Nonlinear analytical solution of nearly incompressible hyperelastic cylinder with variable thickness under non-uniform pressure by perturbation technique

Document Type: Research Paper

Authors

Department of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran

Abstract

In this paper, nonlinear analytical solution of pressurized thick cylindrical shells with variable thickness made of hyperelastic materials is presented. The governing equilibrium equations for the cylindrical shell with variable thickness under non-uniform internal pressure are derived based on first-order shear deformation theory (FSDT). The shell is assumed to be made of isotropic and homogenous hyperelastic material in nearly incompressible condition. Two-term Mooney-Rivlin type material is considered which is a suitable hyperelastic model for rubbers. Boundary Layer Method of the perturbation theory which is known as Match Asymptotic Expansion (MAE) is used for solving the governing equations. In order to validate the results of the current analytical solution in analyzing pressurized hyperelastic thick cylinder with variable thickness, a numerical solution based on Finite Element Method (FEM) have been investigated. Afterwards, for a rubber case study, displacements, stresses and hydrostatic pressure distribution resulting from MAE and FEM solution have been presented. Furthermore, the effects of geometry, loading, material properties and incompressibility parameter have been studied. Considering the applicability of the rubber elasticity theory to aortic soft tissues such as elastin, the behaviour of blood vessels under non-uniform pressure distribution has been investigated. The results prove the effectiveness of FSDT and MAE combination to derive and solve the governing equations of nonlinear problems such as nearly incompressible hyperelastic shells.

Keywords

  [1]      M. C. Boyce, E.M. Arruda, Constitutive models of rubber elasticity: a review, Rubber Chemistry and Technology, Vol. 73, No. 3, pp. 504-523, 2000.

  [2]      W. Ma, B. Qu, F. Guan, Effect of the friction coefficient for contact pressure of packer rubber, Journal of Mechanical Engineering Science, Vol. 228, No. 16, pp. 2881-2887, 2014.

  [3]      T. Sussman, K. J. Bathe, A finite element formulation for nonlinear incompressible elastic and inelastic analysis, Computers & Structures, Vol. 26, No. 112, pp. 357-109, 1987.

  [4]      M. Levinson, I. W. Burgess, A comparison of some simple constitutive relations for slightly compressible rubber-like materials,International Journal of Mechanical Sciences, Vol. 13, No. 6, pp. 563-572, 1971.

  [5]      J. C. Simo, R. L. Taylor, Penalty function formulations for incompressible nonlinear elastostatics, Computer Methods in Applied Mechanics and Engineering, Vol. 35, pp. 107-118, 1982.

  [6]      J. C. Simo, R. L. Taylor, Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms, Computer Methods in Applied Mechanics and Engineering, Vol. 85, No. 3, pp. 273-310, 1991.

  [7]      J. S. Chen, C. Pan, A pressure projection method for nearly incompressible rubber hyperelasticity, Part I: Theory, Journal of Applied Mechanics, Vol. 63, No. 4, pp. 862-868, 1996.

  [8]      J. S. Chen, C. T. Eu, C. Pan, A pressure projection method for nearly incompressible rubber hyperelasticity, Part II: Applications, Journal of Applied Mechanics, Vol. 63, No. 4, pp. 869-876, 1996.

  [9]      I. Bijelonja, I. Demirdžic, S. Muzaferija, A finite volume method for large strain analysis of incompressible hyperelastic materials, International Journal for Numerical methods in Engineering, Vol. 64, pp. 1594-1609, 2005.

[10]      C. A. C. Silva, M. L. Bittencourtb, Structural shape optimization of 3D nearly-incompressible hyperelasticity problems, Latin American Journal of Solids and Structures, Vol. 5, pp. 129-156, 2008.

[11]      S. Doll, K. Schweizerhof, On the development of volumetric strain energy functions, Journal of Applied Mechanics, Vol. 97, pp.17–21, 2000.

[12]      H. Ghaemi, K. Behdinan, A. Spence, On the development of compressible pseudo-strain energy density function for elastomers Part 1. Theory and experiment, Journal of Materials Processing Technology, Vol. 178, pp. 307-316, 2006.

[13]      G. Montella, A. Calabrese, G. Serino, Mechanical characterization of a Tire Derived Material: experiments, hyperelastic modeling and numerical validation, Construction and Building Materials, Vol. 66, pp. 336-347, 2014.

[14]      V. Dias, C. Odenbreit, O. Hechler, F. Scholzen, T. B. Zineb, Development of a constitutive hyperelastic material law for numerical simulations of adhesive steel–glass connections using structural silicone, International Journal of Adhesion and Adhesives, Vol. 48, pp. 194–209, 2014.

[15]      Y. Zhu, X. Y. Luo, R. W. Ogden, Nonlinear axisymmetric deformations of an elastic tube under external pressure, European Journal of Mechanics- A/Solids, Vol. 29, No. 2, pp. 216-229, 2010.

[16]      M. Tanveer, J. W. Zu, Non-linear vibration of hyperelastic axisymmetric solids by a mixed p-type method, International Journal of Non-Linear Mechanics, Vol. 47, pp. 30-41, 2012.

[17]      J. Kiendl, M. C. Hsu, M. C. H. Wu, A. Reali, Isogeometric Kirchhoff–Love shell formulations for general hyperelastic materials. Computer Methods in Applied Mechanics and Engineering., Vol. 291, pp. 280-303, 2015.

