Mechanic of Advanced and Smart Materials
Quarterly

Large deformation analysis of functionally graded cylinder under extension-torsion: analytical closed form and finite element solutions

Document Type : Original Article

Author

ali taheri

Assistant Professor, Faculty of Engineering, University of Larestan

https://doi.org/10.61186/masm.4.1.20
Abstract
In this study, an analytical solution has been developed to examine the mechanical behavior of an incompressible functionally graded hyperelastic cylinder subjected to simultaneous extension and torsion. The recently proposed exp-exp strain energy density is employed to predict the behavior of hyperelastic material, and its related material parameters are assumed to vary along the radial direction in an exponential fashion. Finite element analysis is conducted by preparing a user-defined UHYPER subroutine in ABAQUS to evaluate the proposed analytical solutions. FEM results and those of the analytical solution are in good agreement for various stretches and twists and reveal that the form of stress distributions and the maximum stress depend on the exponential power in the material variation function. In contrast to axial stretch, the effect of twist on the distribution of longitudinal stress is more complicated, and for large twists, two extrema in the stress distribution plot are observed, which move toward the center and outer surface of the cylinder on further twisting. Moreover, the longitudinal stress controls the variation of von-Mises and strain energy density throughout the radial direction. Additionally, considering an axial stretch, a point is identified where the axial force arising from torsion is compressive for stretches below this value, and it brings about the cylinder to elongate under twisting. However, this part of the total axial force varies from a tension state to a compression one for larger stretches, i.e., by increasing the twist, the cylinder first tends to shorten and then elongates on further twisting.

