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 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:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1966.  Google Scholar

[2]

J. D. Achenbach,, Wave Propagation in Elastic Solids, North-Holland Series in Applied Mathematics and Mechanics, 16. North-Holland Publishing Co., Amsterdam, 1976.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[6]

M. A. Biot, The influence of initial stress on elastic waves, J. App. Phy., 11 (1940), 522-530.  doi: 10.1063/1.1712807.  Google Scholar

[7]

C. J. Chuong and Y. C. Fung, On residual stress in arteries, J. Biomech. Eng., 108 (1986), 189-192.   Google Scholar

[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. Google Scholar

[9] M. E. Gurtin, An Introduction to Continuum Mechanics, Mathematics in Science and Engineering, 158. Academic Press, Inc., New York-London, 1981.   Google Scholar
[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.  Google Scholar

[11]

A. Hoger, On the determination of residual stress in an elastic body, J. Elasticity, 16 (1986), 303-324.  doi: 10.1007/BF00040818.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

A. Hoger, The elasticity tensor of a residually stressed material, J. Elasticity, 31 (1991), 219-237.  doi: 10.1007/BF00044971.  Google Scholar

[17]

C. S. Man and W. Y. Lu, Towards an acoustoelastic theory of measurement of residual stress, J. Elasticity, 17 (1987), 159-182.   Google Scholar

[18]

H. D. McNivenA. H. Shah and J. L. Sackman, Axially symmeteric waves in hollow, elastic rods: Part 1, J. Acous. Soc. Am., 40 (1966), 784-792.   Google Scholar

[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. Google Scholar

[20] R. W. Ogden, NonLinear Elastic Deformations, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, Halsted Press, New York, 1984.   Google Scholar
[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.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[26]

E. RodriguezA. Hoger and A. D. McCulloch, Stress-dependent finite growth in soft elastic tissues, J. Biomech., 27 (1994), 455-467.   Google Scholar

[27]

M. Shams, Wave Propagation in Residually-Stressed Materials, PhD thesis, University of Glasgow, Glasgow, UK, 2010. Google Scholar

[28]

M. ShamsM. 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.  Google Scholar

[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.  Google Scholar

[30]

A. J. M. Spencer, Theory of invariants, Continuum Physics, Academic Press, New York, 1 (1971), 239-353.   Google Scholar

[31]

K. Takamizawa and K. Hayashi, Strain energy density function and uniform starin hypothesis for arterial mechanics, J. Biomech., 20 (1987), 7-17.   Google Scholar

[32]

S. Tang, Wave propagation in initially-stressed elastic solids, Acta Mech., 61 (1967), 92-106.   Google Scholar

[33]

X. M. ZhangZ. 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.   Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1966.  Google Scholar

[2]

J. D. Achenbach,, Wave Propagation in Elastic Solids, North-Holland Series in Applied Mathematics and Mechanics, 16. North-Holland Publishing Co., Amsterdam, 1976.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[6]

M. A. Biot, The influence of initial stress on elastic waves, J. App. Phy., 11 (1940), 522-530.  doi: 10.1063/1.1712807.  Google Scholar

[7]

C. J. Chuong and Y. C. Fung, On residual stress in arteries, J. Biomech. Eng., 108 (1986), 189-192.   Google Scholar

[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. Google Scholar

[9] M. E. Gurtin, An Introduction to Continuum Mechanics, Mathematics in Science and Engineering, 158. Academic Press, Inc., New York-London, 1981.   Google Scholar
[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.  Google Scholar

[11]

A. Hoger, On the determination of residual stress in an elastic body, J. Elasticity, 16 (1986), 303-324.  doi: 10.1007/BF00040818.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

A. Hoger, The elasticity tensor of a residually stressed material, J. Elasticity, 31 (1991), 219-237.  doi: 10.1007/BF00044971.  Google Scholar

[17]

C. S. Man and W. Y. Lu, Towards an acoustoelastic theory of measurement of residual stress, J. Elasticity, 17 (1987), 159-182.   Google Scholar

[18]

H. D. McNivenA. H. Shah and J. L. Sackman, Axially symmeteric waves in hollow, elastic rods: Part 1, J. Acous. Soc. Am., 40 (1966), 784-792.   Google Scholar

[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. Google Scholar

[20] R. W. Ogden, NonLinear Elastic Deformations, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, Halsted Press, New York, 1984.   Google Scholar
[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.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[26]

E. RodriguezA. Hoger and A. D. McCulloch, Stress-dependent finite growth in soft elastic tissues, J. Biomech., 27 (1994), 455-467.   Google Scholar

[27]

M. Shams, Wave Propagation in Residually-Stressed Materials, PhD thesis, University of Glasgow, Glasgow, UK, 2010. Google Scholar

[28]

M. ShamsM. 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.  Google Scholar

[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.  Google Scholar

[30]

A. J. M. Spencer, Theory of invariants, Continuum Physics, Academic Press, New York, 1 (1971), 239-353.   Google Scholar

[31]

K. Takamizawa and K. Hayashi, Strain energy density function and uniform starin hypothesis for arterial mechanics, J. Biomech., 20 (1987), 7-17.   Google Scholar

[32]

S. Tang, Wave propagation in initially-stressed elastic solids, Acta Mech., 61 (1967), 92-106.   Google Scholar

[33]

X. M. ZhangZ. 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.   Google Scholar

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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