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 and Continuous Dynamical Systems - S, 2020, 13 (10) : 2877-2904. doi: 10.3934/dcdss.2020123
References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1966. [2] J. D. Achenbach,, Wave Propagation in Elastic Solids, North-Holland Series in Applied Mathematics and Mechanics, 16. North-Holland Publishing Co., Amsterdam, 1976. [3] S. D. Akbarov and A. N. Guz, Axisymmetric longitudinal wave propagation in pre-stressed compound circular cylinders, Int. J. Eng. Sc., 42 (2004), 769-791. [4] S. D. Akbarov and E. T. Bagirov, Axisymmetric longitudinal wave dispersion in a bi-layered circular cylinder with inhomogeneous initial stresses, J. Sound Vib, 450 (2019), 1-27. [5] M. A. Biot, Non-linear theory of elasticity and the linearized case for a body under initial stress, Phil. Mag., 27 (1939), 468-489. [6] M. A. Biot, The influence of initial stress on elastic waves, J. App. Phy., 11 (1940), 522-530.  doi: 10.1063/1.1712807. [7] C. J. Chuong and Y. C. Fung, On residual stress in arteries, J. Biomech. Eng., 108 (1986), 189-192. [8] A. Guillou and R. W. Ogden, Growth in soft biological tissue and residual stress development, Mechanics of Biological Tissue, Springer, Berlin Heidelberg, (2006), 47–62. [9] M. E. Gurtin, An Introduction to Continuum Mechanics, Mathematics in Science and Engineering, 158. Academic Press, Inc., New York-London, 1981. [10] A. Hoger, On the residual stress possible in an elastic body with material symmetry, Arch. Rat. Mech. Anal., 88 (1985), 271-290.  doi: 10.1007/BF00752113. [11] A. Hoger, On the determination of residual stress in an elastic body, J. Elasticity, 16 (1986), 303-324.  doi: 10.1007/BF00040818. [12] A. Hoger, Residual stress in an elastic body: A theory for small strains and arbitrary rotations, J. Elasticity, 31 (1993), 1-24.  doi: 10.1007/BF00041621. [13] A. Hoger, The constitutive equation for finite deformations of a transversely isotropic hyperelastic material with residual stress, J. Elasticity, 33 (1993), 107-118.  doi: 10.1007/BF00705801. [14] B. E. Johnson and A. Hoger, The dependence of the elasticity tensor on residual stress, J. Elasticity, 33 (1993), 145-165.  doi: 10.1007/BF00705803. [15] B. E. Johnson and A. Hoger, The use of strain energy function to quantify the effect of residual stress on mechanical behaviour, Math. and Mech. of Solids, 4 (1993), 447-470. [16] A. Hoger, The elasticity tensor of a residually stressed material, J. Elasticity, 31 (1991), 219-237.  doi: 10.1007/BF00044971. [17] C. S. Man and W. Y. Lu, Towards an acoustoelastic theory of measurement of residual stress, J. Elasticity, 17 (1987), 159-182. [18] H. D. McNiven, A. H. Shah and J. L. Sackman, Axially symmeteric waves in hollow, elastic rods: Part 1, J. Acous. Soc. Am., 40 (1966), 784-792. [19] R. W. Ogden, Nonlinear elasticity, anisotropy and residual stresses in soft tissue, Biomechanics of Soft Tissue in Cardiovasular Systems, Springer, Wien, (2003), 65–108. [20] R. W. Ogden, NonLinear Elastic Deformations, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, Halsted Press, New York, 1984. [21] R. W. Ogden and C. A. J. Schulze-Bauer, Phenomenological and structural aspects of the mechanical response of arteries, Mech. Bio., 242 (2000), 125-140. [22] R. W. Ogden, Nonlinear Elasticity with Application to Material Modelling, Lecture Notes 6, Centre of Excellence for Advanced Materials and Structures, Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, 2003. [23] R. W. Ogden, Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissue, Biomechanics of Soft Tissue in Cardiovasular Systems, Springer, Wien, (2003), 65–108. [24] A. Ozturk and S. D. Akbarov, Torsional wave propagation in a pre-stressed circular cylinder embedded in a pre-stressed elastic medium, Applied Mathematical Modelling, 33 (2009), 3636-3649.  