Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material

Egbuhuzor Udechukwu Peter; Udoh Ndipmong Augustine

doi:doi:10.11648/j.advances.20230401.15

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Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material

Egbuhuzor Udechukwu Peter, Udoh Ndipmong Augustine

Published in Advances (Volume 4, Issue 1)

Received: 27 January 2023 Accepted: 24 February 2023 Published: 15 March 2023

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Abstract

The finite deformation of internally pressurized isotropic compressible synthetic rubber-like material governed by Levinson and Burgess strain energy function is analysed. A second-order nonlinear ordinary differential (Lane-Emden) equationwith shooting boundary value was derivedfor the determination of displacements distributions. Several analytical methods were employed to solve the resulting boundary value problem but no closed form solution was obtained at the moment. Fortunately, a lot of software have been developed to handle such highly nonlinear second order ordinary differential equations with specific values of parameters. Also, the stresses acting on the material were determined. We obtained numerical solution by applying shooting method and validated the result using collocation method on mathematica (ode45 solver). The simulation of the system is made forρ = 14N/m², and the cylindrical symmetric deformation attained its maximum displacements and stresses atr(1) = 1.16638m and σ_rr = (-1.2973e-05)kg/m/s². We were able to develop numerical schemes using shooting and Collocation methods which made it easier to determine position of maximum stresses and pressure in a cylindrical material of Levinson-Burgess strain energy function. These numerical schemes can solve any nonlinear second-order ordinary differential equations with any given boundary conditions on Mathematica Software. The results of the two schemes were statistically compared using t-test and results obtained showed, the two methods have no significant difference which validates the solutions.

Published in	Advances (Volume 4, Issue 1)
DOI	10.11648/j.advances.20230401.15
Page(s)	36-43
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

Deformation, Displacement and Stresses, Levinson and Burgess, Hollow Sphere, Hollow Cylinder

References

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[2]	Huang, Z., On the Finite Displacement Problem of a Hollow Sphere Under Internal and External Pressures. Applied Mathematics and Mechanics, 6, 1097-1102.
[3]	Stange, J. B., Finite Deformation of Hollow Sphere of Linear Elastic Perfectly Plastic Material. Acta Mechanica, 1984, 50: p. 201-209.
[4]	Hill, J. M., Cylindrical and Spherical Inflation in Compressible Finite Elasticity. IMA Journal of Applied Mathematics, 1992. 50: p. 195-201.
[5]	Levinson, M. and I. W. Burgess, A Comparison of some simple constitutive relations for slightly compressible rubber-like materials. International Journal of Mechanical Sciences, 1971. 13: p. 563-572.
[6]	Egbuhuzor, U. P, and E. N. Erumaka, Finite Deformation of Internally Pressurized Spherical Compressible Rubber-like Material. Asian Research Journal of Mathematics, 16 (3), 2020. 16 (3): p. 38-49.
[7]	Aani, Y. and G. H. Rahimi, On the stability of internally pressurized thick-walled spherical and cylindrical shells made of functionally graded incompressible hyperelastic material. Latin American Journal of Solids and Structures, 2018. 15 (4).
[8]	Aani, Y. and G. H. Rahimi, Stress analysis of thick pressure vessel composed of incompressible hyperelastic materials, International Journal of Recent Advances in Mechanical Engineering, 2015. 4 (3): 19-37.
[9]	De Pascalis, R., I. D., Abrahams and W. J. Parnell, Simple shear of a compressible quasilinear viscoelastic material. International Journal of Engineering Science, 2015, 88: p. 64-72.
[10]	Giuseppe, P. and S. Giuseppe, The Gent model for rubber-like materials: An appraisal for an ingeniousand simple idea. International Journal of Non-Linear Mechanics, 2015, 68: p. 17-24.
[11]	Blatz, P. J. and W. L. Ko, Constitutive model in nonlinear finite element analysis. Trans. Soc. Rheol., 1962, 6: p. 223 - 251.
[12]	Chung, D. T., C. O. Horgan and R., Abeyaratne, The finite deformation of internally pressurized hollow cylinders and spheres for a class of compressible elastic materials. International Journal of Solids and Structures, 1986, 22 (12): p. 1557-1570.
[13]	Horgan, C. O., Remarks on ellipticity for the generalized Blatz-Ko constitutive model for compressible nonlinearly elastic solid. Journal of Elasticity, 1986, 42: 165-176.
[14]	Kulcu, I. D., A hyperelastic constitutive model for rubber-like materials. Archive of Applied Mechanics, 2019, 1-8.
[15]	Akhundov, V. M. and Lunev, V. P. (2013). Modeling of the forming of radial tyre carcass based on applied theory of fibre-reinforced materials. International Polymer Science and Technology, 2013, 40 (10): 37-40.
[16]	Fereidoonnezhad, B., Naghdabadi, R. and Arghavani, J., A hyperelastic constitutive model for fibre-reinforced rubber-like materials. International Journal of Engineering Science, 2013, 71: 36-44.
[17]	Moreira, D. C., and Numes, L. C., Comparison of simple and plane shear for an incompressible isotropic hyperelastic material under large deformation. Polymer Testing, 2013, 32 (2): 240-248.
[18]	Pence, T. J., and Gou, K., On compressible versions of the incompressible neo-Hookean material. Mathematics and Mechanics of Solids, 2015, 20 (2): 157-182.

