Modeling of hyperelastic incompressible behavior of functionally graded material under bending load

Document Type : Research Article

Authors

1 Tarbiat Modares University

2 Post. Doc. Researcher

3 PhD Student of Mechanical Engineering at Tarbiat Modares University

Abstract

In this paper, the behavior of inhomogeneous functionally graded rubber with large deformations and under bending loading is modeled by assuming an incompressible hyper-elastic material. In hyper-elastic inhomogeneous functionally graded materials, mechanical properties continuously changes from one point to another in the specified direction. In the other words, they gradually become material from material to another. For modeling the nonlinear behavior of material, hyperelasticity theory and strain energy density functions, which are a function of the invariants of the left deformation Cauchy-Green tensor, are used. In order to be able to apply the existing energy functions to inhomogeneous functionally graded materials, they must be changed, therefore, the changes in the constant of the energy functions are assumed power shape and in the direction of the curvature radius after bending, due to the inhomogeneous functionally graded of the material. Since many materials are inhomogeneous, using the assumption of inhomogeneous functionally graded of the material is one of the most practical methods. In this paper, the modeling of the hyperelastic behavior of inhomogeneous functionally graded rubber is done under bending loading and extraction of the Cauchy stress relations governing the cross-section caused by this loading. For modeling, the generalized Mooney-Rivlin energy function is used and the properties change in radial direction are considered and heterogeneity variations are also investigated.