Highlights

[1] Huang W, Ding O, Wang H., Wu Z, Luo Y, Shi W, Yang L, Liang Y, Liu C, Wu J. Design of stretchable and self-powered sensing device for portable and remote trace biomarkers detection, Nature Communications, 2023; 14:1-13.
[2] Pan X, Xu Z, Bao R, Pan C. Research Progress in Stretchable Circuits: Materials, Methods, and Applications. Advance Sensor Research, 2023; 11:20-28.
[3] Xiang C, Wang Z, Yang C, Yao X, Wang Y, Suo Z. Stretchable and fatigue-resistant materials, Materials today, 2020; 34:7-16.
[4] Yan S, Shan Y, Jun L, Zulmari S, Ting Z, Marvin H, Bo L, Tong L, Long G, Yunhe Z, Yutao D, Bo Y, and Xudong W. Stretchable Encapsulation Materials with High Dynamic Water Resistivity and Tissue-Matching Elasticity, ACS Appl. Mater. Interfaces, 2022; 14: 18935–18943.
[5] Shojaeifard M, Bayat M R, Baghani M. Swelling-induced finite bending of functionally graded pH-responsive hydrogels: a semi-analytical method. Applied Mathematics and Mechanics, 2019; 40:679-694.
[6] Valiollahi A, Shojaeifard M, Baghani M. Closed form solutions for large deformation of cylinders under combined extension-torsion. International Journal of Mechanical Sciences, 2019; 157: 336-347.
[7] Shojaeifard M, Rouhani F, Baghani M. A combined analytical–numerical analysis on multidirectional finite bending of functionally graded temperature-sensitive hydrogels. Journal of Intelligent Material Systems and Structures, 2019; 30: 1882 - 1895.
[8] Xu J, Yuan X, Zhang H, Zhao Z, Zhao W. Combined effects of axial load and temperature on finite deformation of incompressible thermo-hyperelastic cylinder. Applied Mathematics and Mechanics, 2019; 40: 499-514.
[9] Rivlin R. Large elastic deformations of isotropic materials. III. Some simple problems in cyclindrical polar coordinates. Philosophical Transactions of the Royal Society of London Series A, Mathematical and Physical Sciences, 1948; 240:509-525.
[10] Rivlin R S. Large elastic deformations of isotropic materials VI. Further results in the theory of torsion, shear and flexure. Philosophical Transactions of the Royal Society of London Series A, Mathematical and Physical Sciences, 1949; 242:173-195.
[11] Humphrey J D. Cardiovascular solid mechanics: cells, tissues, and organs. New York: Springer; 2013.
[12] Taber L A. Nonlinear theory of elasticity: applications in biomechanics: World Scientific; 2004.
[13] Rivlin R. Large elastic deformations of isotropic materials. I. Fundamental concepts. Philosophical Transactions of the Royal Society of London Series A, Mathematical and Physical Sciences, 1948; 240:459- 490.
[14] Treloar L. The elasticity of a network of long-chain molecules—II. Transactions of the Faraday Society, 1943; 39:241-246.
[15] Ogden R W. Large deformation isotropic elasticity–on the correlation of theory and experiment for incompressible rubberlike solids. Proceedings of the Royal Society of London A Mathematical and Physical Sciences, 1972; 326:565-584.
[16] Gent A. A new constitutive relation for rubber. Rubber chemistry technology, 1996; 69:59-61.
[17] Pan Y, Zhong Z. A viscoelastic constitutive modeling of rubber-like materials with the Payne effect. Applied Mathematical Modelling, 2017; 50,621-632.
[18] López-Campos J, Segade A, Fernández J, Casarejos E, Vilán J. Behavior characterization of viscohyperelastic models for rubber-like materials using genetic algorithms. Applied Mathematical Modelling, 2019; 66:241-255.
[19] Chagnon G, Verron E, Gornet L, Marckmann G, Charrier P. On the relevance of continuum damage mechanics as applied to the Mullins effect in elastomers. Journal of the Mechanics Physics of Solids, 2004; 52:1627- 1650.
[20] Bechir, H., Chevalier, L., Chaouche, M., Boufala, K. Hyperelastic constitutive model for rubber-like materials based on the first Seth strain measures invariant. European Journal of Mechanics-A/Solids, 2006; 25,110-124.
[21] Khajehsaeid H, Arghavani J, Naghdabadi R. A hyperelastic constitutive model for rubber-like materials. European Journal of Mechanics-A/Solids, 2013; 38:144-151.
[22] Darijani H, Naghdabadi R, Kargarnovin M H. Constitutive modeling of rubberlike materials based on consistent strain energy density functions. Polymer Engineering Science, 2010; 50:1058-1066.
[23] Mansouri M, Darijani H. Constitutive modeling of isotropic hyperelastic materials in an exponential framework using a self-contained approach. International Journal of Solids Structures, 2014; 51:4316-4326.
[24] Darijani H, Naghdabadi R. Hyperelastic materials behavior modeling using consistent strain energy density functions. Acta mechanica, 2010; 213:235-254.
[25] Anani Y, Rahimi G H. Stress analysis of thick pressure vessel composed of functionally graded incompressible hyperelastic materials. International Journal of Mechanical Sciences, 2015; 104:1-7.
[26] Bilgili E. Controlling the stress–strain inhomogeneities in axially sheared and radially heated hollow rubber tubes via functional grading. Mechanics Research Communications, 2003; 30:257-266.
[27] Anani Y, Rahimi G. Stress analysis of thick spherical pressure vessel composed of transversely isotropic functionally graded incompressible hyperelastic materials. Latin American Journal of Solids and Structures, 2016; 13:407-434.
[28] Anani Y, Rahimi G H. Stress analysis of rotating cylindrical shell composed of functionally graded incompressible hyperelastic materials. International Journal of Mechanical Sciences, 2016; 108:122-128.
[29] Anani Y, Rahimi G. Modeling of visco‐hyperelastic behavior of transversely isotropic functionally graded rubbers. Polymer Engineering Science, 2016; 56:342-347.
[30] Pascon J, Coda H. High-order tetrahedral finite elements applied to large deformation analysis of functionally graded rubber-like materials. Applied Mathematical Modelling, 2013; 37:8757-8775.
[31] Moallemi A, Baghani M, Almasi A, Zakerzadeh M R, Baniassadi M. Large deformation and stability analysis of functionally graded pressure vessels: An analytical and numerical study. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2018; 232:3300-3314.
[32] Almasi A, Baghani M, Moallemi A, Baniassadi M, Faraji G. Investigation on thermal stresses in FGM hyperelastic thick-walled cylinders. Journal of Thermal Stresses, 2018; 41:204-221.
[33] Humphrey J, Barazotto Jr R, Hunter W. Finite extension and torsion of papillary muscles: a theoretical framework. Journal of Biomechanics, 1992; 25:541-547.
[34] Ogden R, Chadwick P. On the deformation of solid and tubular cylinders of incompressible isotropic elastic material. Journal of the Mechanics Physics of Solids, 1972; 20:77-90.
[35] Kanner L M, Horgan C O. On extension and torsion of strain-stiffening rubber-like elastic circular cylinders. Journal of Elasticity, 2008; 93,39-47.
[36] Fung Y. Elasticity of soft tissues in simple elongation. American Journal of Physiology-Legacy Content, 1967; 213:1532-1544.
[37] Horgan C O, Murphy J G. Extension and torsion of incompressible non-linearly elastic solid circular cylinders. Mathematics Mechanics of Solids, 2011; 16,482-491.
[38] Varga O H. Stress-strain behavior of elastic materials; selected problems of large deformations. NewYork: Interscience 1966.
[39] Horgan C O, Murphy J G. Finite extension and torsion of fiber-reinforced non-linearly elastic circular cylinders. International Journal of Non-Linear Mechanics, 2012; 47:97-104.
[40] Khajehsaeed H, M Baghani M, Naghdabadi R. Finite strain numerical analysis of elastomeric bushings under multi-axial loadings: a compressible visco-hyperelastic approach, International Journal of Mechanics and Materials in Design, 2013; 9:385–399.

