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

• A study is presented to observe the effect of residual stress on waves in an incompressible, hyper-elastic, thick and hollow cylinder of infinite length. The problem is based on the non-linear theory of infinitesimal deformations occurring after a finite deformation. A prototype model of strain energy function is used which adequately includes the effects of residual stress and deformation. The expressions for internal pressure and the axial load are calculated and graphical illustrations are presented. Analysis of infinitesimal wave propagation is carried for the axisymmetric case in the considered cylinder. Numerical solution is obtained in the undeformed configuration and analyzed for the two-point boundary-value problem. Dispersion curves are plotted for varying choice of parameters.

Mathematics Subject Classification: Primary: 74B20, 58F17; Secondary: 74H99.

 Citation: • • 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)

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