Keywords

Main Subjects

References

[1] Y. Anani, G.H. Rahimi, Modeling of hyperelastic behavior of functionally graded rubber under mechanical and thermal load, (2016).
[2] L. R. G. Terloar, The Physics of Rubber Elasticity, Oxford University Press, New York, (2005).
[3] E. M. Arruda, M. C. Boyce, A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials, Journal of the Mechanics and Physics of Solids, 41(2) (1993) 389-412.
[4] H. M. James, E. Guth, Theory of the elastic properties of rubber, The Journal of Chemical Physics, 11(10) (1943) 455-481.
[5]  P. Flory, Theory of elasticity of polymer networks. The effect of local constraints on junctions, The Journal of Chemical Physics, 66(12) (1977) 5720-5729.
[6] F.  T.  Wall, P.  J. Flory, Statistical thermodynamics of rubber  elasticity, The  Journal  of  Chemical Physics, 19(12) (1951) 1435-1439.
[7] L. A., A Constitutive Model for Carbon Black Filled Rubber: Experimental Investigation and Mathematical Representation, j. of Continuum Mechanics and Thermodynamics, 8(3) (1996) 153-169.
[8] T. J. Van Dyke, A. Hoger, A comparison of second- order constitutive theories for hyperelastic materials, International journal of solids and structures, 37(41) (2000) 5873-5917.
[9] B. Meissner, L. Matějka, Comparison of recent rubber- elasticity theories with biaxial  stress–strain  data:  the slip-link theory of Edwards and Vilgis, Polymer, 43(13) (2002) 3803-3809.
[10] M. M. Attard, Finite strain––isotropic hyperelasticity, International Journal of Solids and Structures, 40(17) (2003) 4353-4378.
[11] M. M. Attard, G. W. Hunt, Hyperelastic constitutive modeling under finite strain, International Journal of Solids and Structures, 41(18-19) (2004) 5327-5350.
[12] L. Treloar, Stress-strain data for vulcanised rubber under various types of deformation, Transactions of the Faraday Society, 40 (1944) 59-70.
[13] M. M. Attard, Finite strain––beam theory, International journal of solids and structures, 40(17) (2003) 4563-4584.
[14] M. M. Attard, G. W. Hunt, Column buckling with shear deformations—a hyperelastic formulation, International Journal of Solids and Structures, 45(14- 15) (2008) 4322-4339.
[15] M. M. Attard, M. Y. Kim, Lateral buckling of beams with shear deformations–A hyperelastic formulation, International Journal of Solids and Structures, 47(20) (2010) 2825-2840.
[16] Y. Anani, Behavioral Modeling of Large-Deformed Rubber Based on the Model of Visco-Hyperelastic and Comparison with Experimental Results, Master’s Thesis, Mechanical Engineering of Sharif University of Technology (2007).
[17] A. Z. Kafi M. A., Bazaz M., Use of Hyperelastic materials to increase the stiffness of braces, First National Conference on Structural and Steel, Steel Structures Association of Iran, Tehran, (2010).
[18] M. Saje, G. Jelenić, Finite element formulation of hyperelastic plane frames subjected to nonconservative loads, Computers & structures, 50(2) (1994) 177-189.
[19] H. Altenbach, V. Eremeyev, On the effective stiffness of plates made of hyperelastic materials with initial stresses, International Journal of Non-Linear Mechanics, 45(10) (2010) 976-981.
[20] N. Kumar, A. DasGupta, On the contact problem   of an inflated spherical hyperelastic membrane, International Journal of Non-Linear Mechanics, 57 (2013) 130-139.
[21] A. J. Gil, Structural analysis of prestressed Saint to moderate strains, Computers & structures, 84(15-16) (2006) 1012-1028.
[22]   A. J. Gil, B. J., Wrinkling analysis of prestressed hyperelastic Saint Venant-Kirchhoff membranes, In: Metro R, editor. Shell and spatial structures: from models to realization. IASS, (2004).
[23]   E. S. Flores, S. Adhikari, M. Friswell, F. Scarpa, Hyperelastic finite element model for single wall carbon nanotubes in tension, Computational Materials Science, 50(3) (2011) 1083-1087.
[24]     E. Bilgili, Modelling mechanical behaviour of continuously graded vulcanised rubbers, Plastics, rubber and composites, 33(4) (2004) 163-169.
[25]   R. Batra, Finite plane strain deformations of rubberlike materials, International Journal for Numerical Methods in Engineering, 15(1) (1980) 145- 156.
[26]   E. Bilgili, Controlling the stress–strain inhomogeneities in axially sheared and radially heated hollow rubber tubes via functional grading, Mechanics Research Communications, 30(3) (2003) 257-266.
[27]    R. Batra, Optimal design of functionally graded incompressible linear elastic cylinders and spheres, AIAA journal, 46(8) (2008) 2050-2057.
[28]    G. Nie, R. Batra, Material tailoring and analysis    of functionally graded isotropic and incompressible linear elastic hollow cylinders, Composite structures, 92(2) (2010) 265-274.
[29]    G. Nie, R. Batra, Exact solutions and material tailoring for functionally graded hollow circular cylinders, Journal of Elasticity, 99(2) (2010) 179-201.
[30]   G. Nie, Z. Zhong, R. Batra, Material tailoring for functionally graded hollow cylinders and spheres, Composites Science and Technology, 71(5) (2011) 666-673.
[31]   R. Batra, Material tailoring and universal relations for axisymmetric deformations of functionally graded rubberlike cylinders and spheres, Mathematics and Mechanics of Solids, 16(7) (2011) 729-738.
[32]  R. Batra, Material tailoring in finite torsional Venant–Kirchhoff hyperelastic membranes subjected deformations of axially graded Mooney–Rivlin circular cylinder, Mathematics and Mechanics of Solids, 20(2) (2015) 183-189.
[33] Y. Anani, R. Naghdabadi, R. Avazmohammadi, Modeling of visco-hyperelastic behavior of foams in uniaxial tension, Proceedings of The 16th International Conference on Iranian Society of Mechanical Engineering(ISME 2008) Kerman, Iran. (in Persian), (2008).
[34] Y. Anani, R. Naghdabadi, Modeling of visco- hyperelastic behavior of rubbers in uniaxial tension, Proceedings of 7th Conference of Iranian Aerospace Society(AERO 2008), Tehran, Iran. (in persian) (2008).
[35] Y. Anani, G. H. Rahimi, Field equations and general solution for axisymmetric thick shell composed of functionally graded incompressible hyperelastic materials, International Journal of Mechanical Sciences, (2017).
[36]  L. M. Kanner, C. O. Horgan, Plane strain bending of strain-stiffening rubber-like rectangular beams, in: International Journal of Solids and Structures, 2008, pp. 1713-1729.
[37]  Y. B. Fu, R. W. Ogden, Nonlinear elasticity: theory and applications, Cambridge University Press, 2001.
[38]  L. Meunier, G. Chagnon, D. Favier, L. Orgéas, P. Vacher, Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber, Polymer Testing, 27(6) (2008) 765-777.
[39]  A. A. Khan, M. Naushad Alam, M. Wajid, Finite element modelling for static and free vibration response of functionally graded beam, Latin American Journal of Solids and Structures, 13(4) (2016) 690- 714.
[40]  M. Foroutan, R. Moradi-Dastjerdi, R. Sotoodeh- Bahreini, Static analysis of FGM cylinders by a mesh- free method, Steel & Composite Structures, 12(1) (2012) 1-11.
 