Keywords

Extension-torsion
Rubber-like cylinder
Functionally graded material Hyperelastic material
Finite element method
[1] Huang W, Ding O, Wang H., Wu Z, Luo Y, Shi W, Yang L, Liang Y, Liu C, Wu J. Design of stretchable and self-powered sensing device for portable and remote trace biomarkers detection, Nature Communications, 2023; 14:1-13.
[2] Pan X, Xu Z, Bao R, Pan C. Research Progress in Stretchable Circuits: Materials, Methods, and Applications. Advance Sensor Research, 2023; 11:20-28.
[3] Xiang C, Wang Z, Yang C, Yao X, Wang Y, Suo Z. Stretchable and fatigue-resistant materials, Materials today, 2020; 34:7-16.
[4] Yan S, Shan Y, Jun L, Zulmari S, Ting Z, Marvin H, Bo L, Tong L, Long G, Yunhe Z, Yutao D, Bo Y, and Xudong W. Stretchable Encapsulation Materials with High Dynamic Water Resistivity and Tissue-Matching Elasticity, ACS Appl. Mater. Interfaces, 2022; 14: 18935–18943.
[5] Shojaeifard M, Bayat M R, Baghani M. Swelling-induced finite bending of functionally graded pH-responsive hydrogels: a semi-analytical method. Applied Mathematics and Mechanics, 2019; 40:679-694.
[6] Valiollahi A, Shojaeifard M, Baghani M. Closed form solutions for large deformation of cylinders under combined extension-torsion. International Journal of Mechanical Sciences, 2019; 157: 336-347.
[7] Shojaeifard M, Rouhani F, Baghani M. A combined analytical–numerical analysis on multidirectional finite bending of functionally graded temperature-sensitive hydrogels. Journal of Intelligent Material Systems and Structures, 2019; 30: 1882 - 1895.
[8] Xu J, Yuan X, Zhang H, Zhao Z, Zhao W. Combined effects of axial load and temperature on finite deformation of incompressible thermo-hyperelastic cylinder. Applied Mathematics and Mechanics, 2019; 40: 499-514.
[9] Rivlin R. Large elastic deformations of isotropic materials. III. Some simple problems in cyclindrical polar coordinates. Philosophical Transactions of the Royal Society of London Series A, Mathematical and Physical Sciences, 1948; 240:509-525.
[10] Rivlin R S. Large elastic deformations of isotropic materials VI. Further results in the theory of torsion, shear and flexure. Philosophical Transactions of the Royal Society of London Series A, Mathematical and Physical Sciences, 1949; 242:173-195.
[11] Humphrey J D. Cardiovascular solid mechanics: cells, tissues, and organs. New York: Springer; 2013.
[12] Taber L A. Nonlinear theory of elasticity: applications in biomechanics: World Scientific; 2004.
[13] Rivlin R. Large elastic deformations of isotropic materials. I. Fundamental concepts. Philosophical Transactions of the Royal Society of London Series A, Mathematical and Physical Sciences, 1948; 240:459- 490.
[14] Treloar L. The elasticity of a network of long-chain molecules—II. Transactions of the Faraday Society, 1943; 39:241-246.
[15] Ogden R W. Large deformation isotropic elasticity–on the correlation of theory and experiment for incompressible rubberlike solids. Proceedings of the Royal Society of London A Mathematical and Physical Sciences, 1972; 326:565-584.
[16] Gent A. A new constitutive relation for rubber. Rubber chemistry technology, 1996; 69:59-61.
[17] Pan Y, Zhong Z. A viscoelastic constitutive modeling of rubber-like materials with the Payne effect. Applied Mathematical Modelling, 2017; 50,621-632.
[18] López-Campos J, Segade A, Fernández J, Casarejos E, Vilán J. Behavior characterization of viscohyperelastic models for rubber-like materials using genetic algorithms. Applied Mathematical Modelling, 2019; 66:241-255.
[19] Chagnon G, Verron E, Gornet L, Marckmann G, Charrier P. On the relevance of continuum damage mechanics as applied to the Mullins effect in elastomers. Journal of the Mechanics Physics of Solids, 2004; 52:1627- 1650.
[20] Bechir, H., Chevalier, L., Chaouche, M., Boufala, K. Hyperelastic constitutive model for rubber-like materials based on the first Seth strain measures invariant. European Journal of Mechanics-A/Solids, 2006; 25,110-124.