[18]      H. R. Eipakchi, Third-order shear deformation theory for stress analysis of a thick conical shell under pressure, Journal of Mechanics of materials and structures, Vol. 5, No. 1, 1-17, 2010.

[19]      M. Ghannad, G. H. Rahimi, M. Z. Nejad, Elastic analysis of pressurized thick cylindrical shells with variable thickness made of functionally graded materials, Composites: Part B, Vol. 45, pp. 388-396, 2013.

[20]      M. Jabbari, M. Z. Nejad, M. Ghannad, Thermo-elastic analysis of axially functionally graded rotating thick truncated conical shells with varying thickness, Composites: Part B, Vol. 96, pp. 20-34, 2016.

[21]      H. Gharooni, M. Ghannad, M. Z. Nejad, Thermo-elastic analysis of clamped-clamped thick FGM cylinders by using third-order shear deformation theory, Latin American Journal of Solids and Structures, Vol. 13, No. 4, pp. 750-774, 2016.

[22]      J. Vossoughi, A. Tozeren, Determination of an effective shear modulus of aorta, Russian Journal of Biomechanics, Vol. 1-2, pp. 20-36, 1998.

[23]      T. E. Carew, R. N. Vaishnav, D. J. Patel, Compressibility of the arterial wall, Circulation Research, Vol. 23, No. 1, pp. 61–68, 1968.

[24]      K. L. Dorrington, N. G. McCrum, Elastin as a rubber, Biopolymers, Vol. 16, No. 6, pp. 1201-1222, 1977.

[25]      L. A. Mihai, A. Goriely, How to characterize a nonlinear elastic material? A review on nonlinear constitutive parameters in isotropic finite elasticity, Journal of Royal Society A, Vol.473, No. 2207, 20170607, 2017.

[26]      J. D. Humphrey, S. L. O’Rourke, 2015, An Introduction to Biomechanics Solids and Fluids, Analysis and Design, 2nd ed., Springer, New York.

[27]      D. Azar, D. Ohadi, A. Rachev, J. F. Eberth, M. J. Uline, T. Shazly, Mechanical and geometrical determinants of wall stress in abdominal aortic aneurysms: A computational study, PLoS ONE, Vol. 13, No. 2, e0192032, 2018.

[28]      J. N. Reddy, 2002, Energy principles and variational methods in applied mechanics, Wiley, New York.

[29]      J. T. Oden, A theory of penalty methods for finite element approximations of highly nonlinear problems in continuum mechanics, Computers and Structures, Vol. 8, pp. 445-449, 1978.

[30]      G. A. Holzapfel, 2000, Nonlinear Solid Mechanics, a Continuum Approach for Engineering, Wiley, New York.

[31]      Y. Başar, D. Weichert, 2000, Nonlinear Continuum Mechanics of Solids, Springer, Berlin.

[32]      I. Doghri, 2000, Mechanics of Deformable Solids: Linear, Nonlinear, Analytical and Computational Aspects, Springer, Berlin.

[33]      J. N. Reddy, 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd ed., CRC Press, New York.

[34]      A. H. Nayfeh, 1981, Introduction to Perturbation Techniques, Wiley, New York.

[35]      Y. Payan, J. Ohayon, (Eds.) 2017, Biomechanics of Living Organs: Hyperelastic Constitutive Laws for Finite Element Modeling, World Bank Publications, London.

[36]      G. A. Holzapfel, R. W. Ogden, (Eds.) 2003, Biomechanics of soft tissue in cardiovascular systems, Springer-Verlag, Austria.

[37]      J. H. Kim, S. Avril, A. Duprey, J. P. Favre, Experimental characterization of rupture in human aortic aneurysms using full-field measurement technique, Biomechanics and Modeling in Mechanobiology, Vol. 11, No. 6, pp. 841-854, 2012.

[38]      G. A. Holzapfel, T. C. Gasser, Computational stress–deformation analysis of arterial walls including high-pressure response International Journal of Cardiology, Vol. 116, pp. 78-85, 2007.

[39]      R. Mihara, A. Takasu, K. Maemura, T. Minami, Prolonged severe hemorrhagic shock at a mean arterial pressure of 40 mmHg does not lead to brain damage in rats, Acute Medicine & Surgery, Vol. 5, pp. 350-357, 2018.

[40]      M. Cecconi, D. D. Backer, M. Antonelli, R. Beale, J. Bakker, C. Hofer, R. Jaeschke, A. Mebazaa, M. R. Pinsky, J. L. Teboul, J. L. Vincent, A. Rhodes, Consensus on circulatory shock and hemodynamic monitoring. Task force of the European Society of Intensive Care Medicine, Intensive Care Medicine, Vol. 40, pp. 1795-1815, 2014.

[41]      B. R. Simon, M. V.  Kaufmann, M. A.  McAfee, A. L.  Baldwin,  L. M. Wilson, Identification and determination of material properties for porohyperelastic analysis of large arteries, Journal of Biomechanical Engineering, Vol. 120, No. 2, pp. 188-194, 1998.


Volume 50, Issue 2
December 2019
Pages 395-412
  • Receive Date: 18 February 2019
  • Revise Date: 15 April 2019
  • Accept Date: 15 April 2019