# American Institute of Mathematical Sciences

October  2020, 13(10): 2877-2904. doi: 10.3934/dcdss.2020123

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

 Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Islamabad, Pakistan

Received  February 2019 Revised  July 2019 Published  October 2020 Early access  October 2019

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.

Citation: Moniba Shams. Wave-propagation in an incompressible hollow elastic cylinder with residual stress. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2877-2904. doi: 10.3934/dcdss.2020123
##### References:

show all references

##### References:
Plot of $\zeta_1$(continuous graph) from Eq. (58) and $\zeta_3$ (dashed graph) from Eq. (59) for $B/A = 1.2$
$\frac{1}{\mu}\frac{dP}{d\lambda_a}$ 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) $\beta_1 = 8, \beta_2 = 1, \lambda_{z} = 1.3, B/A = 1.5$
$P^{*}$ versus $\lambda_a$ for different wall thickness $B/A$ and zero residual stress with ($\lambda_{z} = 1.3$)
$P^{*}$ versus $\lambda_a$ for different $B/A$ with $\beta_1 = 2 = \beta_2$ and $\lambda_{z} = 1.3$
$P^{*}$ versus $\lambda_a$ for different $B/A$ with $B/A$, $\beta_1 = -2 = \beta_2$ and $\lambda_{z} = 2$
$P^{*}$ 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) $\beta_1 = 0.5, \beta_2 = 2$
$N/A^{'}$ 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) $B/A = 2, \beta_1 = 0.3, \beta_2 = 0.8$
$N/A^{'}$ 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) $\beta_1 = -0.2, \beta_2 = 0.8$
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 ${k}$
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) $\hat\beta = 1.5$
First modes from Eqs. (152)–(156) for $\beta_1 = 7, \beta_2 = 2$ and (a) $\hat\beta = 1.5$, (b) $\hat\beta = 2$, (c) $\hat\beta = 2.5$
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) $\beta_1 = -5, \beta_2 = -1$
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)
 [1] Gunther Uhlmann, Jenn-Nan Wang. Unique continuation property for the elasticity with general residual stress. Inverse Problems & Imaging, 2009, 3 (2) : 309-317. doi: 10.3934/ipi.2009.3.309 [2] Rebecca Vandiver. Effect of residual stress on peak cap stress in arteries. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1199-1214. doi: 10.3934/mbe.2014.11.1199 [3] Victor Isakov, Nanhee Kim. Weak Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 799-825. doi: 10.3934/dcds.2010.27.799 [4] Nanhee Kim. Uniqueness and Hölder type stability of continuation for the linear thermoelasticity system with residual stress. Evolution Equations & Control Theory, 2013, 2 (4) : 679-693. doi: 10.3934/eect.2013.2.679 [5] Hong Zhou, M. Gregory Forest. Anchoring distortions coupled with plane Couette & Poiseuille flows of nematic polymers in viscous solvents: Morphology in molecular orientation, stress & flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 407-425. doi: 10.3934/dcdsb.2006.6.407 [6] Donatella Donatelli, Corrado Lattanzio. On the diffusive stress relaxation for multidimensional viscoelasticity. Communications on Pure & Applied Analysis, 2009, 8 (2) : 645-654. doi: 10.3934/cpaa.2009.8.645 [7] Tuan Hiep Pham, Jérôme Laverne, Jean-Jacques Marigo. Stress gradient effects on the nucleation and propagation of cohesive cracks. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 557-584. doi: 10.3934/dcdss.2016012 [8] Shuyang Dai, Fengru Wang, Jerry Zhijian Yang, Cheng Yuan. A comparative study of atomistic-based stress evaluation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4999-5021. doi: 10.3934/dcdsb.2020322 [9] V. Torri. Numerical and dynamical analysis of undulation instability under shear stress. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 423-460. doi: 10.3934/dcdsb.2005.5.423 [10] Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153 [11] Christian Meyer, Stephan Walther. Optimal control of perfect plasticity part I: Stress tracking. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021022 [12] Alexandre Caboussat, Roland Glowinski. Numerical solution of a variational problem arising in stress analysis: The vector case. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1447-1472. doi: 10.3934/dcds.2010.27.1447 [13] Hong-Ming Yin. A free boundary problem arising from a stress-driven diffusion in polymers. Discrete & Continuous Dynamical Systems, 1996, 2 (2) : 191-202. doi: 10.3934/dcds.1996.2.191 [14] Muhammad Mansha Ghalib, Azhar Ali Zafar, Zakia Hammouch, Muhammad Bilal Riaz, Khurram Shabbir. Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 683-693. doi: 10.3934/dcdss.2020037 [15] L. Chupin. Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn-Hilliard formulation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 45-68. doi: 10.3934/dcdsb.2003.3.45 [16] Roger E. Khayat, Martin Ostoja-Starzewski. On the objective rate of heat and stress fluxes. Connection with micro/nano-scale heat convection. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 991-998. doi: 10.3934/dcdsb.2011.15.991 [17] Jan Burczak, Josef Málek, Piotr Minakowski. Stress-diffusive regularizations of non-dissipative rate-type materials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1233-1256. doi: 10.3934/dcdss.2017067 [18] Paschalis Karageorgis. Small-data scattering for nonlinear waves with potential and initial data of critical decay. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 87-106. doi: 10.3934/dcds.2006.16.87 [19] Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843 [20] Fei Guo, Bao-Feng Feng, Hongjun Gao, Yue Liu. On the initial-value problem to the Degasperis-Procesi equation with linear dispersion. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1269-1290. doi: 10.3934/dcds.2010.26.1269

2020 Impact Factor: 2.425