 
[9]  
 
 
[1] Y. Anani, G.H. Rahimi, Modeling of hyperelastic behavior of functionally graded rubber under mechanical and thermal load, (2016).
[2] L. R. G. Terloar, The Physics of Rubber Elasticity, Oxford University Press, New York, (2005).
[3] E. M. Arruda, M. C. Boyce, A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials, Journal of the Mechanics and Physics of Solids, 41(2) (1993) 389-412.
[4] H. M. James, E. Guth, Theory of the elastic properties of rubber, The Journal of Chemical Physics, 11(10) (1943) 455-481.
[5]P. Flory, Theory of elasticity of polymer networks. The effect of local constraints on junctions, The Journal of Chemical Physics, 66(12) (1977) 5720-5729.
[6]F.  T.  Wall, P.  J. Flory, Statistical thermodynamics of rubber  elasticity, The  Journal  of  Chemical Physics, 19(12) (1951) 1435-1439.
[7] L. A., A Constitutive Model for Carbon Black Filled Rubber: Experimental Investigation and Mathematical Representation, j. of Continuum Mechanics and Thermodynamics, 8(3) (1996) 153-169.
[8] T. J. Van Dyke, A. Hoger, A comparison of second- order constitutive theories for hyperelastic materials, International journal of solids and structures, 37(41) (2000) 5873-5917.
[9] B. Meissner, L. Matějka, Comparison of recent rubber- elasticity theories with biaxial  stress–strain  data:  the slip-link theory of Edwards and Vilgis, Polymer, 43(13) (2002) 3803-3809.
[10] M. M. Attard, Finite strain––isotropic hyperelasticity, International Journal of Solids and Structures, 40(17)(2003) 4353-4378.
[11] M. M. Attard, G. W. Hunt, Hyperelastic constitutive modeling under finite strain, International Journal of Solids and Structures, 41(18-19) (2004) 5327-5350.
[12] L. Treloar, Stress-strain data for vulcanised rubber under various types of deformation, Transactions of the Faraday Society, 40 (1944) 59-70.
[13] M. M. Attard, Finite strain––beam theory, International journal of solids and structures, 40(17) (2003) 4563-4584.
[14] M. M. Attard, G. W. Hunt, Column buckling with shear deformations—a hyperelastic formulation, International Journal of Solids and Structures, 45(14- 15) (2008) 4322-4339.
[15] M. M. Attard, M. Y. Kim, Lateral buckling of beams with shear deformations–A hyperelastic formulation, International Journal of Solids and Structures, 47(20) (2010) 2825-2840.
[16] Y. Anani, Behavioral Modeling of Large-Deformed Rubber Based on the Model of Visco-Hyperelastic and Comparison with Experimental Results, Master’s Thesis, Mechanical Engineering of Sharif University of Technology (2007).
[17] A. Z. Kafi M. A., Bazaz M., Use of Hyperelastic materials to increase the stiffness of braces, First National Conference on Structural and Steel, Steel Structures Association of Iran, Tehran, (2010).
[18] M. Saje, G. Jelenić, Finite element formulation of hyperelastic plane frames subjected to nonconservative loads, Computers & structures, 50(2) (1994) 177-189.
[19] H. Altenbach, V. Eremeyev, On the effective stiffness of plates made of hyperelastic materials with initial stresses, International Journal of Non-Linear Mechanics, 45(10) (2010) 976-981.
[20] N. Kumar, A. DasGupta, On the contact problem   of an inflated spherical hyperelastic membrane, International Journal of Non-Linear Mechanics, 57 (2013) 130-139.
[21] A. J. Gil, Structural analysis of prestressed Saint to moderate strains, Computers & structures, 84(15-16) (2006) 1012-1028.
[22]   A. J. Gil, B. J., Wrinkling analysis of prestressed hyperelastic Saint Venant-Kirchhoff membranes, In: Metro R, editor. Shell and spatial structures: from models to realization. IASS, (2004).
