Biomedical Engineering Reference
In-Depth Information
40
30
20
10
node 2
node 3
node 4
node 5
0
−10
−20
−1.5
−1.4
−1.3
−1.2
−1.1
−1
−0.9
−0.8
−0.7
y (m)
x 10
−4
Fig. 18
The same curves of Fig.
17
, highlighting the central nodes 4 and 5
are inevitably produced by the Newton-Raphson method in these soft stiffness
configurations. A big displacement of one degree of freedom may unduly drag
other degrees of freedom to either inside or outside the adhesion zones. Abrupt
variations of derivative magnitudes usually lead to either numerical instability or
wrong results.
Controlling step evolution allows to find intermediate equilibrium positions to
snap-through and snap-backs. These results are almost impossible to be observed
experimentally. To reconcile the numerical results with ones experimentally ob-
served, Fig.
19
shows only some stable equilibrium configurations found for desired
transmural pressure levels (points in evidence) and disregards the equilibrium path
that might show unstable configurations. If the whole arc-length trajectories were
regarded, path morphology would be like in Fig.
20
. It would not be a closed curve
and the energy barriers of the adhesive contact would be visible.
The open curve in Fig.
20
indicates that the adhesion work with continuous
surface tension is conservative, which is consistent with the hypothesis of h6.
Hysteresis is a characteristic of non-conservative systems and it just appears in
the results when snap-through is considered. The snap-through occurs when the
equilibrium of the whole system is not quasi-statically controlled. Since it is
impracticable to completely control a continuum medium, a theoretical conservative
system end up becoming non-conservative. Both the attached and detached condi-
tions are conditions of minimum potential energy, with an energy barrier between
them. In Fig.
20
it is possible to see the nodes 4 and 5 getting up and down not
through an energy barrier but through a transmural pressure barrier, with peaks near
y
D
10
4
m.
1:43