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Egbuhuzor Udechukwu Peter, Udoh Ndipmong Augustine. (2023). Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material. Advances, 4(1), 36-43. https://doi.org/10.11648/j.advances.20230401.15

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Egbuhuzor Udechukwu Peter; Udoh Ndipmong Augustine. Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material. Advances. 2023, 4(1), 36-43. doi: 10.11648/j.advances.20230401.15

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AMA Style

Egbuhuzor Udechukwu Peter, Udoh Ndipmong Augustine. Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material. Advances. 2023;4(1):36-43. doi: 10.11648/j.advances.20230401.15

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@article{10.11648/j.advances.20230401.15,
  author = {Egbuhuzor Udechukwu Peter and Udoh Ndipmong Augustine},
  title = {Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material},
  journal = {Advances},
  volume = {4},
  number = {1},
  pages = {36-43},
  doi = {10.11648/j.advances.20230401.15},
  url = {https://doi.org/10.11648/j.advances.20230401.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.advances.20230401.15},
  abstract = {The finite deformation of internally pressurized isotropic compressible synthetic rubber-like material governed by Levinson and Burgess strain energy function is analysed. A second-order nonlinear ordinary differential (Lane-Emden) equationwith shooting boundary value was derivedfor the determination of displacements distributions. Several analytical methods were employed to solve the resulting boundary value problem but no closed form solution was obtained at the moment. Fortunately, a lot of software have been developed to handle such highly nonlinear second order ordinary differential equations with specific values of parameters. Also, the stresses acting on the material were determined. We obtained numerical solution by applying shooting method and validated the result using collocation method on mathematica (ode45 solver). The simulation of the system is made forρ = 14N/m2, and the cylindrical symmetric deformation attained its maximum displacements and stresses atr(1) = 1.16638m and σrr = (-1.2973e-05)kg/m/s2. We were able to develop numerical schemes using shooting and Collocation methods which made it easier to determine position of maximum stresses and pressure in a cylindrical material of Levinson-Burgess strain energy function. These numerical schemes can solve any nonlinear second-order ordinary differential equations with any given boundary conditions on Mathematica Software. The results of the two schemes were statistically compared using t-test and results obtained showed, the two methods have no significant difference which validates the solutions.},
 year = {2023}
}

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TY  - JOUR
T1  - Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material
AU  - Egbuhuzor Udechukwu Peter
AU  - Udoh Ndipmong Augustine
Y1  - 2023/03/15
PY  - 2023
N1  - https://doi.org/10.11648/j.advances.20230401.15
DO  - 10.11648/j.advances.20230401.15
T2  - Advances
JF  - Advances
JO  - Advances
SP  - 36
EP  - 43
PB  - Science Publishing Group
SN  - 2994-7200
UR  - https://doi.org/10.11648/j.advances.20230401.15
AB  - The finite deformation of internally pressurized isotropic compressible synthetic rubber-like material governed by Levinson and Burgess strain energy function is analysed. A second-order nonlinear ordinary differential (Lane-Emden) equationwith shooting boundary value was derivedfor the determination of displacements distributions. Several analytical methods were employed to solve the resulting boundary value problem but no closed form solution was obtained at the moment. Fortunately, a lot of software have been developed to handle such highly nonlinear second order ordinary differential equations with specific values of parameters. Also, the stresses acting on the material were determined. We obtained numerical solution by applying shooting method and validated the result using collocation method on mathematica (ode45 solver). The simulation of the system is made forρ = 14N/m2, and the cylindrical symmetric deformation attained its maximum displacements and stresses atr(1) = 1.16638m and σrr = (-1.2973e-05)kg/m/s2. We were able to develop numerical schemes using shooting and Collocation methods which made it easier to determine position of maximum stresses and pressure in a cylindrical material of Levinson-Burgess strain energy function. These numerical schemes can solve any nonlinear second-order ordinary differential equations with any given boundary conditions on Mathematica Software. The results of the two schemes were statistically compared using t-test and results obtained showed, the two methods have no significant difference which validates the solutions.
VL  - 4
IS  - 1
ER  -