[21] Khajehsaeid H, Arghavani J, Naghdabadi R. A hyperelastic constitutive model for rubber-like materials. European Journal of Mechanics-A/Solids, 2013; 38:144-151.
[22] Darijani H, Naghdabadi R, Kargarnovin M H. Constitutive modeling of rubberlike materials based on consistent strain energy density functions. Polymer Engineering Science, 2010; 50:1058-1066.
[23] Mansouri M, Darijani H. Constitutive modeling of isotropic hyperelastic materials in an exponential framework using a self-contained approach. International Journal of Solids Structures, 2014; 51:4316-4326.
[24] Darijani H, Naghdabadi R. Hyperelastic materials behavior modeling using consistent strain energy density functions. Acta mechanica, 2010; 213:235-254.
[25] Anani Y, Rahimi G H. Stress analysis of thick pressure vessel composed of functionally graded incompressible hyperelastic materials. International Journal of Mechanical Sciences, 2015; 104:1-7.
[26] Bilgili E. Controlling the stress–strain inhomogeneities in axially sheared and radially heated hollow rubber tubes via functional grading. Mechanics Research Communications, 2003; 30:257-266.
[27] Anani Y, Rahimi G. Stress analysis of thick spherical pressure vessel composed of transversely isotropic functionally graded incompressible hyperelastic materials. Latin American Journal of Solids and Structures, 2016; 13:407-434.
[28] Anani Y, Rahimi G H. Stress analysis of rotating cylindrical shell composed of functionally graded incompressible hyperelastic materials. International Journal of Mechanical Sciences, 2016; 108:122-128.
[29] Anani Y, Rahimi G. Modeling of visco‐hyperelastic behavior of transversely isotropic functionally graded rubbers. Polymer Engineering Science, 2016; 56:342-347.
[30] Pascon J, Coda H. High-order tetrahedral finite elements applied to large deformation analysis of functionally graded rubber-like materials. Applied Mathematical Modelling, 2013; 37:8757-8775.
[31] Moallemi A, Baghani M, Almasi A, Zakerzadeh M R, Baniassadi M. Large deformation and stability analysis of functionally graded pressure vessels: An analytical and numerical study. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2018; 232:3300-3314.
[32] Almasi A, Baghani M, Moallemi A, Baniassadi M, Faraji G. Investigation on thermal stresses in FGM hyperelastic thick-walled cylinders. Journal of Thermal Stresses, 2018; 41:204-221.
[33] Humphrey J, Barazotto Jr R, Hunter W. Finite extension and torsion of papillary muscles: a theoretical framework. Journal of Biomechanics, 1992; 25:541-547.
[34] Ogden R, Chadwick P. On the deformation of solid and tubular cylinders of incompressible isotropic elastic material. Journal of the Mechanics Physics of Solids, 1972; 20:77-90.
[35] Kanner L M, Horgan C O. On extension and torsion of strain-stiffening rubber-like elastic circular cylinders. Journal of Elasticity, 2008; 93,39-47.
[36] Fung Y. Elasticity of soft tissues in simple elongation. American Journal of Physiology-Legacy Content, 1967; 213:1532-1544.
[37] Horgan C O, Murphy J G. Extension and torsion of incompressible non-linearly elastic solid circular cylinders. Mathematics Mechanics of Solids, 2011; 16,482-491.
[38] Varga O H. Stress-strain behavior of elastic materials; selected problems of large deformations. NewYork: Interscience 1966.
[39] Horgan C O, Murphy J G. Finite extension and torsion of fiber-reinforced non-linearly elastic circular cylinders. International Journal of Non-Linear Mechanics, 2012; 47:97-104.
[40] Khajehsaeed H, M Baghani M, Naghdabadi R. Finite strain numerical analysis of elastomeric bushings under multi-axial loadings: a compressible visco-hyperelastic approach, International Journal of Mechanics and Materials in Design, 2013; 9:385–399.
Mechanic of Advanced and Smart Materials
Volume 4, Issue 1
Spring 2024
Pages 20-39

  • Receive Date 03 February 2024
  • Revise Date 13 February 2024
  • Accept Date 12 March 2024

Home

Guide for Authors

Submit Manuscript

Contact Us