Wave-propagation in an incompressible hollow elastic cylinder with residual stress

Figure 1.  Plot of $ \zeta_1 $(continuous graph) from Eq. (58) and $ \zeta_3 $ (dashed graph) from Eq. (59) for $ B/A = 1.2 $

Figure 2.  $ \frac{1}{\mu}\frac{dP}{d\lambda_a} $ versus $ \lambda_a $, (a) $ \beta_1 = 0 = \beta_2, \lambda_{z} = 1.3 = B/A $, (b) $ \beta_1 = 0 = \beta_2, \lambda_{z} = 1.3, B/A = 1.5 $, (c) $ \beta_1 = 8, \beta_2 = 1, \lambda_{z} = 1.3 = B/A $, (d) $ \beta_1 = 8, \beta_2 = 1, \lambda_{z} = 1.3, B/A = 1.5 $

Figure 3.  $ P^{*} $ versus $ \lambda_a $ for different wall thickness $ B/A $ and zero residual stress with ($ \lambda_{z} = 1.3 $)

Figure 4.  $ P^{*} $ versus $ \lambda_a $ for different $ B/A $ with $ \beta_1 = 2 = \beta_2 $ and $ \lambda_{z} = 1.3 $

Figure 5.  $ P^{*} $ versus $ \lambda_a $ for different $ B/A $ with $ B/A $, $ \beta_1 = -2 = \beta_2 $ and $ \lambda_{z} = 2 $

Figure 6.  $ P^{*} $ versus $ \lambda_a $ with $ B/A = 1.2, \lambda_{z} = 1.2 $ and (a) $ \beta_1 = 0 = \beta_2 $, (b) $ \beta_1 = 0.2, \beta_2 = 0.3 $, (c) $ \beta_1 = 0.7, \beta_2 = 0.3 $, (d) $ \beta_1 = 0.5, \beta_2 = 0.5 $, (e) $ \beta_1 = 0.5, \beta_2 = -0.5 $, (f) $ \beta_1 = 2, \beta_2 = 0.5 $, (g) $ \beta_1 = 0.5, \beta_2 = 2 $

Figure 7.  $ N/A^{'} $ versus $ \lambda_a $ for $ \lambda_{z} = 1.2 $ and (a) $ B/A = 1.2, \beta_1 = 0 = \beta_2 $, (b) $ B/A = 1.5, \beta_1 = 0 = \beta_2 $, (c) $ B/A = 2, \beta_1 = 0 = \beta_2 $, (d) $ B/A = 1.2, \beta_1 = -0.5, \beta_2 = 0.8 $, (e) $ B/A = 1.4, \beta_1 = -0.5, \beta_2 = 0.8 $, (f) $ B/A = 1.5, \beta_1 = -0.8, \beta_2 = 1.5 $, (g) $ B/A = 2, \beta_1 = 0.3, \beta_2 = 0.8 $

Figure 8.  $ N/A^{'} $ versus $ \lambda_a $ for $ B/A = 1.2, \lambda_{z} = 1.2 $ and (a) $ \beta_1 = 0.2, \beta_2 = 0.8 $, (b) $ \beta_1 = 0.8, \beta_2 = -0.2 $, (c) $ \beta_1 = 0 = \beta_2 $, (d) $ \beta_1 = -0.2, \beta_2 = 0.8 $

Figure 9.  Comparison of first three modes between the linear elasticity case from Eq. (165) (continuous curve) and numerical results for $ \beta_1 = 0 = \beta_2, \beta = \hat\beta = 2.5 $ and zero residual stress, from Eq. (152)–(156) with (a) $ \omega $ with respect $ {k} $, (b) $ c $ with respect to $ {k} $

Figure 10.  First modes from Eqs. (152)–(156) in the absence of residual stress, $ \beta_1 = 0 = \beta_2 $, (a) $ \hat\beta = 3 $, (b) $ \hat\beta = 2.5 $, (c) $ \hat\beta = 2 $, (d) $ \hat\beta = 1.5 $

Figure 11.  First modes from Eqs. (152)–(156) for $ \beta_1 = 7, \beta_2 = 2 $ and (a) $ \hat\beta = 1.5 $, (b) $ \hat\beta = 2 $, (c) $ \hat\beta = 2.5 $

Figure 12.  First modes from Eqs. (152)–(156) for $ \hat\beta = 2.5 $ and (a) $ \beta_1 = 0 = \beta_2 $, (b) $ \beta_1 = 4, \beta_2 = 1 $, (c) $ \beta_1 = 7, \beta_2 = 2 $, (d) $ \beta_1 = -4, \beta_2 = 1 $, (e) $ \beta_1 = -5, \beta_2 = -1 $

Figure 13.  Four initial modes from Eqs. (152)–(156) for $ \hat\beta = 2.5 $, with $ \beta_1 = 2, \beta_2 = 1 $ (Continuous graph) and $ \beta_1 = 0 = \beta_2 $ (dashed graph)