A gel of thickness h is squeezed by two porous walls wit
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A gel of thickness h is squeezed by two porous walls with normal stress w (see Fig. 8.12). Assume that the gel is thin a
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A gel of thickness h is squeezed by two porous walls with normal stress w (see Fig. 8.12). Assume that the gel is thin and their surfaces are mechanically fixed to the wall. In this situation, the displacement of the gel network can take place only in the direction normal to the wall. Let u(x, t) be the displacement of the gel network, where x (0 < x < h) is the coordinate taken normal to the wall. Answer the following questions. (a) Show that the elastic energy of deformation of the gel is written as A = 0h dxK2e ?u ?x2 ? [u(h) ? u(0)]w (8.145) where Ke = K + (4/3)G. The last term represents the energy due to the force by the wall. (b) Assume that the volume average velocity is zero: ?u? + (1 ? ?)vs = 0. Show that the energy dissipation function is written as ? = 1 2 0h dx2u??2 (8.146) (c) Using the Onsager principle, show that u(x, t) satisfies the following diffusion equation ?u ?t = ?Ke ??x 2u2 (8.147) with the boundary conditions Ke ?u ?x = ?w at x = 0, h (8.148) (d) Solve the above equation under the initial condition u(x, 0) = 0, and obtain the time variation of the gap distance ?h(t) = u(h, t) ? u(0, t).
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