modulus for the sphere.
However, there is a problem, as even though the actual buckling displacement was relatively insensitive
to mesh shape producing errors of less than 3%, the maximum force, Fb (shown in Figure 2) achieved
at this displacement was sensitive to discretisation details and produced much more significant errors of
about 15%. An approach was adopted here to reduce this artefact of the modelling; this used the force,
f , at the point half way to buckling, εb/2. As this point is far from the buckling instability, it is much
less sensitive to small changes in the mesh geometry or the precise detail of the discretisation of the time
step.
The wide range of geometric variations shown in Figure 5 are again plotted in Figure 6, but this time
to show how the normalised force at half of the buckling displacement varies with Poisson’s ratio and
for different values of normalised wall thickness. Again, there is a clear geometric dependence that
is not predicted in [Pauchard and Rica 1998]. Knowledge of this dependence, as shown in the graph,