Thin-walled tube, inflation

In a thin-walled tube of large radius the inflating pressure P is much smaller than the stresses t and t2 that it generates, and thus P can be neglected in comparison with the stress ts in determining p.) Inserting this result for p in Eqs. (1.21) and (1.22) yields a relation between the extension ratio k2 and the expansion ratio v (=A.jA.2) of the internal volume of the tube ... 

The veins are thin-walled tubular structures that may collapse (i.e., the cross-sectional area does not maintain its circular shape and becomes less than in the unstressed geometry) when subjected to negative transmural pressures P (internal minus external pressures). Experimental studies (Moreno et al., 1970) demonstrated that the structural performance of veins is similar to that of thin-walled elastic tubes (Fig. 3.10). Three regions may be identified in a vein subjected to a transmural pressure When P > 0, the tube is inflated, its cross section increases and maintains a circular shape when P < 0, the tube cross section collapses first to an ellipse shape and at a certain negative transmural pressure, a contact is obtained between opposite walls, thereby generating two lumens. Structural analysis of the stability of thin elastic rings and their postbuckling shape (Flaherty et al., 1972), as well as experimental studies (Thiriet et al., 2001) revealed the different complex modes of collapsed cross sections. In order to facilitate at least a one-dimensional fluid flow analysis, it is useful to represent the mechanical characteristics of the vein wall by a tube law relationship that locally correlates between the transmural pressure and the vein cross-sectional area. 

A long, thin-walled rubber tube (Figure 1.8.S) is inflated with a gas pressure pa with its length Wo held constant. Assuming that the material obeys the neo-Hookean model, determine how much the tube will inflate. Show that this is a planar extension, identical to Example 1.8.2. 

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