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Author Information

Egbuhuzor Udechukwu Peter

Mathematics and Statistics Department, Faculty of Science, Federal University, Otuoke, Nigeria
Udoh Ndipmong Augustine

Mathematics and Statistics Department, Faculty of Science, Federal University, Otuoke, Nigeria

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APA Style

Egbuhuzor Udechukwu Peter, Udoh Ndipmong Augustine. (2023). Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material. Advances, 4(1), 36-43. https://doi.org/10.11648/j.advances.20230401.15

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ACS Style

Egbuhuzor Udechukwu Peter; Udoh Ndipmong Augustine. Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material. Advances. 2023, 4(1), 36-43. doi: 10.11648/j.advances.20230401.15

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AMA Style

Egbuhuzor Udechukwu Peter, Udoh Ndipmong Augustine. Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material. Advances. 2023;4(1):36-43. doi: 10.11648/j.advances.20230401.15

Copy | Download

@article{10.11648/j.advances.20230401.15,
  author = {Egbuhuzor Udechukwu Peter and Udoh Ndipmong Augustine},
  title = {Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material},
  journal = {Advances},
  volume = {4},
  number = {1},
  pages = {36-43},
  doi = {10.11648/j.advances.20230401.15},
  url = {https://doi.org/10.11648/j.advances.20230401.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.advances.20230401.15},
  abstract = {The finite deformation of internally pressurized isotropic compressible synthetic rubber-like material governed by Levinson and Burgess strain energy function is analysed. A second-order nonlinear ordinary differential (Lane-Emden) equationwith shooting boundary value was derivedfor the determination of displacements distributions. Several analytical methods were employed to solve the resulting boundary value problem but no closed form solution was obtained at the moment. Fortunately, a lot of software have been developed to handle such highly nonlinear second order ordinary differential equations with specific values of parameters. Also, the stresses acting on the material were determined. We obtained numerical solution by applying shooting method and validated the result using collocation method on mathematica (ode45 solver). The simulation of the system is made forρ = 14N/m2, and the cylindrical symmetric deformation attained its maximum displacements and stresses atr(1) = 1.16638m and σrr = (-1.2973e-05)kg/m/s2. We were able to develop numerical schemes using shooting and Collocation methods which made it easier to determine position of maximum stresses and pressure in a cylindrical material of Levinson-Burgess strain energy function. These numerical schemes can solve any nonlinear second-order ordinary differential equations with any given boundary conditions on Mathematica Software. The results of the two schemes were statistically compared using t-test and results obtained showed, the two methods have no significant difference which validates the solutions.},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - Finite Deformations of Internally Pressurized Synthetic Compressible Cylindrical Rubber-Like Material
AU  - Egbuhuzor Udechukwu Peter
AU  - Udoh Ndipmong Augustine
Y1  - 2023/03/15
PY  - 2023
N1  - https://doi.org/10.11648/j.advances.20230401.15
DO  - 10.11648/j.advances.20230401.15
T2  - Advances
JF  - Advances
JO  - Advances
SP  - 36
EP  - 43
PB  - Science Publishing Group
SN  - 2994-7200
UR  - https://doi.org/10.11648/j.advances.20230401.15
AB  - The finite deformation of internally pressurized isotropic compressible synthetic rubber-like material governed by Levinson and Burgess strain energy function is analysed. A second-order nonlinear ordinary differential (Lane-Emden) equationwith shooting boundary value was derivedfor the determination of displacements distributions. Several analytical methods were employed to solve the resulting boundary value problem but no closed form solution was obtained at the moment. Fortunately, a lot of software have been developed to handle such highly nonlinear second order ordinary differential equations with specific values of parameters. Also, the stresses acting on the material were determined. We obtained numerical solution by applying shooting method and validated the result using collocation method on mathematica (ode45 solver). The simulation of the system is made forρ = 14N/m2, and the cylindrical symmetric deformation attained its maximum displacements and stresses atr(1) = 1.16638m and σrr = (-1.2973e-05)kg/m/s2. We were able to develop numerical schemes using shooting and Collocation methods which made it easier to determine position of maximum stresses and pressure in a cylindrical material of Levinson-Burgess strain energy function. These numerical schemes can solve any nonlinear second-order ordinary differential equations with any given boundary conditions on Mathematica Software. The results of the two schemes were statistically compared using t-test and results obtained showed, the two methods have no significant difference which validates the solutions.
VL  - 4
IS  - 1
ER  -

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