[23]   E. S. Flores, S. Adhikari, M. Friswell, F. Scarpa, Hyperelastic finite element model for single wall carbon nanotubes in tension, Computational Materials Science, 50(3) (2011) 1083-1087.
[24]     E. Bilgili, Modelling mechanical behaviour of continuously graded vulcanised rubbers, Plastics, rubber and composites, 33(4) (2004) 163-169.
[25] R. Batra, Finite plane strain deformations of rubberlike materials, International Journal for Numerical Methods in Engineering, 15(1) (1980) 145- 156.
[26] E. Bilgili, Controlling the stress–strain inhomogeneities in axially sheared and radially heated hollow rubber tubes via functional grading, Mechanics Research Communications, 30(3) (2003) 257-266.
[27] R. Batra, Optimal design of functionally graded incompressible linear elastic cylinders and spheres, AIAA journal, 46(8) (2008) 2050-2057.
[28]    G. Nie, R. Batra, Material tailoring and analysis    of functionally graded isotropic and incompressible linear elastic hollow cylinders, Composite structures, 92(2) (2010) 265-274.
[29]    G. Nie, R. Batra, Exact solutions and material tailoring for functionally graded hollow circular cylinders, Journal of Elasticity, 99(2) (2010) 179-201.
[30]   G. Nie, Z. Zhong, R. Batra, Material tailoring for functionally graded hollow cylinders and spheres, Composites Science and Technology, 71(5) (2011) 666-673.
[31]   R. Batra, Material tailoring and universal relations for axisymmetric deformations of functionally graded rubberlike cylinders and spheres, Mathematics and Mechanics of Solids, 16(7) (2011) 729-738.
[32] R. Batra, Material tailoring in finite torsional Venant–Kirchhoff hyperelastic membranes subjected deformations of axially graded Mooney–Rivlin circular cylinder, Mathematics and Mechanics of Solids, 20(2) (2015) 183-189.
[33] Y. Anani, R. Naghdabadi, R. Avazmohammadi, Modeling of visco-hyperelastic behavior of foams in uniaxial tension, Proceedings of The 16th International Conference on Iranian Society of Mechanical Engineering(ISME 2008) Kerman, Iran. (in Persian), (2008).
[34] Y. Anani, R. Naghdabadi, Modeling of visco- hyperelastic behavior of rubbers in uniaxial tension, Proceedings of 7th Conference of Iranian Aerospace Society(AERO 2008), Tehran, Iran. (in persian) (2008).
[35] Y. Anani, G. H. Rahimi, Field equations and general solution for axisymmetric thick shell composed of functionally graded incompressible hyperelastic materials, International Journal of Mechanical Sciences, (2017).
[36]  L. M. Kanner, C. O. Horgan, Plane strain bending of strain-stiffening rubber-like rectangular beams, in: International Journal of Solids and Structures, 2008, pp. 1713-1729.
[37]  Y. B. Fu, R. W. Ogden, Nonlinear elasticity: theory and applications, Cambridge University Press, 2001.
[38]  L. Meunier, G. Chagnon, D. Favier, L. Orgéas, P. Vacher, Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber, Polymer Testing, 27(6) (2008) 765-777.
[39]  A. A. Khan, M. Naushad Alam, M. Wajid, Finite element modelling for static and free vibration response of functionally graded beam, Latin American Journal of Solids and Structures, 13(4) (2016) 690- 714.
[40]  M. Foroutan, R. Moradi-Dastjerdi, R. Sotoodeh- Bahreini, Static analysis of FGM cylinders by a mesh- free method, Steel & Composite Structures, 12(1) (2012) 1-11.
 
 
Amirkabir Journal of Mechanical Engineering
Volume 52, Issue 8
November 2020
Pages 2045-2062

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