doi: 10.1016/j.apm.2008.12.003. [25] A. Rachev and K. Hayashi, Theoratical study of the effects of vascular smooth muscle contraction and strain and stress distribution in arteries, Ann. Bio. Eng., 27 (1999), 459-468. [26] E. Rodriguez, A. Hoger and A. D. McCulloch, Stress-dependent finite growth in soft elastic tissues, J. Biomech., 27 (1994), 455-467. [27] M. Shams, Wave Propagation in Residually-Stressed Materials, PhD thesis, University of Glasgow, Glasgow, UK, 2010. [28] M. Shams, M. Destrade and R. W. Ogden, Initial stresses in elastic solids: Constitutive laws and acoustoelasticity, J. Wave Motion, 48 (2011), 552-567.  doi: 10.1016/j.wavemoti.2011.04.004. [29] M. Shams, Reflection of plane waves from the boundary of an initially stressed incompressible half-space, Mathematics and Mechanics of Solids, 24 (2019), 406-433.  doi: 10.1177/1081286517741524. [30] A. J. M. Spencer, Theory of invariants, Continuum Physics, Academic Press, New York, 1 (1971), 239-353. [31] K. Takamizawa and K. Hayashi, Strain energy density function and uniform starin hypothesis for arterial mechanics, J. Biomech., 20 (1987), 7-17. [32] S. Tang, Wave propagation in initially-stressed elastic solids, Acta Mech., 61 (1967), 92-106. [33] X. M. Zhang, Z. H. Niu and J. G. Yu, Effect of initial stress on axisymmetric torsional wave in a unidirectional composite hollow cylinder, Proceedings of the 2015 Symposium on Piezoelectricity, Acoustic Waves and Device Applications, SPAWDA, 2015 (2015), 349-352.

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References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1966. [2] J. D. Achenbach,, Wave Propagation in Elastic Solids, North-Holland Series in Applied Mathematics and Mechanics, 16. North-Holland Publishing Co., Amsterdam, 1976. [3] S. D. Akbarov and A. N. Guz, Axisymmetric longitudinal wave propagation in pre-stressed compound circular cylinders, Int. J. Eng. Sc., 42 (2004), 769-791. [4] S. D. Akbarov and E. T. Bagirov, Axisymmetric longitudinal wave dispersion in a bi-layered circular cylinder with inhomogeneous initial stresses, J. Sound Vib, 450 (2019), 1-27. [5] M. A. Biot, Non-linear theory of elasticity and the linearized case for a body under initial stress, Phil. Mag., 27 (1939), 468-489. [6] M. A. Biot, The influence of initial stress on elastic waves, J. App. Phy., 11 (1940), 522-530.  doi: 10.1063/1.1712807. [7] C. J. Chuong and Y. C. Fung, On residual stress in arteries, J. Biomech. Eng., 108 (1986), 189-192. [8] A. Guillou and R. W. Ogden, Growth in soft biological tissue and residual stress development, Mechanics of Biological Tissue, Springer, Berlin Heidelberg, (2006), 47–62. [9] M. E. Gurtin, An Introduction to Continuum Mechanics, Mathematics in Science and Engineering, 158. Academic Press, Inc., New York-London, 1981. [10] A. Hoger, On the residual stress possible in an elastic body with material symmetry, Arch. Rat. Mech. Anal., 88 (1985), 271-290.  doi: 10.1007/BF00752113. [11] A. Hoger, On the determination of residual stress in an elastic body, J. Elasticity, 16 (1986), 303-324.  doi: 10.1007/BF00040818. [12] A. Hoger, Residual stress in an elastic body: A theory for small strains and arbitrary rotations, J. Elasticity, 31 (1993), 1-24.  