?When we inflate a tube-like rubber balloon, we often se
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?When we inflate a tube-like rubber balloon, we often see the situation shown in Fig. 3.12: the strongly inflated part c
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?When we inflate a tube-like rubber balloon, we often see the situation shown in Fig. 3.12: the strongly inflated part coexists with the less inflated part. Answer the following questions. (a) The deformation free energy density of the rubber can be written as f(?r, ? ), where ?r and ? are the elongational ratios in the radial and the axial directions. Let h, R, and L be the initial thickness, radius, and axial length of the tube-like balloon. Suppose that part (of fraction x) is weakly inflated, and the remaining part (of fraction (1 ? x)) is strongly inflated. The total free energy of the system is written as Ftot = 2?RLh[xf(?r1, ? 1) + (1?x)f(?r2, ? 2)] ? ?R2L?P[x?2 r1? 1 + (1?x)?2 r2? 2 ? 1] (3.95) where ?P is the extra pressure in the balloon. Show that the equilibrium state is determined by the following set of equations: ?f ??ri = 2p?ri? i for i = 1, 2 (3.96) ?f ?? i = p?2 ri for i = 1, 2 (3.97) and f(?r1, ? 1) ? f(?r2, ? 2) = p(?2 r1? 1 ? ?2 r2? 2) (3.98) where p = R?P 2h (3.99) (b) Discuss whether such coexistence is possible or not for the deformation free energy given by eq. (3.49).
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