Consider the uniaxial elongation shown in Fig. 3.3. The
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Consider the uniaxial elongation shown in Fig. 3.3. The material point at (x0, y0, z0) is displaced to x = 1? ? x0, y =
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Consider the uniaxial elongation shown in Fig. 3.3. The material point at (x0, y0, z0) is displaced to x = 1? ? x0, y = 1? ? y0. z = ?z0 (9.144) The Henkey strain (t) is defined by (t) = ln ?(t) (9.145) Answer the following questions. (a) Show that the velocity at point (x, y, z) is written as vx = ? 1 2 x, v ? y = ?1 2 y, v ? z = ? z (9.146) (b) Show that for a Newtonian fluid of viscosity ?, the elongational stress ?(t) = ?zz(t)??xx(t) is given by ?(t) = 3? ?(t) (9.147) (c) Suppose that a weight W is hung on the bottom of a cylindrical sample of cross-section A. Show that the elongation of the cylinder is described by ?? = W 3A? ?2 (9.148) Show that ? goes to infinity at a certain time t?. (d) For a viscoelastic fluid, eq. (9.147) is written as ?(t) = 3 ?? t dt G(t ? t)? (t) (9.149) Obtain the time evolution equation for ? in the same situation discussed above.
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