doi: 10.1007/BF00041621. [13] A. Hoger, The constitutive equation for finite deformations of a transversely isotropic hyperelastic material with residual stress, J. Elasticity, 33 (1993), 107-118.  doi: 10.1007/BF00705801. [14] B. E. Johnson and A. Hoger, The dependence of the elasticity tensor on residual stress, J. Elasticity, 33 (1993), 145-165.  doi: 10.1007/BF00705803. [15] B. E. Johnson and A. Hoger, The use of strain energy function to quantify the effect of residual stress on mechanical behaviour, Math. and Mech. of Solids, 4 (1993), 447-470. [16] A. Hoger, The elasticity tensor of a residually stressed material, J. Elasticity, 31 (1991), 219-237.  doi: 10.1007/BF00044971. [17] C. S. Man and W. Y. Lu, Towards an acoustoelastic theory of measurement of residual stress, J. Elasticity, 17 (1987), 159-182. [18] H. D. McNiven, A. H. Shah and J. L. Sackman, Axially symmeteric waves in hollow, elastic rods: Part 1, J. Acous. Soc. Am., 40 (1966), 784-792. [19] R. W. Ogden, Nonlinear elasticity, anisotropy and residual stresses in soft tissue, Biomechanics of Soft Tissue in Cardiovasular Systems, Springer, Wien, (2003), 65–108. [20] R. W. Ogden, NonLinear Elastic Deformations, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, Halsted Press, New York, 1984. [21] R. W. Ogden and C. A. J. Schulze-Bauer, Phenomenological and structural aspects of the mechanical response of arteries, Mech. Bio., 242 (2000), 125-140. [22] R. W. Ogden, Nonlinear Elasticity with Application to Material Modelling, Lecture Notes 6, Centre of Excellence for Advanced Materials and Structures, Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, 2003. [23] R. W. Ogden, Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissue, Biomechanics of Soft Tissue in Cardiovasular Systems, Springer, Wien, (2003), 65–108. [24] A. Ozturk and S. D. Akbarov, Torsional wave propagation in a pre-stressed circular cylinder embedded in a pre-stressed elastic medium, Applied Mathematical Modelling, 33 (2009), 3636-3649.  doi: 10.1016/j.apm.2008.12.003. [25] A. Rachev and K. Hayashi, Theoratical study of the effects of vascular smooth muscle contraction and strain and stress distribution in arteries, Ann. Bio. Eng., 27 (1999), 459-468. [26] E. Rodriguez, A. Hoger and A. D. McCulloch, Stress-dependent finite growth in soft elastic tissues, J. Biomech., 27 (1994), 455-467. [27] M. Shams, Wave Propagation in Residually-Stressed Materials, PhD thesis, University of Glasgow, Glasgow, UK, 2010. [28] M. Shams, M. Destrade and R. W. Ogden, Initial stresses in elastic solids: Constitutive laws and acoustoelasticity, J. Wave Motion, 48 (2011), 552-567.  doi: 10.1016/j.wavemoti.2011.04.004. [29] M. Shams, Reflection of plane waves from the boundary of an initially stressed incompressible half-space, Mathematics and Mechanics of Solids, 24 (2019), 406-433.  doi: 10.1177/1081286517741524. [30] A. J. M. Spencer, Theory of invariants, Continuum Physics, Academic Press, New York, 1 (1971), 239-353. [31] K. Takamizawa and K. Hayashi, Strain energy density function and uniform starin hypothesis for arterial mechanics, J. Biomech., 20 (1987), 7-17. [32] S. Tang, Wave propagation in initially-stressed elastic solids, Acta Mech., 61 (1967), 92-106. [33] X. M. Zhang, Z. H. Niu and J. G. Yu, Effect of initial stress on axisymmetric torsional wave in a unidirectional composite hollow cylinder, Proceedings of the 2015 Symposium on Piezoelectricity, Acoustic Waves and Device Applications, SPAWDA, 2015 (2015), 349-352.
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)
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